Exact Support Recovery for Sparse Spikes Deconvolution

Abstract

This paper studies sparse spikes deconvolution over the space of measures. We focus on the recovery properties of the support of the measure (i.e., the location of the Dirac masses) using total variation of measures (TV) regularization. This regularization is the natural extension of the $\ell ^1$ norm of vectors to the setting of measures. We show that support identification is governed by a specific solution of the dual problem (a so-called dual certificate) having minimum $L^2$ norm. Our main result shows that if this certificate is non-degenerate (see the definition below), when the signal-to-noise ratio is large enough TV regularization recovers the exact same number of Diracs. We show that both the locations and the amplitudes of these Diracs converge toward those of the input measure when the noise drops to zero. Moreover, the non-degeneracy of this certificate can be checked by computing a so-called vanishing derivative pre-certificate. This proxy can be computed in closed form by solving a linear system. Lastly, we draw connections between the support of the recovered measure on a continuous domain and on a discretized grid. We show that when the signal-to-noise level is large enough, and provided the aforementioned dual certificate is non-degenerate, the solution of the discretized problem is supported on pairs of Diracs which are neighbors of the Diracs of the input measure. This gives a precise description of the convergence of the solution of the discretized problem toward the solution of the continuous grid-free problem, as the grid size tends to zero.

1 Introduction

1.1 Sparse Spikes Deconvolution

Super-resolution is a central problem in imaging science and loosely speaking corresponds to recovering fine-scale details from a possibly noisy input signal or image. This thus encompasses the problems of data interpolation (recovering missing sampling values on a regular grid) and deconvolution (removing acquisition blur). We refer to the review articles [24, 27] and the references therein for an overview of these problems.

We consider in our article an idealized super-resolution problem, known as sparse spikes deconvolution. It corresponds to recovering 1D spikes (i.e., both their positions and amplitudes) from blurry and noisy measurements. These measurements are obtained by a convolution of the spikes train against a known kernel. This setup can be seen as an approximation of several imaging devices. A method of choice to perform this recovery is to introduce a sparsity-enforcing prior, among which the most popular is a $\ell ^1$-type norm, which favors the emergence of spikes in the solution.

1.2 Previous Works

Discrete $\ell ^1$ regularization $\ell ^1$-type techniques were initially proposed in geophysics [10, 23, 28] to recover the location of density changes in the underground for seismic exploration. They were later studied in depth by David Donoho and co-workers, see for instance [14]. Their popularity in signal processing and statistics can be traced back to the development of the basis pursuit method [9] for approximation in redundant dictionaries and the Lasso method [31] for statistical estimation.

The theoretical analysis of the $\ell ^1$-regularized deconvolution was initiated by Donoho [14]. Assessing the performance of discrete $\ell ^1$ regularization methods is challenging and requires to take into account both the specific properties of the operator to invert and of the signal that is aimed at being recovered. A popular approach is to assess the recovery of the positions of the nonzero coefficients. This requires to impose a well-conditioning constraint that depends on the signal of interest, as initially introduced by Fuchs [20], and studied in the statistics community under the name of “irrepresentability condition,” see [34]. A similar approach is used by Dossal and Mallat in [15] to study the problem of support stability over a discrete grid.

Imposing the exact recovery of the support of the signal to recover might be a too strong assumption. The inverse problem community rather focuses on the $L^2$ recovery error, which typically leads to a linear convergence rate with respect to the noise amplitude. The seminal paper of Grasmair et al. [21] gives a necessary and sufficient condition for such a convergence, which corresponds to the existence of a non-saturating dual certificate (see Sect. 2 for a precise definition of certificates). This can be understood as an abstract condition, which is often difficult to check on practical problems such as deconvolution.

Note that the continuous setting adopted in the present paper might be seen as a limit of such discrete problems, and in Sect. 5, we relate our results to well-known results on discrete grids.

Let us also note that, although we focus here on $\ell ^1$-based methods, there is a vast literature on various nonlinear super-resolution schemes. This includes for instance greedy [25, 26], root finding [3, 11], matrix pencils [13] and compressed sensing [16, 18] approaches.

Inverse problems regularization with measures Working over a discrete grid makes the mathematical analysis difficult. Following recent proposals [2, 4, 8, 12], we consider here this sparse deconvolution over a continuous domain, i.e., in a grid-free setting. This shift from the traditional discrete domain to a continuous one offers considerable advantages in term of mathematical analysis, allowing for the first time the emergence of almost sharp signal-dependent criteria for stable spikes recovery (see references below). Note that while the corresponding continuous recovery problem is infinite dimensional in nature, it is possible to find its solution using either provably convergent algorithms [4] or root finding methods for ideal low-pass filters [8].

Inverse problem regularization over the space of measures is now well understood (see for instance [4, 29]) and requires to perform variational analysis over a non-reflexive Banach space (as in [22]), which leads to some mathematical technicalities. We capitalize on these earlier works to build our analysis of the recovery performance.

Theoretical analysis of deconvolution over the space of measures For deconvolution from ideal low-pass measurements, the groundbreaking paper [8] shows that it is indeed possible to construct a dual certificate by solving a linear system when the input Diracs are well separated. This work is further refined in [7] that studies the robustness to noise. In a series of paper [2, 30], the authors study the prediction (i.e., denoising) error using the same dual certificate, but they do not consider the reconstruction error (recovery of the spikes). In our work, we use a different certificate to assess the exact recovery of the spikes when the noise is small enough.

In view of the applications of super-resolution, it is crucial to understand the precise location of the recovered Diracs locations when the measurements are noisy. Partial answers to this questions are given in [19] and [1], where it is shown (under different conditions on the signal-to-noise level) that the recovered spikes are clustered tightly around the initial measure’s Diracs. In this article, we fully answer the question of the position of the recovered Diracs in the setting where the signal-to-noise ratio is large enough.

1.3 Formulation of the Problem and Contributions

Let $m_0=\sum _{i=1}^N a_{0,i}\delta _{x_{0,i}}$ be a discrete measure defined on the torus $\mathbb {T}=\mathbb {R}/\mathbb {Z}$, where $a_0 \in \mathbb {R}^N$ and $x_0 \in \mathbb {T}^N$. We assume we are given some low-pass filtered observation $y_0=\varPhi m_0 \in L^2(\mathbb {T})$. Here, $\varPhi$ denotes a convolution operator with some kernel $\varphi \in C^2(\mathbb {T})$. The observation might be noisy, in which case we are given $y_0+w= \varPhi m_0 + w$, with $w\in L^2(\mathbb {T})$, instead of $y_0$.

Following [8, 12], we hope to recover $m_0$ by solving the problem

among all Radon measures, where $|| m ||_{\text {TV}}$ refers to the total variation (defined below) of $m$. Note that in our setting, the total variation is the natural extension of the $\ell ^1$ norm of finite-dimensional vectors to the setting of Radon measures, and it should not be mistaken for the total variation of functions, which is routinely used to recover signals or images.

We may also consider reconstructing $m_0$ by solving the following penalized problem for $\lambda >0$, also known as the Beurling LASSO (see for instance [1]):

This is especially useful if the observation is noisy, in which case $y_0$ should be replaced with $y_0+w$.

Four questions immediately arise:

• 1. Does the resolution of $(\mathcal {P}_0(y_0))$ for $y_0=\varPhi m_0$ actually recover interesting measures $m_0$?

• 2. How close is the solution of $(\mathcal {P}_\lambda (y_0))$ to the solution of $(\mathcal {P}_0(y_0))$ when $\lambda$ is small enough?

• 3. How close is the solution of $(\mathcal {P}_\lambda (y_0+w))$ to the solution of $(\mathcal {P}_\lambda (y_0))$ when both $\lambda$ and $w/\lambda$ are small enough?

• 4. What can be said about the above questions when solving $(\mathcal {P}_\lambda (y_0))$ with measures supported on a fixed finite grid?

The first question is addressed in the landmark paper [8] in the case of ideal low-pass filtering: Measures $m_0$ whose spikes are separated enough are the unique solution of $(\mathcal {P}_0(y_0))$ (for data $y_0=\varPhi m_0$). Several other cases (using observations different from convolutions) are also tackled in [12], particularly in the case of non-negative measures.

The second and third questions receive partial answers in [1, 4, 7, 19]. In [4], it is shown that if the solution of $(\mathcal {P}_0(y_0))$ is unique, the measures recovered by $(\mathcal {P}_\lambda (y_0+w))$ converge to the solution of $(\mathcal {P}_0(y_0))$ in the sense of the weak-* convergence when $\lambda \rightarrow 0$ and $\frac{|| w ||_2^2}{\lambda }\rightarrow 0$. In [7], the authors measure the reconstruction error using the $L^2$ norm of a low-pass filtered version of recovered measures. In [1], error bounds are derived from the amplitudes of the reconstructed measure. In [19], bounds are given in terms of the original measure. However, those works provide little information about the structure of the measures recovered by $(\mathcal {P}_\lambda (y_0+w))$: Are they made of less spikes than $m_0$ or, in the contrary, do they present lots of parasitic spikes? What happens if one compels the spikes to belong to a finite grid?

The fourth question is of primary importance since most numerical schemes for sparse regularization solve a finite-dimensional optimization problem over a fixed discretization grid. Following [8], one can remark that in the noiseless setting, if $m_0$ is recovered over the continuous domain and if its support is included in the grid, $m_0$ is also guaranteed to be recovered by the discretized problem. But this is of little interest in practice because the noise is likely to impact in a different manner the discrete problem and the input measure might fall outside the grid locations. Dossal and Mallat in [15] study the stability of the position of the Diracs on the grid, which leads to overly pessimistic conclusions because noise typically forces the spikes to translate over the domain. Studying the convergence of the discretized problem toward the continuous one is thus important to obtain a precise description of the discretized solution. To the best of our knowledge, the work of  [2] is the only one to provide some conclusion about this convergence in term of denoising error. No previous work has studied the capability of the discretized problem to estimate in a precise manner the location of the spikes of the input measure.

Contributions The present paper studies in detail the structure of the recovered measure. For this purpose, we define the minimal $L^2$-norm certificate. This certificate fully governs the behavior of the regularization when both $\lambda$ and $|| w ||_2/\lambda$ are small.

Our first contribution is a set of results indicating that the regions of saturation of the certificate (when it reaches $+1$ or $-1$) are approximately stable when $\lambda$ and $|| w ||_2/\lambda$ are small enough. This means that the recovered measures are supported closely to the support of the input measure if the latter is identifiable (solution of the noiseless problem $(\mathcal {P}_0(y_0))$).

Our second contribution introduces the non-degenerate source condition, which imposes that the second derivative of the minimal norm certificate does not vanish on the saturation points. Under this condition, we show that for $\lambda$ and $|| w ||_2/\lambda$ small enough, the reconstructed measure has exactly the same number of spikes as the original measure and that their locations and amplitudes converge to those of the original one.

Our third contribution shows that under the non-degenerate source condition, the minimal norm certificate can actually be computed in closed form by simply solving a linear system. This in turn also implies that the errors in the amplitudes and locations decay linearly with respect to the noise level.

Our fourth and last contribution focuses on the regularization over a discrete finite grid, which corresponds to the so-called Lasso or basis pursuit denoising problem. We show that when $\lambda$ and $|| w ||_2/\lambda$ are small enough, and provided that the non-degenerate source condition holds, the discretized solution is located on pairs of Diracs adjacent to the input Diracs location. This gives a precise description of how the solution to the discretized problem converges to the one of the continuous problem when the stepsize of the grid vanishes.

Throughout the paper, the proposed definitions and results are illustrated in the case of the ideal low-pass filter, showing that the assumptions are actually relevant. Note that the code to reproduce the figures of this article is available online.¹

Outline of the paper Section 2 defines the framework for the recovery of Radon measures using total variation minimization. We also expose basic results that are used throughout the paper. Section 3 is devoted to the main result of the paper: We define the non-degenerate source condition, and we show that it implies the robustness of the reconstruction using $(\mathcal {P}_\lambda (y_0+w))$. In Sect. 4, we show how the specific dual certificate involved in the non-degenerate source condition can be computed numerically by solving a linear system. Lastly, Sect. 5 focuses on the recovery of measures on a discrete grid.

1.4 Notations

For any Radon measure $m$ defined on $\mathbb {T}$, we denote its support by ${{\mathrm{Supp}}}(m)$. If ${{\mathrm{Supp}}}(m)$ is a finite set (in which case we say that $m$ is a discrete measure) and $m\ne 0$, then $m$ is of the form $m= \sum _{i=1}^N a_i \delta _{x_i}$, where $N\in \mathbb {N}^*$, $a\in \mathbb {R}^N$, $x\in \mathbb {T}^N$ and $a_i\ne 0$ and $x_i\ne x_j$ for all $1\leqslant i,j\leqslant N$. In the rest of the paper, we shall write $m=m_{a,x}$ to hint that $m$ has the above decomposition (implying that $a_i\ne 0$ and $x_i\ne x_j$ for all $1\leqslant i,j\leqslant N$).

We also define the signed support:

$$\begin{aligned} {{\mathrm{Supp^{\pm }}}}m&= ({{\mathrm{Supp}}}m_+) \times \{1\}\cup ({{\mathrm{Supp}}}m_-)\times \{-1\} \subset \mathbb {T}\times \{+1,-1 \} \end{aligned}$$

where $m_+$ (respectively $m_-$) denotes the positive (respectively negative) part of $m$. For a discrete measure $m=m_{a,x}$,

$$\begin{aligned} {{\mathrm{Supp^{\pm }}}}m&=\left\{ (t,v)\in \mathbb {T}\times \{+1,-1\}, \ m(\{t \})\ne 0 \text{ and } {{\mathrm{sign}}}m(\{t \})=v \right\} \\&= \{(x_i,{{\mathrm{sign}}}a_i), \ 1\leqslant i\leqslant N\}. \end{aligned}$$

We shall consider restrictions of measures and functions to subsets of $\mathbb {T}$. For $m \in \mathcal {M}(\mathbb {T})$ a discrete measure and $J=\{x_1,\ldots ,x_k\} \subset \mathbb {T}$ a finite set, we define

$$\begin{aligned} {m\vert }_{J} = a \in \mathbb {T}^{k} \quad \text {where} \quad \forall \,i=1,\ldots ,k, \quad a_i = m(\{x_i\}). \end{aligned}$$

For $\eta \in C(\mathbb {T})$ a continuous function defined on $\mathbb {T}$, we define

$$\begin{aligned} {\eta \vert }_{J} = ( \eta (x_j) )_{j=1}^k \in \mathbb {T}^{k}. \end{aligned}$$

Given a convolution operator $\varPhi$ with kernel $t\mapsto \varphi (-t)$, we define $\varPhi _x : \mathbb {R}^N \rightarrow L^2(\mathbb {T})$ (respectively $\varPhi _x'$, $\varPhi _x''$) by

$$\begin{aligned} \forall a\in \mathbb {R}^N,\ \varPhi _x(a)&= \varPhi ( m_{a,x} ) = \sum _{i=1}^N a_i \varphi (x_i - \cdot ), \end{aligned}$$

(1)

$$\begin{aligned} \varPhi '_x(a)&= (\varPhi _x(a))' = \sum _{i=1}^N a_i \varphi '(x_i - \cdot ), \end{aligned}$$

(2)

$$\begin{aligned} {\varPhi ''_x}(a)&= (\varPhi _x(a))'' = \sum _{i=1}^N a_i \varphi ''(x_i - \cdot ). \end{aligned}$$

(3)

We define

$$\begin{aligned} \varGamma _x&= (\varPhi _x, \varPhi _x') : (u,v) \in \mathbb {R}^N \times \mathbb {R}^N \mapsto \varPhi _x u + \varPhi '_x v \in \text {L}^2(\mathbb {T}), \end{aligned}$$

(4)

$$\begin{aligned} \varGamma _x'&= (\varPhi '_x, \varPhi ''_x): (u,v) \in \mathbb {R}^N \times \mathbb {R}^N \mapsto \varPhi _x' u + \varPhi ''_x v \in \text {L}^2(\mathbb {T}). \end{aligned}$$

(5)

Eventually, in order to study small noise regimes, we shall consider domains $D_{\alpha ,\lambda _0}$, for $\alpha >0$, $\lambda _0>0$, where:

$$\begin{aligned} D_{\alpha ,\lambda _0}= \left\{ (\lambda ,w)\in \mathbb {R}_+\times L^2(\mathbb {T}) \;;\; 0\leqslant \lambda \leqslant \lambda _0 \quad \text {and} \quad || w ||_2\leqslant \alpha \lambda \right\} . \end{aligned}$$

(6)

2 Preliminaries

In this section, we precise the framework and we state the basic results needed in the next sections. We refer to [5] for aspects regarding functional analysis and to [17] as far as duality in optimization is concerned.

2.1 Topology of Radon Measures

Since $\mathbb {T}$ is compact, the space of Radon measures $\mathcal {M}(\mathbb {T})$ can be defined as the dual of the space $C(\mathbb {T})$ of continuous functions on $\mathbb {T}$, endowed with the uniform norm. It is naturally a Banach space when endowed with the dual norm (also known as the total variation), defined as

$$\begin{aligned} \forall m \in \mathcal {M}(\mathbb {T}), \quad || m ||_{\text {TV}}= \sup \left\{ \int \psi \mathrm {d}m \;;\; \psi \in C(\mathbb {T}), || \psi ||_{\infty } \leqslant 1 \right\} . \end{aligned}$$

(7)

In that case, the dual of $\mathcal {M}(\mathbb {T})$ is a complicated space, and it is strictly larger than $C(\mathbb {T})$ as $C(\mathbb {T})$ is not reflexive.

However, if we endow $\mathcal {M}(\mathbb {T})$ with its weak-* topology (i.e., the coarsest topology such that the elements of $C(\mathbb {T})$ define continuous linear forms on $\mathcal {M}(\mathbb {T})$), then $\mathcal {M}(\mathbb {T})$ is a locally convex space whose dual is $C(\mathbb {T})$.

In the following, we endow $C(\mathbb {T})$ (respectively $\mathcal {M}(\mathbb {T})$) with its weak (respectively its weak-*) topology so that both have symmetrical roles: One is the dual of the other and conversely. Moreover, since $C(\mathbb {T})$ is separable, the set $\left\{ m \in \mathcal {M}(\mathbb {T}) \;;\; || m ||_{\text {TV}} \leqslant 1 \right\}$ endowed with the weak-* topology is metrizable.

Given a function $\varphi \in C^{2}(\mathbb {T}, \mathbb {R})$, we define an operator $\varPhi : \mathcal {M}(\mathbb {T}) \rightarrow \text {L}^2(\mathbb {T})$ as

$$\begin{aligned} \forall \,m \in \mathcal {M}(\mathbb {T}), \quad \varPhi (m) : t \mapsto \int _{\mathbb {T}} \varphi (x-t) \mathrm {d}m(x). \end{aligned}$$

It can be shown using Fubini’s theorem that $\varPhi$ is weak-* to weak continuous. Moreover, its adjoint operator $\varPhi ^* : \text {L}^2(\mathbb {T}) \rightarrow C(\mathbb {T})$ is defined as

$$\begin{aligned} \forall \,y \in \text {L}^2(\mathbb {T}), \quad \varPhi ^*(y) : t \mapsto \int _{\mathbb {T}} \varphi (t-x) y(x) \mathrm {d}x. \end{aligned}$$

2.2 Subdifferential of the Total Variation

It is clear from the definition of the total variation in (7) that it is convex lower semi-continuous with respect to the weak-* topology. Its subdifferential is defined as

$$\begin{aligned} \partial || m ||_{\text {TV}} = \left\{ \eta \in C(\mathbb {T}) \;;\; \forall \tilde{m}\in \mathcal {M}(\mathbb {T}), || \tilde{m} ||_{\text {TV}} \geqslant || m ||_{\text {TV}} + \int \eta \, \mathrm {d}(\tilde{m}-m) \right\} , \end{aligned}$$

(8)

for any $m\in \mathcal {M}(\mathbb {T})$ such that $|| m ||_{\text {TV}}<+\infty$.

Since the total variation is a sublinear function, its subgradient has a special structure. One may show (see Proposition 12 in “Appendix 1”) that

$$\begin{aligned} \partial || m ||_{\text {TV}} = \left\{ \eta \in C(\mathbb {T}) \;;\; || \eta ||_{\infty } \leqslant 1 \quad \text {and} \quad \int \eta \, \mathrm {d}m =|| m ||_{\text {TV}} \right\} . \end{aligned}$$

(9)

In particular, when $m$ is a measure with finite support, i.e., $m=\sum _{i=1}^N a_i \delta _{x_i}$ for some $N\in \mathbb {N}$, with $(a_i)_{1\leqslant i\leqslant N}\in (\mathbb {R}^*)^N$ and distinct $(x_i)_{1\leqslant i \leqslant N}\in \mathbb {T}^N$

$$\begin{aligned} \partial || m ||_{\text {TV}} = \left\{ \eta \in C(\mathbb {T}) \;;\; || \eta ||_{\infty } \leqslant 1 \;\text {and}\; \forall \,i=1,\ldots ,N, \quad \eta (x_i)={{\mathrm{sign}}}(a_i) \right\} . \end{aligned}$$

(10)

2.3 Primal and Dual Problems

Given an observation $y_0=\varPhi m_0 \in \text {L}^2(\mathbb {T})$ for some $m_0\in \mathcal {M}(\mathbb {T})$, we consider reconstructing $m_0$ by solving either the relaxed problem for $\lambda >0$

or the constrained problem

If $m_0$ is the unique solution of $(\mathcal {P}_0(y_0))$, we say that $m_0$ is identifiable.

In the case where the observation is noisy (i.e., the observation $y_0$ is replaced with $y_0+w$ for $w\in L^2(\mathbb {T})$), we attempt to reconstruct $m_0$ by solving $\mathcal {P}_\lambda (y_0+w)$ for a well-chosen value of $\lambda >0$.

Existence of solutions for $(\mathcal {P}_\lambda (y_0))$ is shown in [4], and existence of solutions for $(\mathcal {P}_0(y_0))$ can be checked using the direct method of the calculus of variations (recall that for $(\mathcal {P}_0(y_0))$, we assume that the observation is $y_0=\varPhi m_0$).

A straightforward approach to studying the solutions of Problem $(\mathcal {P}_\lambda (y_0))$ is then to apply Fermat’s rule: A discrete measure $m=m_{a,x}=\sum _{i=1}^N a_i\delta _{x_i}$ is a solution of $(\mathcal {P}_\lambda (y_0))$ if and only if there exists $\eta \in C(\mathbb {T})$ such that

$$\begin{aligned} \varPhi ^*(\varPhi m -y_0) +\lambda \eta =0, \end{aligned}$$

with $|| \eta ||_{\infty } \leqslant 1$ and $\eta (x_i)={{\mathrm{sign}}}(a_i)$ for $1\leqslant i\leqslant N$.

Another source of information for the study of Problems $(\mathcal {P}_\lambda (y_0))$ and $(\mathcal {P}_0(y_0))$ is given by their associated dual problems. In the case of the ideal low-pass filter, this approach is also the key to the numerical algorithms used in [1, 2, 8]: The dual problem can be recast into a finite-dimensional problem.

The Fenchel dual problem to $(\mathcal {P}_\lambda (y_0))$ is given by

which may be reformulated as a projection on a closed convex set (see [1, 4])

This formulation immediately yields existence and uniqueness of a solution to $(\mathcal {D}_\lambda (y_0))$.

The dual problem to $(\mathcal {P}_0(y_0))$ is given by

Contrary to $(\mathcal {D}_\lambda (y_0))$, the existence of a solution to $(\mathcal {D}_0(y_0))$ is not always guaranteed, so that in the following (see Definition 5) we make this assumption.

Existence is guaranteed when for instance ${{\mathrm{Im}}}\varPhi ^*$ is finite dimensional (as is the case in the framework of [8]). If a solution to $(\mathcal {D}_0(y_0))$ exists, the unique solution of $(\mathcal {D}_\lambda (y_0))$ converges to a certain solution of $(\mathcal {D}_0(y_0))$ for $\lambda \rightarrow 0^+$ as shown in Proposition 1 below.

2.4 Dual Certificates

The strong duality between $(\mathcal {P}_\lambda (y_0))$ and $(\mathcal {D}_\lambda (y_0))$ is proved in [4, Prop. 2] by seeing $(\mathcal {D}'_\lambda (y))$ as a predual problem for $(\mathcal {P}_\lambda (y_0))$. As a consequence, both problems have the same value and any solution $m_\lambda$ of $(\mathcal {P}_\lambda (y_0))$ is linked with the unique solution $p_\lambda$ of $(\mathcal {D}_\lambda (y_0))$ by the extremality condition

$$\begin{aligned} \left\{ \begin{array}{c} \varPhi ^*p_\lambda \in \partial || m_\lambda ||_{\text {TV}}, \\ -p_\lambda = \frac{1}{\lambda }(\varPhi m_\lambda - y_0). \end{array} \right. \end{aligned}$$

(11)

Moreover, given a pair $(m_\lambda ,p_\lambda )\in \mathcal {M}(\mathbb {T}) \times L^2(\mathbb {T})$, if relations (11) hold, then $m_\lambda$ is a solution to Problem $(\mathcal {P}_\lambda (y_0))$ and $p_\lambda$ is the unique solution to Problem $(\mathcal {D}_\lambda (y_0))$.

As for $(\mathcal {P}_0(y_0))$, a proof of strong duality is given in “Appendix 1” (see Proposition 13). If a solution $p^\star$ to $(\mathcal {D}_0(y_0))$ exists, then it is linked to any solution $m^\star$ of $(\mathcal {P}_0(y_0))$ by

$$\begin{aligned} \varPhi ^* p^\star \in \partial {|| m^\star ||_{\text {TV}}}, \end{aligned}$$

(12)

and similarly, given a pair $(m^\star ,p^\star )\in \mathcal {M}(\mathbb {T})\times L^2(\mathbb {T})$, if relation (12) hold, then $m^\star$ is a solution to Problem $(\mathcal {P}_0(y_0))$ and $p^\star$ is a solution to Problem $(\mathcal {D}_0(y_0))$).

Since finding $\eta = \varPhi ^* p^\star$ which satisfies (12) gives a quick proof that $m^\star$ is a solution of $(\mathcal {P}_0(y_0))$, we call $\eta$ a dual certificate for $m^\star$. We may also use a similar terminology for $\eta _\lambda =\varPhi ^* p_\lambda$ and Problem $(\mathcal {P}_\lambda (y_0))$.

In general, dual certificates for $(\mathcal {P}_0(y_0))$ are not unique, but we consider in the following definition a specific one, which is crucial for our analysis.

Definition 1

(Minimal norm certificate) When it exists, the minimal norm dual certificate associated with $(\mathcal {P}_0(y_0))$ is defined as $\eta _0=\varPhi ^* p_0$ where $p_0 \in L^2(\mathbb {T})$ is the solution of $(\mathcal {D}_0(y_0))$ with minimal norm, i.e.,

$$\begin{aligned} \eta _0=\varPhi ^* p_0,&\quad \text {where} \quad p_0=\underset{p}{{{\mathrm{argmin}}}}\; \left\{ || p ||_2 \;;\; p \text{ is } \text{ a } \text{ solution } \text{ of } (\mathcal {D}_0(y_0)) \right\} . \end{aligned}$$

(13)

Observe that in the above definition, $p_0$ is well defined provided there exists a solution to Problem $(\mathcal {D}_0(y_0))$, since $p_0$ is then the projection of 0 onto the non-empty closed convex set of solutions. Moreover, in view of the extremality conditions (12), given any solution $m^\star$ to $(\mathcal {P}_0(y_0))$, it may be expressed as

$$\begin{aligned} p_0=\underset{ p }{{{\mathrm{argmin}}}}\; \left\{ || p ||_2 \;;\; \varPhi ^* p \in \partial {|| m^\star ||_{\text {TV}}} \right\} . \end{aligned}$$

(14)

Proposition 1

(Convergence of dual certificates) Let $p_\lambda$ be the unique solution of Problem $(\mathcal {D}_\lambda (y_0))$, and $p_0$ be the solution of Problem $(\mathcal {D}_0(y_0))$ with minimal norm defined in (13). Then,

$$\begin{aligned} \lim _{\lambda \rightarrow 0^+} p_\lambda = p_0 \quad \text{ for } \text{ the } L^2 \text{ strong } \text{ topology. } \end{aligned}$$

Moreover, the dual certificates $\eta _\lambda = \varPhi ^*p_\lambda$ for Problem $(\mathcal {P}_\lambda (y_0))$ converge to the minimal norm certificate $\eta _0 = \varPhi ^*p_0$. More precisely,

$$\begin{aligned} \forall k\in \{0,1,2\}, \quad \lim _{\lambda \rightarrow 0^+} \eta _\lambda ^{(k)} = \eta _0^{(k)}, \end{aligned}$$

(15)

in the sense of the uniform convergence.

Proof

Let $p_\lambda$ be the unique solution of $(\mathcal {D}_\lambda (y_0))$. By optimality of $p_\lambda$ (respectively $p_0$) for $(\mathcal {D}_\lambda (y_0))$ (respectively $(\mathcal {D}_0(y_0))$)

$$\begin{aligned} \langle y_0,\,p_\lambda \rangle - \lambda || p_\lambda ||_2^2&\geqslant \langle y_0,\,p_0\rangle -\lambda || p_0 ||_2^2, \end{aligned}$$

(16)

$$\begin{aligned} \langle y_0,\,p_0\rangle&\geqslant \langle y_0,\,p_\lambda \rangle . \end{aligned}$$

(17)

As a consequence $|| p_0 ||_2^2\geqslant || p_\lambda ||_2^2$ for all $\lambda >0$.

Now, let $(\lambda _n)_{n\in \mathbb {N}}$ be any sequence of positive parameters converging to 0. The sequence $p_{\lambda _n}$ being bounded in $L^2(\mathbb {T})$, we may extract a subsequence (denoted $\lambda _{n'}$) such that $p_{\lambda _{n'}}$ weakly converges to some $p^\star \in L^2(\mathbb {T})$. Passing to the limit in (16), we get $\langle y_0,\,p^\star \rangle \geqslant \langle y_0,\,p_0\rangle$. Moreover, $\varPhi ^*p_{\lambda _n}$ weakly converges to $\varPhi ^* p^\star$ in $C(\mathbb {T})$, so that $|| \varPhi ^* p^\star ||_{\infty } \leqslant \liminf _{n'} || \varPhi ^*p_{\lambda _{n'}} ||_{\infty } \leqslant 1$, and $p^\star$ is therefore a solution of $(\mathcal {D}_0(y_0))$.

But one has

$$\begin{aligned} || p^\star ||_2\leqslant \liminf _{n'} || p_{\lambda _{n'}} ||_2\leqslant || p_0 ||_2, \end{aligned}$$

hence $p^\star =p_0$ and in fact $\lim _{{n'}\rightarrow +\infty } || p_{\lambda _{n'}} ||_2=|| p_0 ||_2$. As a consequence, $p_{\lambda _{n'}}$ converges to $p_0$ for the $L^2(\mathbb {T})$ strong topology as well. This being true any sequence $\lambda _n\rightarrow 0^+$, we get the result claimed for $p_\lambda$: Assume by contradiction that there exists $\varepsilon _0>0$ and a sequence $\lambda _n \searrow 0$ such that $\Vert p_0-p_{\lambda _n}\Vert _2 \geqslant \varepsilon _0$ for all $n\in \mathbb {N}$. By the above argument, we may extract a subsequence $\lambda _{n'}$ which converges toward $p_0$, which contradicts $\Vert p_0-p_{\lambda _n'}\Vert _2 \geqslant \varepsilon _0$. Hence, $\lim _{\lambda \rightarrow 0}p_\lambda =p_0$ strongly in $L^2$.

It remains to prove the convergence of the dual certificates. Observing that $\eta _\lambda ^{(k)}(t)=\int \varphi ^{(k)}(t-x)p_\lambda (x) \mathrm {d}x$, we get

$$\begin{aligned} |\eta _\lambda ^{(k)}(t)-\eta _0^{(k)} (t)|&= \Big | \int _{\mathbb {T}} \varphi ^{(k)}(t-x)(p_\lambda -p_0)(x) \mathrm {d}x \Big |\\&\leqslant \sqrt{\int _{\mathbb {T}} |\varphi ^{(k)}(t-x)|^2 \mathrm {d}x} \sqrt{\int _{\mathbb {T}} |(p_\lambda -p_0)(x)|^2 \mathrm {d}x}\\&\leqslant C || p_\lambda - p_0 ||_2, \end{aligned}$$

where $C>0$ does not depend on $t$ nor $k$, hence the uniform convergence. $\square$

2.5 Application to the Ideal Low-pass Filter

In this paragraph, we apply the above duality results to the particular case of the Dirichlet kernel, defined as

$$\begin{aligned} \varphi (t) = \sum _{k=-f_c}^{f_c} e^{2i\pi kt} = \frac{\sin \left( (2f_c+1)\pi t\right) }{\sin (\pi t)}. \end{aligned}$$

(18)

It is well known that in this case, the spaces ${{\mathrm{Im}}}\varPhi$ and ${{\mathrm{Im}}}\varPhi ^*$ are finite dimensional, being the space of real trigonometric polynomials with degree less than or equal to $f_c$.

We first check that a solution to $(\mathcal {D}_0(y_0))$ always exists. As a consequence, given any measure $m_0$, the minimal norm certificate is well defined.

Proposition 2

Existence of $p_0$ Let $m_0\in \mathcal {M}(\mathbb {T})$ and $y_0=\varPhi m_0\in L^2(\mathbb {T})$. There exists a solution of $(\mathcal {D}_0(y_0))$. As a consequence, $p_0\in L^2(\mathbb {T})$ is well defined.

Proof

We rewrite $(\mathcal {D}_0(y_0))$ as

$$\begin{aligned} \underset{|| \eta ||_{\infty } \leqslant 1, \eta \in {{\mathrm{Im}}}\varPhi ^*}{\sup }\; \langle m_0,\,\eta \rangle . \end{aligned}$$

Let $(\eta _n)_{n\in \mathbb {N}}$ be any maximizing sequence. Then, $(\eta _n)_{n\in \mathbb {N}}$ is bounded in the finite-dimensional space of trigonometric polynomials with degree $f_c$ or less. We may extract a subsequence converging to $\eta ^\star \in C(\mathbb {T})$. But $|| \eta ^\star ||_{\infty } \leqslant 1$ and $\eta ^\star \in {{\mathrm{Im}}}\varPhi ^*$, so that $\eta ^\star =\varPhi ^* p^\star$ for some $p^\star$ solution of $(\mathcal {D}_0(y_0))$. $\square$

A striking result of [8] is that discrete measures are identifiable provided that their support is separated enough, i.e., $\Delta (m_0) \geqslant \frac{C}{f_c}$ for some $C>0$, where $\Delta (m_0)$ is the so-called minimum separation distance.

Definition 2

(Minimum separation) The minimum separation of the support of a discrete measure $m$ is defined as

$$\begin{aligned} \Delta (m)=\inf _{(t,t')\in {{\mathrm{Supp}}}(m)} |t-t'|, \end{aligned}$$

where $|t-t'|$ is the distance on the torus between $t$ and $t'\in \mathbb {T}$, and we assume $t\ne t'$.

In [8], it is proved that $C \leqslant 2$ for complex measures (i.e., of the form $m_{a,x}$ for $a \in \mathbb {C}^N$ and $x \in \mathbb {T}^N$) and $C \leqslant 1.87$ for real measures (i.e., of the form $m_{a,x}$ for $a \in \mathbb {R}^N$ and $x \in \mathbb {T}^N$). Extrapolating from numerical simulations on a finite grid, the authors conjecture that for complex measures, one has $C\geqslant 1$. In this section, we apply results from Sect. 2.4 to show that for real measures, necessarily $C\geqslant \frac{1}{2}$.

We rely on the following theorem, proved by P. Turán [32].

Theorem 1

(Turán) Let $P(z)$ be a non-trivial polynomial of degree $n$ such that $|P(1)|=\max _{|z|=1} |P(z)|$. Then for any root $z_0$ of $P$ on the unit circle, $|\arg (z_0)|\geqslant \frac{\pi }{n}$. Moreover, if $|\arg (z_0)|= \frac{\pi }{n}$, then $P(z)=c(1+z^n)$ for some $c\in \mathbb {C}^*$.

From this theorem, we derive necessary conditions for measures that can be reconstructed by $(\mathcal {P}_0(y_0))$.

Corollary 1

(Non-identifiable measures) There exists a discrete measure $m_0$ with $\Delta (m_0)=\frac{1}{2f_c}$ such that $m_0$ is not a solution of $(\mathcal {P}_0(y_0))$ for $y_0=\varPhi m_0$.

Proof

Let $m_0=\delta _{-\frac{1}{2f_c}}+ \delta _0 -\delta _{\frac{1}{2f_c}}$, assume by contradiction that $m$ is a solution of $(\mathcal {P}_0(y_0))$, and let $\eta \in C(\mathbb {T})$ be an associated dual certificate (which exists since ${{\mathrm{Im}}}\varPhi ^*$ is finite dimensional). Then necessarily $\eta (-\frac{1}{2f_c})=\eta (0)=1$ and $\eta \left( \frac{1}{2f_c}\right) \!=\!-1$ and by the intermediate value theorem, there exists $t_0\!\in \! (0, \frac{1}{2f_c})$ such that $\eta (t_0)\!=\!0$.

Writing $\eta (t)=\sum _{k=-f_c}^{f_c}d_k e^{2i\pi kt}$, the polynomial $P(z) = \sum _{k=0}^{2f_c}d_{k-f_c}z^k$ satisfies $P(1)=1=\sup _{|z|=1}|P(z)|=|P(e^{\frac{2i\pi }{2f_c}})|$, and $P(e^{2i\pi t_0})=0$.

By Theorem 1, we cannot have $|2\pi t_0 - 0 |<\frac{\pi }{2f_c}$ nor $|2\pi t_0-\frac{2\pi }{2f_c}|<\frac{\pi }{2f_c}$, hence $t_0=\frac{1}{4f_c}$ and $P(z)=c(1+z^{2f_c})$, so that $\eta (t)=\cos (2\pi f_ct)$. But this implies $\eta (-\frac{1}{2f_c})=-1$, which contradicts the optimality of $\eta$. $\square$

In a similar way, we may also deduce the following corollary.

Corollary 2

(Opposite spikes separation) Let $m^\star \in \mathcal {M}(\mathbb {T})$ be any discrete measure solution of Problem $\mathcal {P}_\lambda (y_0+w)$ or $\mathcal {P}_0(y_0)$ where $y_0=\varPhi m_0$ for any data $m_0\in \mathcal {M}(\mathbb {T})$ and any noise $w\in L^2(\mathbb {T})$. If there exists $x^\star _0\in \mathbb {T}$ (respectively $x^\star _1\in \mathbb {T}$) such that $m^\star (\{x^\star _0\})>0$ (respectively $m^\star (\{x^\star _1\})<0$), then $|x^\star _0 - x^\star _1 |\geqslant \frac{1}{2f_c}$.

3 Noise Robustness

This section is devoted to the study of the behavior of solutions to $\mathcal {P}_\lambda (y_0+w)$ for small values of $\lambda$ and $|| w ||$. In order to study such regimes, as already defined in (6), we consider sets of the form

$$\begin{aligned} D_{\alpha ,\lambda _0}= \left\{ (\lambda ,w)\in \mathbb {R}_+\times L^2(\mathbb {T}) \;;\; 0\leqslant \lambda \leqslant \lambda _0 \quad \text {and} \quad || w ||_2\leqslant \alpha \lambda \right\} , \end{aligned}$$

for $\alpha >0$ and $\lambda _0>0$.

First, we introduce the notion of extended support of a measure. Then, we show that this concept governs the structure of solutions at small noise regime. After introducing the non-degenerate source condition, we state the main result of the paper, i.e., that under this assumption, the solutions of $\mathcal {P}_\lambda (y_0+w)$ have the same number of spikes as the original measure, and that these spikes converge smoothly to those of the original measure.

3.1 Extended Signed Support

Our first step in understanding the behavior of solutions to $\mathcal {P}_\lambda (y_0+w)$ at low noise regime is to introduce the notion of extended signed support.

Definition 3

(Extended signed support) Let $m_0\in \mathcal {M}(\mathbb {T})$ such that there exists a solution to $(\mathcal {D}_0(y_0))$ (where as usual $y_0=\varPhi m_0$), and let $\eta _0\in C(\mathbb {T})$ be the associated minimal norm certificate.

The extended support of $m_0$ is defined as:

$$\begin{aligned} {{\mathrm{Ext}}}(m_0) = \left\{ t\in \mathbb {T} \;;\; \eta _0(t)=\pm 1 \right\} , \end{aligned}$$

(19)

and the extended signed support of $m_0$ as:

$$\begin{aligned} {{\mathrm{Ext^\pm }}}(m_0) = \left\{ (t,v)\in \mathbb {T}\times \{+1,-1\} \;;\; \eta _0(t)=v \right\} . \end{aligned}$$

(20)

Notice that ${{\mathrm{Ext}}}m_0$ and ${{\mathrm{Ext^\pm }}}m_0$ actually depend on $y_0=\varPhi m_0$ rather than on $m_0$ itself. For any measure $m_0\in \mathcal {M}(\mathbb {T})$, the (signed) support and the extended (signed) support of $m_0$ are in general not related. Yet, from the optimality conditions (12), we observe:

Proposition 3

Let $m_0\in \mathcal {M}(\mathbb {T})$ and $y_0=\varPhi m_0$ such that there exists a solution to $(\mathcal {D}_0(y_0))$. Then:

• $m_0$ is a solution to $(\mathcal {P}_0(y_0))$ if and only if ${{\mathrm{Supp^{\pm }}}}m_0 \subset {{\mathrm{Ext^\pm }}}m_0$.

• In any case, if $\varPhi _{{{\mathrm{Ext}}}m_0}$ has full rank, the solution to $(\mathcal {P}_0(y_0))$ is unique.

Here, following the notation (1), we have denoted by $\varPhi _{{{\mathrm{Ext}}}m_0}$ the restriction of $\varPhi$ to the space of measures with support in ${{\mathrm{Ext}}}m_0$. The link between Proposition 3 and the source condition [6] is discussed in Sect. 3.3

3.2 Local Behavior of the Support

In this paragraph, we focus on the local properties of the support of solutions to $\mathcal {P}_\lambda (y_0+w)$ at low noise regime. As usual, we denote $y_0=\varPhi m_0$ for some $m_0\in \mathcal {M}(\mathbb {T})$. For now, we make as few assumptions as possible on $m_0$. In particular, we do not assume that $\varPhi _{{{\mathrm{Ext}}}m_0}$ has full rank. Any solution to $\mathcal {P}_\lambda (y_0+w)$ (which is not necessarily unique) is denoted by $\tilde{m}_\lambda$.

Lemma 1

Assume that there exists a solution to $(\mathcal {D}_0(y_0))$ and let $\varepsilon >0$. Then, there exists $\alpha >0$, $\lambda _0>0$ such that for all $(\lambda , w)\in D_{\alpha , \lambda _0}$,

$$\begin{aligned} {{\mathrm{Supp^{\pm }}}}\tilde{m}_\lambda \subset \left( {{\mathrm{Ext^\pm }}}m_0\right) \oplus \left( (-\varepsilon ,+\varepsilon )\times \{0\}\right) , \end{aligned}$$

(21)

where given two sets $A$ and $B$, $A\oplus B= \left\{ a+b \;;\; a\in A, b\in B \right\}$ denotes their Minkowski sum.

In particular, if ${{\mathrm{Ext}}}m_0$ consists in isolated points $x_{0,1}, \ldots x_{0,N}$, Lemma 1 states that all the mass of $\tilde{m}_\lambda$ is concentrated in boxes $(x_{i,0}-\varepsilon , x_{i,0}+\varepsilon )$, where $\varepsilon \rightarrow 0$ when $\lambda , || w || \rightarrow 0$. Moreover, in each box, $\tilde{m}_\lambda$ has the sign of $\eta _0(x_{0,i})$.

Also, if ${{\mathrm{Ext^\pm }}}m_0=\emptyset$ (i.e., $y=0$), we see that $\tilde{m}_\lambda =0$ for $\lambda$ and $\frac{|| w ||_2}{\lambda }$ small enough [in fact, any $\lambda _0>0$ and $\alpha = \frac{1}{ || \varPhi ^* ||_{2,\infty }}$ suffices, as can be seen from (11)].

Proof

We split the proof in several parts.

Behavior of the minimal norm certificate Let us consider the sets:

$$\begin{aligned} {{\mathrm{Ext}}}^+&= \left\{ t\in \mathbb {T} \;;\; \eta _0(t)=1 \right\} ,\quad {{\mathrm{Ext}}}^- = \left\{ t\in \mathbb {T} \;;\; \eta _0(t)=-1 \right\} ,\\ {{\mathrm{Ext}}}^{+,\varepsilon }&= {{\mathrm{Ext}}}^+ \oplus (-\varepsilon ,\varepsilon ),\quad {{\mathrm{Ext}}}^{-,\varepsilon } = {{\mathrm{Ext}}}^- \oplus (-\varepsilon ,\varepsilon ). \end{aligned}$$

From the uniform continuity of $\eta _0$, for $\varepsilon$ small enough, $\eta _0>\frac{1}{2}$ in ${{\mathrm{Ext}}}^{+,\varepsilon }$ and $\eta _0<-\frac{1}{2}$ in ${{\mathrm{Ext}}}^{-,\varepsilon }$, so that ${{\mathrm{Ext}}}^{+,\varepsilon } \cap {{\mathrm{Ext}}}^{-,\varepsilon }=\emptyset$.

If ${{\mathrm{Ext}}}^{+,\varepsilon } \cup {{\mathrm{Ext}}}^{-,\varepsilon } \subsetneq \mathbb {T}$, the set $K_\varepsilon = \mathbb {T}\setminus \left( {{\mathrm{Ext}}}^{+,\varepsilon } \cup {{\mathrm{Ext}}}^{-,\varepsilon }\right)$ being compact, $\sup _{K_\varepsilon } |\eta _0|<1$. We define $r= 1 -\sup _{K_\varepsilon }|\eta _0|$.

If ${{\mathrm{Ext}}}^{+,\varepsilon } \cup {{\mathrm{Ext}}}^{-,\varepsilon } = \mathbb {T}$, the connectedness of $\mathbb {T}$ implies that ${{\mathrm{Ext}}}^{+,\varepsilon } =\mathbb {T}$ and ${{\mathrm{Ext}}}^{-,\varepsilon }=\emptyset$, or conversely. In that case, we define $r=1$.

In any case, we see that for all $g\in C(\mathbb {T})$, if $|| g-\eta _0 ||_\infty < r$, then

$$\begin{aligned} \left\{ t\in \mathbb {T} \;;\; g(t)=1 \right\} \subset {{\mathrm{Ext}}}^{+,\varepsilon } \quad \text {and} \quad \left\{ t\in \mathbb {T} \;;\; g(t)=-1 \right\} \subset {{\mathrm{Ext}}}^{-,\varepsilon }. \end{aligned}$$

(22)

Variations of dual certificates Let $p_\lambda$ be the solution of the noiseless problem $(\mathcal {D}_\lambda (y_0))$ and $\tilde{p}_\lambda$ be the solution of the noisy dual problem $\mathcal {D}_\lambda (y_0+w)$ for $w\in L^2(\mathbb {T})$. Since the mapping $\frac{y_0}{\lambda } \mapsto p_\lambda$ is a projection onto a convex set (see $(\mathcal {D}'_\lambda (y))$), it is non-expansive, i.e.,

$$\begin{aligned} || p_\lambda -\tilde{p}_{\lambda } ||_2 \leqslant \frac{|| w ||_2}{\lambda }. \end{aligned}$$

(23)

As a consequence, if $\eta _\lambda = \varPhi ^* p_\lambda$ (respectively $\tilde{\eta }_\lambda = \varPhi ^* \tilde{p}_\lambda$) is the dual certificate of the noiseless (respectively noisy) problem, we have

$$\begin{aligned} || \eta _\lambda -\tilde{\eta }_\lambda ||_\infty \leqslant M \frac{|| w ||_2}{\lambda } \end{aligned}$$

(24)

for some $M>0$ (in fact $M=\sqrt{\int _\mathbb {T}|\varphi (t)|^2dt}= || \varPhi ^* ||_{\infty , 2}$).

From now on, we set $\alpha = \frac{r}{2M}$ and we impose $\frac{|| w ||_2}{\lambda }\leqslant \alpha$. Writing

$$\begin{aligned} || \eta _0-\tilde{\eta }_\lambda ||_{\infty }&\leqslant || \eta _0-\eta _\lambda ||_{\infty } + || \eta _\lambda -\tilde{\eta }_\lambda ||_{\infty },\\&\leqslant || \eta _0-\eta _\lambda ||_{\infty } + \frac{r}{2}, \end{aligned}$$

we see using Proposition 1 that for $\lambda$ small enough $\tilde{\eta }_\lambda$ satisfies (22).

Structure of the reconstructed measure By (22) for $g=\tilde{\eta }_\lambda$ and using the extremality conditions, we obtain that $|\tilde{m}_\lambda |(K_\varepsilon )=0$ and that $\tilde{m}_\lambda$ (respectively $-\tilde{m}_\lambda$) is non-negative in ${{\mathrm{Ext}}}^{+,\varepsilon }$ (respectively ${{\mathrm{Ext}}}^{-,\varepsilon }$). Indeed, the extremality conditions impose that $\tilde{\eta }_\lambda = {{\mathrm{sign}}}\frac{d\tilde{m}_\lambda }{d|\tilde{m}_\lambda |}$, $\tilde{m}_\lambda$-almost everywhere, hence the claimed result. $\square$

Lemma 1 does not make any assumption on the local structure of ${{\mathrm{Ext^\pm }}}m_0$ and does not provide any information on the local structure of $\tilde{m}_\lambda$ either (it might even not be discrete). If we assume that $\eta _0'' (x)\ne 0$ for some $x\in {{\mathrm{Ext}}}m_0$, then the reconstructed measure has at most one spike in the neighborhood of $x$.

Lemma 2

Assume that there exists a solution to $(\mathcal {D}_0(y_0))$ and that $\eta _0''(x)\ne 0$ for some $x\in {{\mathrm{Ext}}}m_0$. Then for $\varepsilon >0$ small enough, there exists $\alpha >0$, $\lambda _0>0$ such that for all $(\lambda , w)\in D_{\alpha , \lambda _0}$, the restriction of $\tilde{m}_\lambda$ to $(x-\varepsilon ,x+\varepsilon )$ is

• either the null measure,

• or of the form $\tilde{a}_{\lambda , w}\delta _{\tilde{x}_{\lambda ,w}}$ where ${{\mathrm{sign}}}\tilde{a}_{\lambda , w}= \eta _0(x)$ and $\tilde{x}_{\lambda ,w}\in (x-\varepsilon ,x+\varepsilon )$.

If, in addition, $m_0$ is identifiable and $|m_0|((x-\varepsilon ,x+\varepsilon ))\ne 0$, only the second case may happen.

Proof

The proof follows the same steps as those of Lemma 1.

Behavior of the minimal norm certificate First, observe that if $\eta _0''(x)\ne 0$ and $\eta _0(x)=1$ (respectively $-1$) for $x\in {{\mathrm{Ext}}}m_0$, then $\eta _0''(x)<0$ (respectively $>0$). As a consequence, $x$ is an isolated point of ${{\mathrm{Ext}}}m_0$. For $\varepsilon >0$ small enough, ${{\mathrm{Ext}}}m_0\cap (x-\varepsilon ,x+\varepsilon )=\{x\}$ and $|\eta _0''(t)|\geqslant \frac{|\eta _0''(x)|}{2}>0$ for all $t\in (x-\varepsilon ,x+\varepsilon )$.

Variations of dual certificates From (23), we infer that

$$\begin{aligned} || \eta _\lambda '' -\tilde{\eta }_\lambda '' ||_\infty \leqslant M \frac{|| w ||_2}{\lambda } \end{aligned}$$

(25)

with $M>0$ (here, $M=\sqrt{\int _\mathbb {T}|\varphi ''(t)|^2 \mathrm {d}t}= || (\varPhi '')^* ||_{\infty , 2}$).

We set $\alpha = \frac{r}{2M}$ with $r=\frac{|\eta _0''(x)|}{2}$ and we impose $\frac{|| w ||_2}{\lambda }\leqslant \alpha$, so that

$$\begin{aligned} || \eta _0''-\tilde{\eta }_\lambda '' ||_{\infty }&\leqslant || \eta _0''-\eta _\lambda '' ||_{\infty } + || \eta _\lambda ''-\tilde{\eta }_\lambda '' ||_{\infty },\\&\leqslant || \eta _0''-\eta _\lambda '' ||_{\infty } + \frac{r}{2}, \end{aligned}$$

thus $|| \eta _0''-\tilde{\eta }_\lambda '' ||_{\infty }<\frac{|\eta _0''(x)|}{2}$ for $\lambda$ small enough.

Structure of the reconstructed measure From the above inequality, we know that $\tilde{\eta }_\lambda$ is strictly concave (respectively strictly convex) in $(x-\varepsilon ,x+\varepsilon )$. As a result, there is at most one point $\tilde{x}_{\lambda , w}$ in $(x-\varepsilon ,x+\varepsilon )$ such that $\tilde{\eta }_\lambda (\tilde{x}_{\lambda , w})=1$ (respectively $-1$).

If $m_0$ is identifiable, it remains to prove that there is indeed one spike in $(x-\varepsilon ,x+\varepsilon )$. This is obtained by relying on a result by Bredies and Pikkarainen [4] which is an application of [22, Th. 3.5]. It guarantees that $\tilde{m}_{\lambda }$ converges to $m$ for the weak-* topology when $\lambda , || w ||_2\rightarrow 0$. We recall the result below (see Proposition 4) for the convenience of the reader.

By weak-* convergence of $\tilde{m}_\lambda$ to $m$ for $\lambda \rightarrow 0^+$ and $|| w ||_2\rightarrow 0$, $\tilde{m}_{\lambda }((x-\varepsilon ,x+\varepsilon ))$ must converge to $m_0((x-\varepsilon ,x+\varepsilon ))$. By the optimality conditions, we see that $|m_0((x-\varepsilon ,x+\varepsilon ))|=|m_0(\{x\})|$, so that $m_0(\{x\})\ne 0$ and ${{\mathrm{sign}}}m_0(\{x\})=\eta _0(x)$, hence the result. $\square$

In the Proof of Lemma 2, we have relied on the following result.

Proposition 4

([22, Th. 3.5],[4, Prop. 5]) Let $m_0$ be an identifiable measure, if $\lambda \rightarrow 0$ and $|| w ||\rightarrow 0$ with $\frac{|| w ||_2^2}{\lambda }\rightarrow 0$, then $\tilde{m}_\lambda$ converges to $m_0$ with respect to the weak-* topology.

3.3 Non-Degenerate Source Condition

The notion of extended signed support has strong connections with the source condition introduced in [6] to derive convergence rates for the Bregman distance.

Definition 4

(Source condition) A measure $m_0$ satisfies the source condition if there exists $p\in L^2(\mathbb {T})$ such that

$$\begin{aligned} \varPhi ^* p \in \partial {|| m_0 ||_{\text {TV}}}. \end{aligned}$$

In a finite-dimensional framework, the source condition is simply equivalent to the optimality of $m_0$ for $(\mathcal {P}_0(y_0))$ given $y_0=\varPhi m_0$. In the framework of Radon measures, the source condition amounts to assuming that $m_0$ is a solution of $(\mathcal {P}_0(y_0))$ and that there exists a solution to $(\mathcal {D}_0(y_0))$. In fact, the source condition simply means that the conditions of Proposition 3 hold.

If one is interested in $m_0$ being the unique solution of $(\mathcal {P}_0(y_0))$ for $y_0=\varPhi m_0$ (in which case we say that $m_0$ is identifiable), the source condition may be strengthened to give a sufficient condition.

Proposition 5

([12, Lemma 1.1]) Let $m_0 = m_{x_0,a_0}$ be a discrete measure. If $\varPhi _{x_0}$ has full rank, and if

• there exists $\eta \in {{\mathrm{Im}}}\varPhi ^*$ such that $\eta \in \partial {|| m_0 ||_{\text {TV}}}$,

• $\forall \,s \notin {{\mathrm{Supp}}}(m_0), \quad |\eta (s)| < 1$,

then $m_0$ is the unique solution of $(\mathcal {P}_0(y_0))$.

In this paper, in view of Lemma 2, we strengthen a bit more the source condition so as to derive a global stability result concerning the support of the solutions of $\mathcal {P}(y_0+w)$ (see Theorem 2).

Definition 5

(Non-degenerate source condition) Let $m_0=m_{x_0,a_0}$ be a discrete measure, and $\{x_{0,1},\ldots x_{0,N}\}={{\mathrm{Supp}}}m_0$. We say that $m_0$ satisfies the non-degenerate source condition (NDSC) if

• there exists $\eta \in {{\mathrm{Im}}}\varPhi ^*$ such that $\eta \in \partial {|| m_0 ||_{\text {TV}}}$.

• the minimal norm certificate $\eta _0$ satisfies

  $$\begin{aligned} \begin{array}{ll} \forall \,s \in \mathbb {T}\setminus \{x_{0,1},\ldots x_{0,N}\}, &{}\quad |\eta _0(s)| < 1, \\ \forall \,i\in \{1,\ldots N\}, &{}\quad \eta _0''(x_{0,i}) \ne 0. \end{array} \end{aligned}$$

In that case, we say that $\eta _0$ is not degenerate.

The first assumption in the above definition is the standard source condition. The last two assumptions impose conditions on the extended signed support, namely that ${{\mathrm{Supp^{\pm }}}}m_0 = {{\mathrm{Ext^\pm }}}(m_0)$ and $\eta _0''(t)\ne 0$ for all $t\in {{\mathrm{Supp}}}m_0$.

When $\varPhi$ is an ideal low-pass filter with cutoff frequency $f_c$, there are numerical evidences that measures having a large enough separation distance (proportional to $f_c$) satisfy the non-degenerate source condition, see Sect. 4.

3.4 Main Result

The following theorem, which is the main result of this paper, gives a global result on the precise structure of the solution when the signal-to-noise ratio is large enough and $\lambda$ is small enough.

Theorem 2

(Noise robustness) Let $m_0=m_{a_0,x_0}=\sum _{i=1}^N a_{0,i} \delta _{x_{0,i}}$ be a discrete measure. Assume that $\varGamma _{x_0}$ (defined in (4)) has full rank and that $m_0$ satisfies the non-degenerate source condition.

Then, there exists $\alpha >0, \lambda _0>0$, such that for $(\lambda ,w) \in D_{\alpha ,\lambda _0}$, the solution $\tilde{m}_\lambda$ of $\mathcal {P}_\lambda (y+w)$ is unique and is composed of exactly $N$ spikes.

Moreover, up to a permutation of indices, we may write $\tilde{m}_\lambda =\sum _{i=1}^N \tilde{a}_{\lambda , i}\delta _{\tilde{x}_{\lambda ,i}}$ with $\tilde{a}_{\lambda , i} \ne 0$ and ${{\mathrm{sign}}}(\tilde{a}_{\lambda , i})={{\mathrm{sign}}}(a_{0, i})$ (for $1\leqslant i \leqslant N$), and writing $(\tilde{a}_0,\tilde{x}_0)=(a_0,x_0)$, the mapping

$$\begin{aligned} (\lambda ,w) \in D_{\alpha ,\lambda _0} \mapsto (\tilde{a}_\lambda , \tilde{x}_\lambda ) \in \mathbb {R}^N\times \mathbb {T}^N, \end{aligned}$$

is $C^{k-1}$ whenever $\varphi \in C^{k}(\mathbb {T})$ ($k\geqslant 2$).

In particular, for $\lambda =\frac{1}{\alpha }|| w ||_2$, we have

$$\begin{aligned} \forall i\in \{1,\ldots N\},\quad |\tilde{x}_{\lambda ,i}-x_{0,i}|=O(|| w ||_2) \quad \text {and} \quad |\tilde{a}_{\lambda ,i}-a_{0,i}|=O(|| w ||_2). \end{aligned}$$

(26)

Proof

Applying Lemma 2 at each point $x_{0,i}$ for $1\leqslant i\leqslant N$ and Lemma 1, we see that for $\varepsilon >0$ small enough, there exists $\alpha >0$, $\lambda _0>0$ such that $\tilde{m}_\lambda$ has at most one spike in each interval $(x_{i,0}-\varepsilon ,x_{i,0}+\varepsilon )$, and

$$\begin{aligned} |\tilde{m}_\lambda |\left( \mathbb {T}\setminus \bigcup _{i=1}^N (x_{i,0}-\varepsilon ,x_{i,0}+\varepsilon )\right) =0. \end{aligned}$$

In fact, since $\varGamma _{x_0}$ has full rank, $\varPhi _{{{\mathrm{Ext}}}m_0}$ has full rank as well and $m_0$ is identifiable (by Proposition 3). Therefore, Lemma 2 ensures that there is indeed one spike in each interval, with sign equal to $\eta _0(x_{0,i})$.

It remains to prove the uniqueness of the amplitudes and locations $(\tilde{a}_\lambda ,\tilde{x}_\lambda )$ and their smoothness as function of $(\lambda , w)$. To this end, we observe that they satisfy the following implicit equation

$$\begin{aligned} E_{s_0}(\tilde{a}_\lambda ,\tilde{x}_\lambda ,\lambda ,w)=0 \end{aligned}$$

where $s_0={{\mathrm{sign}}}(a_0)=(\eta _0(x_{i,0}))_{1\leqslant i\leqslant N}$, and

$$\begin{aligned} E_{s_0}(a,x,\lambda ,w) = \begin{pmatrix} \varPhi _x^* ( \varPhi _x a - y_0 - w ) + \lambda s_0 \\ {\varPhi '_x}^* ( \varPhi _x a - y_0 - w ) \end{pmatrix} = \varGamma _x^* ( \varPhi _x a - y_0 - w ) + \lambda \begin{pmatrix} s_0\\ 0 \end{pmatrix}. \end{aligned}$$

Indeed, this implicit equation simply states that $\tilde{\eta }_\lambda (\tilde{x}_{\lambda ,i})= {{\mathrm{sign}}}(a_{0,i})={{\mathrm{sign}}}(\tilde{a}_{\lambda ,i})$ and that $\tilde{\eta }_\lambda '(\tilde{x}_{\lambda ,i})= 0$.

Since $((a,x),(\lambda ,w)) \mapsto E_{s_0}(a,x,\lambda ,w)$ is a $C^1$ function defined on $(\mathbb {R}^{N}\times \mathbb {T}^N) \times (\mathbb {R}\times L^2(\mathbb {T}^N))$, we may apply the implicit functions theorem.

The derivative of $E_{s_0}$ with respect to $x$ and $a$ reads

$$\begin{aligned} \frac{ \partial E}{\partial a} (a,x,\lambda ,w)&= \varGamma _x^* \varPhi _x \\ \frac{ \partial E}{\partial x} (a,x,\lambda ,w)&= \begin{pmatrix} {{\mathrm{diag}}}({\varPhi _x^{*}}' (\varPhi _x a - y_0-w ))\\ {{\mathrm{diag}}}({\varPhi _x^{*}}'' (\varPhi _x a - y_0-w )) \end{pmatrix} + \varGamma _x^* {\varPhi '_x} {{\mathrm{diag}}}(a). \end{aligned}$$

so that for $\lambda = 0$, $w=0$ and using $y_0 = \varPhi _{x_0}a_0$, one obtains

$$\begin{aligned} \frac{ \partial E_s}{\partial (a,x)} (a_0,x_0,0,0)&= \varGamma _{x_0}^* \begin{pmatrix} \varPhi _{x_0}, \, \varPhi '_{x_0} {{\mathrm{diag}}}(a_0) \end{pmatrix} \\&= (\varGamma _{x_0}^* \varGamma _{x_0}) \begin{pmatrix} \mathrm {Id}&{} 0 \\ 0 &{} {{\mathrm{diag}}}(a_0) \end{pmatrix}. \end{aligned}$$

Since we assume $\varGamma _{x_0}$ has full rank, then $\frac{ \partial E_{s_0}}{\partial (a,x)} (a_0,x_0,0,0)$ is invertible and the implicit functions theorem applies: There is a neighborhood $V\times W$ of $(a_0,x_0)\times \{(0,0)\}$ in $(\mathbb {R}^{N}\times \mathbb {T}^N)\times (\mathbb {R}\times L^2(\mathbb {T}))$ and a function $f : W\rightarrow V$ such that

$$\begin{aligned}&((a,x),\lambda ,w) \in V\times W \quad \text {and} \quad E_{s_0}(a,x,\lambda ,w)=0 \\&\quad \Longleftrightarrow \quad (\lambda ,w) \in W \quad \text {and} \quad (a,x)=f(\lambda ,w). \end{aligned}$$

Moreover, writing $(\hat{a}_{\lambda ,w}, \hat{x}_{\lambda ,w}) = f(\lambda ,w)\in \mathbb {R}^N\times \mathbb {T}^N$, we have

• $(\hat{a}_{0,0}, \hat{x}_{0,0}) = (a_0, x_0)$,

• for any $(\lambda ,w) \in W$, ${{\mathrm{sign}}}(\hat{a}_{\lambda ,w}) = s_0$,

• if $\varphi \in C^k(\mathbb {T})$ (for $k\geqslant 2$), then $f\in C^{k-1}(W)$.

The constructed amplitudes and locations $(\hat{a}_{\lambda ,w}, \hat{x}_{\lambda ,w})$ coincide with those of the solutions of $\mathcal {P}_\lambda (y_0+w)$ for all $(\lambda , w)\in W$ such that $|| w ||_2\leqslant \alpha \lambda$. Possibly changing the value of $\lambda _0$ so that $D_{\alpha , \lambda _0}\subset W$, we obtain the desired result. $\square$

Remark 1

Although this paper focuses on identifiable measures, Theorem 2 describes the evolution of the solutions of $\mathcal {P}_\lambda (y_0+w)$ for any input measure $m_1$ such that there exists $m_0$ which satisfies the non-degenerate source condition and $y_0=\varPhi m_1= \varPhi m_0$. Instead of converging toward $m_1$, the solutions will converge toward $m_0$.

3.5 Extensions

Theorem 2 extends in a straightforward manner to higher dimensions, i.e., when replacing $\mathbb {T}$ by $\mathbb {T}^d$ for $d \geqslant 1$. In the NDSC introduced in Definition 5, one should replace, for $i=1,\ldots ,N$, the constraint $\eta _0''(x_{0,i}) \ne 0$ by the constraint that the Hessian $D^2 \eta _0(x_{0,i}) \in \mathbb {R}^{d \times d}$ is invertible.

The proof also extends to non-stationary filtering operators, i.e., which can be written as

$$\begin{aligned} \forall \,t \in \mathbb {T}^d, \quad \varPhi m(t) = \int _{\mathbb {T}^d} \varphi (x,t) \mathrm {d}m(x) \end{aligned}$$

where $\varphi \in C^2(\mathbb {T}^d \times \mathbb {T}^d)$.

3.6 Application to the ideal Low-pass filter

We first observe that the injectivity condition on $\varGamma _x$ assumed in Theorem 2 always holds.

Proposition 6

(Injectivity of $\varGamma _{x}$ Let $x=(x_1,\ldots x_N)\in \mathbb {T}^N$ with $x_i\ne x_j$ for $i\ne j$ and $N\leqslant f_c$. Then, $\varGamma _x=(\varPhi _x, \varPhi _x')$ has full rank.

The proof is given in “Appendix 2.”

As to whether or not the non-degenerate source condition holds for discrete measures, we will discuss this matter in Sect. 4 more in depth. For now, let us mention that we have observed empirically that this condition holds under the hypotheses of Theorem $1.2$ in [8], namely that $\Delta (m)>\frac{1.87}{f_c}$, but also with measures with far smaller values of $\Delta (m)$.

Figure 1 shows the whole solution path $\lambda \mapsto \tilde{m}_\lambda$ of the solutions of $\mathcal {P}_\lambda (\varPhi m_0 + w)$ when $f_c=10$ and the input measure is identifiable and has three spikes separated by $\Delta (m_0)=0.7/f_c$. Such a measure satisfies the non-degenerate source condition as shown in plot (a). The plots (b,c,d) illustrate the conclusion of Theorem 2. For values of $\lambda$ which are too small with respect to $|| w ||$, the solution $\tilde{m}_\lambda$ is perturbed with spurious spikes, but as soon as $\lambda$ is large enough, $\tilde{m}_\lambda$ has a support that closely (but not exactly) matches the one of $m_0$. For large value of $\lambda$, spikes starts disappearing, and the support is not correctly estimated. Figure 2 shows the solutions of $\mathcal {P}_\lambda (\varPhi m_0 + \lambda w_0)$, i.e., the noise $w=\lambda w_0$ is scaled by the regularization parameter $\lambda$. In accordance with Theorem 2, this shows that for $\Vert w\Vert _2/\lambda = \Vert w_0\Vert _2 \leqslant 0.07$, the support of the spikes is precisely estimated.

Fig. 1

a Input measure $m_0$, and corresponding minimal norm certificate. b–d Regularization paths $\lambda \mapsto \tilde{m}_\lambda$ that are solutions of $\mathcal {P}_\lambda (\varPhi m_0 + w)$ for three different noise levels $|| w ||$. Each “strip” represents the evolution of a spike as $\lambda$ varies. The color refers to the sign of the spike (blue for negative and red for positive) and the (vertical) width is proportional to its amplitude. The exact location is given by the middle of each band (Color figure online)

Fig. 2

Same plots as Fig. 1 except that the solutions of $\mathcal {P}(\varPhi m_0 + \lambda w_0)$ are displayed instead of those of $\mathcal {P}_\lambda (\varPhi m_0 + w)$

4 Vanishing Derivatives Pre-certificate

We show in this section that, if the non-degenerate source condition holds, the minimal norm certificate $\eta _0$ is characterized by its values on the support of $m_0$ and the fact that its derivative must vanish on the support of $m_0$. Thus, one may compute the minimal norm certificate simply by solving a linear system, without handling the cumbersome constraint $|| \eta _0 ||_{\infty } \leqslant 1$.

4.1 Dual Pre-certificates

Loosely speaking, we call pre-certificate any “good candidate” for a solution of (12). Typically, a pre-certificate is built by solving a linear system (with possibly a condition on its norm). The following pre-certificate appears naturally in our analysis.

Definition 6

(Vanishing derivative pre-certificate) The vanishing derivative pre-certificate associated with a measure $m_0 = m_{a_0,x_0}$ is $\eta _\mathrm{V}= \varPhi ^* p_{\mathrm{V}}$ where

$$\begin{aligned} p_{\mathrm{V}}=\underset{p \in L^2(\mathbb {T})}{{{\mathrm{argmin}}}}\;|| p ||_2 \quad \text {subj. to} \quad \forall \,1\leqslant i\leqslant N,\quad \left\{ \begin{array}{l} (\varPhi ^* p)(x_{0,i})={{\mathrm{sign}}}(a_{0,i}), \\ (\varPhi ^* p)'(x_{0,i})=0. \end{array} \right. \end{aligned}$$

(27)

It is clear that if the source condition (see Definition 4) holds, then $p_{\mathrm{V}}$ exists (since Problem (27) is feasible). Observe that, in general, $\eta _\mathrm{V}$ is not a certificate for $m_0$ since it does not satisfy the constraint $\Vert \eta _\mathrm{V}\Vert _\infty \leqslant 1$. The following proposition gathers several facts about the vanishing derivative pre-certificate which show that it is indeed a good candidate for the minimal norm certificate.

Proposition 7

Let $m_0=m_{a_0,x_0}=\sum _{i=1}^N a_{0,i} \delta _{x_{0,i}}$ be a discrete measure. The following assertions hold.

• Problem (27) is feasible and $\Vert \eta _\mathrm{V}\Vert _\infty \leqslant 1$ if and only if the source condition holds and $\eta _\mathrm{V}=\eta _0$.

• If Problem (27) is feasible and $\varGamma _{x_0}$ has full rank, i.e., $\varGamma _{x_0}^* \varGamma _{x_0} \in \mathbb {R}^{2N \times 2N}$ is invertible, then

  $$\begin{aligned} \eta _\mathrm{V}= \varPhi ^* \varGamma _{x_0}^{+,*} \begin{pmatrix} {{\mathrm{sign}}}(a_0) \\ 0 \end{pmatrix} \quad \text {where} \quad \varGamma _{x_0}^{+,*} = \varGamma _{x_0}(\varGamma _{x_0}^* \varGamma _{x_0})^{-1}. \end{aligned}$$

• If $\varGamma _{x_0}$ has full rank, then $m_0$ satisfies the non-degenerate source condition if and only if Problem (27) is feasible and

  $$\begin{aligned} \begin{array}{ll} \forall \,s \in \mathbb {T}\setminus \{x_{0,1},\ldots x_{0,N}\}, &{}\quad | \eta _\mathrm{V}(s)| < 1, \\ \forall \,i\in \{1,\ldots N\}, &{}\quad \eta _\mathrm{V}''(x_{0,i}) \ne 0. \end{array} \end{aligned}$$

The third assertion of Proposition 7 states that it is equivalent to check the non-degenerate source condition on $\eta _0$ (Definition 5) or to check the same conditions on $\eta _\mathrm{V}$. In case those conditions hold, one even has $\eta _\mathrm{V}=\eta _0$ (first assertion). The main point of this equivalence is that the second assertion yields a practical expression to compute $\eta _\mathrm{V}$ which may be used in numerical experiments (see Sect. 4.3).

Proof

For the first assertion, we observe that if Problem (27) is feasible (and thus $p_{\mathrm{V}}$ exists) and $\Vert \eta _\mathrm{V}\Vert _\infty \leqslant 1$, then $\eta _\mathrm{V}\in \partial {|| m_0 ||_{\text {TV}}}$ and the source condition holds. Hence, $\Vert p_{\mathrm{V}}\Vert _2\geqslant \Vert p_0\Vert _2$. On the other hand, the minimal norm certificate $\eta _0$ must satisfy all the constraints of (27); thus, the minimality of the norms of both $\eta _\mathrm{V}$ and $\eta _0$ implies that $\eta _\mathrm{V}=\eta _0$. The converse implication is obvious.

For the second assertion, Problem (27) can be written as

$$\begin{aligned} \eta _\mathrm{V}= \underset{ \eta = \varPhi ^* p }{{{\mathrm{argmin}}}}\; || p ||_2. \quad \text {subj. to} \quad \left\{ \begin{array}{l} \varPhi _{x_0}^*p = {{\mathrm{sign}}}(a_0), \\ \varPhi _{x_0}'^*p = 0, \end{array} \right. \end{aligned}$$

which is a quadratic optimization problem in a Hilbert space with a finite number of affine equality constraints. Moreover, the assumption that $\varGamma _{x_0}$ has full rank implies that the constraints are qualified. Hence, it can be solved by introducing Lagrange multipliers $u$ and $v$ for the constraints. One should therefore solve the following linear system to obtain the value of $p= p_{\mathrm{V}}$

$$\begin{aligned} \begin{pmatrix} \mathrm {Id}&{} \varPhi _{x_0} &{} \varPhi '_{x_0} \\ \varPhi _{x_0}^* &{} 0 &{}0 \\ {\varPhi '_{x_0}}^* &{} 0 &{}0 \\ \end{pmatrix} \begin{pmatrix} p \\ u \\ v \end{pmatrix} = \begin{pmatrix} 0 \\ s \\ 0 \end{pmatrix}. \end{aligned}$$

Solving for $(u, v)$ in these equations gives the result.

For the third assertion, if the non-degenerate source condition holds, we apply Theorem 2 which yields a $C^1$ path $\lambda \mapsto (\tilde{a}_\lambda ,\tilde{x}_\lambda )$ of solutions of $\mathcal {P}_\lambda (y_0)$ (we consider the case $w=0$). Then from Proposition 8 below, we obtain that $\eta _\mathrm{V}$ is a valid certificate and $\eta _\mathrm{V}=\eta _0$; hence, $\eta _\mathrm{V}$ is non-degenerate. The converse implication is a straightforward consequence of the first assertion. $\square$

4.2 Necessary Condition for Support Recovery

There is a priori no reason for the vanishing derivative pre-certificate $\eta _\mathrm{V}$ to satisfy $|| \eta _\mathrm{V} ||_{\infty }\leqslant 1$. Here, we prove that that is in fact a necessary condition for (noiseless) exact support recovery to hold on some interval $[0,\lambda _0)$ with $\lambda _0>0$, i.e., the solutions of $\mathcal {P}_\lambda (y_0)$ having exactly $N$ spikes which converge smoothly toward those of the original measure.

Proposition 8

Let $m_0=m_{a_0,x_0}=\sum _{i=1}^N a_{0,i} \delta _{x_{0,i}}$ be a discrete measure such that $\varGamma _{x_0}$ has full rank. Assume that there exists $\lambda _0>0$ and a $C^1$ path $[0,\lambda _0)\rightarrow \mathbb {R}^N\times \mathbb {T}^N$, $\lambda \mapsto (a_\lambda , x_\lambda )$ such that for all $\lambda \in [0,\lambda _0)$ the measure $m_\lambda =m_{a_\lambda ,x_\lambda }$ is a solution to $\mathcal {P}_\lambda (y_0)$ (the noiseless problem).

Then $\eta _\mathrm{V}$ exists, $\Vert \eta _\mathrm{V}\Vert _\infty \leqslant 1$ and $\eta _\mathrm{V}=\eta _0$.

Proof

Let $p_\lambda =\frac{1}{\lambda } (y_0-\varPhi m_\lambda )=\frac{1}{\lambda } (\varPhi _{x_0}a_0 -\varPhi _{x_\lambda } a_\lambda )$ be the certificate defined by the optimality conditions (11). We show that $\varPhi ^* p_\lambda$ converges toward $\varPhi ^* \varGamma _{x_0}^{+,*} \begin{pmatrix} {{\mathrm{sign}}}(a_0) \\ 0 \end{pmatrix}= \eta _\mathrm{V}$ (and that the latter exists).

Writing

$$\begin{aligned} a_\lambda ' = \frac{\mathrm {d}a_\lambda }{\mathrm {d}\lambda } \in \mathbb {R}^N \quad \text {and} \quad x_\lambda ' = \frac{\mathrm {d}x_\lambda }{\mathrm {d}\lambda } \in \mathbb {R}^N, \end{aligned}$$

we observe that for any $i\in \{1,\ldots N\}$ and any $x\in \mathbb {T}$,

$$\begin{aligned}&\frac{a_{\lambda ,i}\varphi (x_{\lambda , i}-x) - a_{0,i}\varphi (x_{0, i}-x)}{\lambda }- \left[ a_{0,i}\varphi ' ( x_{0, i}-x) x_{0,i}' + a_{0,i}'\varphi ( x_{0, i}-x) \right] \\&\quad = \int _{0}^1 \left[ a_{\lambda t,i}\varphi ' ( x_{\lambda t, i}-x) x_{\lambda t,i}' + a_{\lambda t,i}'\varphi ( x_{\lambda t, i}-x) \right] \\&\qquad \qquad - \left[ a_{0,i}\varphi ' ( x_{0, i}-x) x_{0,i}' + a_{0,i}'\varphi ( x_{0, i}-x) \right] \mathrm {d}t, \end{aligned}$$

and the latter integral converges (uniformly in $x$) to zero when $\lambda \rightarrow 0^+$ by uniform continuity of its integrand (since $a$, $x$ and $\varphi$ are $C^1$). As a consequence, we obtain that $\frac{y_0-\varPhi _{ x_\lambda }{a}_\lambda }{\lambda }$ converges uniformly to $-\varGamma _{x_0} \begin{pmatrix} \mathrm {Id}&{} 0 \\ 0 &{} {{\mathrm{diag}}}(a_0) \end{pmatrix} \begin{pmatrix} a_0' \\ x_0' \end{pmatrix}$.

On the other hand, we observe that for $\lambda$ small enough, ${{\mathrm{sign}}}(a_\lambda )={{\mathrm{sign}}}(a_0)$, and using the notations of the Proof of Theorem 2, the implicit equation $E_{s_0}(a_\lambda ,x_\lambda ,\lambda ,0)=0$ holds. Differentiating that equation at $\lambda =0$, we obtain:

$$\begin{aligned} \left( \frac{ \partial E_{s_0}}{\partial (a,x)} (a_0,x_0,0,0) \right) \begin{pmatrix} a_0' \\ x_0' \end{pmatrix} + \frac{ \partial E_{s_0}}{\partial \lambda } (a_0,x_0,0,0)&=0, \end{aligned}$$

or equivalently

$$\begin{aligned} (\varGamma _{x_0}^* \varGamma _{x_0})\begin{pmatrix} \mathrm {Id}&{} 0 \\ 0 &{} {{\mathrm{diag}}}(a_0) \end{pmatrix} \begin{pmatrix} a_0' \\ x_0' \end{pmatrix}&= - \begin{pmatrix} s_0\\ 0 \end{pmatrix}. \end{aligned}$$

As a consequence, Problem (27) is feasible and we see that $\frac{y_0-\varPhi _{ x_\lambda }{a}_\lambda }{\lambda }$ converges uniformly (and thus in the $L^2$ strong topology) to $\varGamma _{x_0}(\varGamma _{x_0}^* \varGamma _{x_0})^{-1}\begin{pmatrix} {{\mathrm{sign}}}(a_0) \\ 0 \end{pmatrix}$ and $\varPhi ^*\left( \frac{y_0-\varPhi _{ x_\lambda }}{\lambda }\right)$ converges uniformly to $\varPhi ^* \varGamma _{x_0}^{+,*} \begin{pmatrix} {{\mathrm{sign}}}(a_0) \\ 0 \end{pmatrix}$ (which is precisely $\eta _\mathrm{V}$ from the second assertion of Proposition 7).

Since $\Vert \varPhi ^* \left( \frac{y_0-\varPhi _{ x_\lambda }a_\lambda }{\lambda }\right) \Vert _\infty = \Vert \varPhi ^*p_\lambda \Vert _\infty \leqslant 1$ for all $\lambda >0$, we obtain that $\Vert \eta _\mathrm{V}\Vert _\infty \leqslant 1$, hence the claimed result. $\square$

4.3 Application to the Ideal Low-pass Filter

In order to prove their identifiability result for measures, the authors of [8] also introduce a “good candidate” for a dual certificate associated with $m=m_{a,x}$ for $a \in \mathbb {C}^N$ and $x \in \mathbb {R}^N$. For $K$ being the square of the Fejer kernel, they build a trigonometric polynomial

$$\begin{aligned} \eta _\mathrm{CF}(t)=\sum _{i=1}^N \left( \alpha _{i}K(t-x_i) + \beta _{i}K'(t-x_i) \right) \text{ with } K(t)=\left( \frac{\sin \left( \left( \frac{f_c}{2}+1 \right) \pi t\right) }{\left( \frac{f_c}{2}+1\right) \sin \pi t}\right) ^4 \end{aligned}$$

and compute $(\alpha _i,\beta _i)_{i=1}^N$ by imposing that $\eta _\mathrm{CF}(x_i)={{\mathrm{sign}}}(a_i)$ and $( \eta _\mathrm{CF})' (x_i)=0$.

They show that the constructed pre-certificate is indeed a certificate, i.e., that $|| \eta _\mathrm{CF} ||_{\infty } \leqslant 1$, provided that the support is separated enough (i.e., when $\Delta (m)\geqslant C/f_c$). This result is important since it proves that measures that have sufficiently separated spikes are identifiable. Furthermore, using the fact that $\eta _\mathrm{CF}$ is not degenerate (i.e., $( \eta _\mathrm{CF})''(x_i) \ne 0$ for all $i=1,\ldots ,N$), the same authors derive an $L^2$ robustness to noise result in [7], and Fernandez-Granda and Azais et al. use the constructed certificate to analyze finely the local averages of the spikes in [1, 19].

From a numerical perspective, we have investigated how this pre-certificate compares with the vanishing derivative pre-certificate that appears naturally in our analysis, by generating real-valued measures for different separation distances and observing when each pre-certificate $\eta$ satisfies $|| \eta ||_{\infty } \leqslant 1$.

As predicted by the result of [8], we observe numerically that the pre-certificate $\eta _\mathrm{CF}$ is a certificate (i.e., $|| \eta _\mathrm{CF} ||_{\infty } \leqslant 1$) for any measure with $\Delta (m_0) \geqslant 1.87/f_c$. We also observe that this continues to hold up to $\Delta (m_0) \geqslant 1/f_c$. Yet, below $1/f_c$, it may happen that some measures are still identifiable (as asserted using the vanishing derivative pre-certificate $\eta _\mathrm{V}$) but $\eta _\mathrm{CF}$ stops being a certificate, i.e., $|| \eta _\mathrm{CF} ||_{\infty } > 1$. A typical example is shown in Fig. 3, where, for $f_c=6$ we have used three equally spaced masses as an input, their separation distance being $\Delta (m_0)\in \{\frac{0.8}{f_c},\frac{0.7}{f_c},\frac{0.6}{f_c},\frac{0.5}{f_c}\}$. Here, we have computed an approximation of the minimal norm certificate $\eta _0$ by solving $(\mathcal {D}_\lambda (y_0))$ with very small $\lambda$.

Fig. 3

Pre-certificates for three equally spaced masses. The blue curves with dots is the Fejer pre-certificate $\eta _\mathrm{CF}$, while red continuous line is the vanishing derivative $\eta _\mathrm{V}$. The black dashed line is the minimal norm pre-certificate $\eta _0$ (Color figure online)

For $\Delta (m_0)=\frac{0.8}{f_c}$, both $\eta _\mathrm{V}$ and $\eta _\mathrm{CF}$ are certificates, so that the vanishing derivatives pre-certificate $\eta _\mathrm{V}$ is equal to the minimal norm certificate $\eta _0$. For $\Delta (m_0)=\frac{0.7}{f_c}$, $\eta _\mathrm{CF}$ violates the constraint $|| \eta _\mathrm{CF} ||_{\infty }\leqslant 1$ but the vanishing derivative pre-certificate is still a certificate (even showing that the measure is identifiable). For $\Delta (m_0)=\frac{0.6}{f_c}$ and $\frac{0.5}{f_c}$, neither $\eta _\mathrm{V}$ nor $\eta _\mathrm{CF}$ satisfy the constraint, hence $\eta _\mathrm{V}\ne \eta _0$. Yet, $\eta _0$ ensures that $m_0$ is a solution to $(\mathcal {P}_0(y_0))$.

From the experiments we have carried out, we have observed that the vanishing derivative pre-certificate $\eta _\mathrm{V}$ behaves in general at least as well as the square Fejer $\eta _\mathrm{CF}$. The only exceptions we have noticed is for a large number of peaks (when $N$ is close to $f_c$), with $\Delta (m_0)\leqslant \frac{1.5}{f_c}$. This is illustrated in Fig. 4 which shows a measure $m_0$ for which $\eta _\mathrm{CF}$ is a non-degenerate certificate (which shows that it is identifiable), but for which $\eta _0 \ne \eta _\mathrm{V}$ since $|| \eta _\mathrm{V} ||_{\infty }>1$ (thus $\eta _\mathrm{V}$ is not a certificate). Typically, we have in this case ${{\mathrm{Supp^{\pm }}}}m_0 \subsetneq {{\mathrm{Ext^\pm }}}(m_0)$. Such a measure is identifiable but there is no support recovery for $\lambda >0$ (in the sense of Proposition 8), hence its support is not stable.

Fig. 4

Example of measure for which $\eta _\mathrm{V}\ne \eta _0$

Such pathological cases are relatively rare. An intuitive explanation for this is the fact that having $\eta _0(x)=\pm 1$ for $x\in \mathbb {T}\setminus {{\mathrm{Supp}}}(m_0)$ or $\eta _0''(x)=0$ for some $x\in {{\mathrm{Supp}}}(m_0)$ tend to impose a large $L^2$ norm, thus contradicting the minimality of $p_0$ (recall that when $\varphi$ is an ideal low- pass filter $|| \eta ||_2=|| p ||_2$).

5 Discrete Sparse Spikes Deconvolution

5.1 Finite-Dimensional $\ell ^1$ Regularization

A popular way to compute approximate solutions to $(\mathcal {P}_\lambda (y_0))$ with fast algorithms is to solve this problem on a finite discrete grid $\mathcal {G}\subset \mathbb {T}$. Denoting by $P$ the cardinal of the grid $\mathcal {G}$, and by $g\in \mathbb {T}^P$ the finite sequence of elements of $\mathcal {G}$, the idea is to solve $\mathcal {P}_\lambda (y_0)$ (or $(\mathcal {P}_0(y_0))$) with the additional constraint that $m=\sum _{i=1}^P a_i \delta _{g_i}$ for some $a\in \mathbb {R}^P$.

This is nothing but the so-called basis pursuit denoising problem [9], also known as the Lasso [31] in statistics. Indeed, defining the linear operator $\Psi$ through

$$\begin{aligned} \Psi a = \varPhi m= \sum _{i=1}^P (\varPhi \delta _{g_i})a_i, \end{aligned}$$

the problem amounts to:

where $\Psi :\mathbb {R}^P \rightarrow L^2(\mathbb {T})$ is a linear operator ($L^2(\mathbb {T})$ may as well be replaced with $\mathbb {R}^Q$ or any Hilbert space), and $a_i$ denotes the mass at each point $i$ of the grid. In the noiseless case, the exact reconstruction problem reads:

The aim of the present section is to study the asymptotic of Problems $(\tilde{\mathcal {P}}_\lambda ^\mathcal {G}(y_0))$ and $(\tilde{\mathcal {P}}_0^\mathcal {G}(y_0))$ as the stepsize of the grid $\mathcal {G}$ vanishes. To this end, we keep the framework of measures and we reformulate the constraint that ${{\mathrm{Supp}}}(m) \subset \mathcal {G}$, i.e., that $m$ can be written as $m=m_{a,x}$, where $x = (x_1,\ldots ,x_N)\in \mathcal {G}^N$. Recall that the notation $m_{a,x}$ hints that $a_i\ne 0$ for all $i$ and that the $x_i$’s are all distinct, so that in general $N\leqslant P$. We thus adopt the following penalization term

$$\begin{aligned} || m ||_{\text {TV},\mathcal {G}} = \sup \left\{ \int \psi \mathrm {d}m \;;\; \psi \in C(\mathbb {T}), \forall t\in \mathcal {G}\ |\psi (t)| \leqslant 1 \right\} , \end{aligned}$$

(28)

so that $|| m ||_{\text {TV},\mathcal {G}} =+\infty$ when ${{\mathrm{Supp}}}(m) \not \subset \mathcal {G}$, and $\sum _{i=1}^N |a_i|$ otherwise.

Problems $(\tilde{\mathcal {P}}_\lambda ^\mathcal {G}(y_0))$ and $(\tilde{\mathcal {P}}_0^\mathcal {G}(y_0))$ are then, respectively, equivalent to:

and

Let us stress the fact that the results of Sects. 5.2 and 5.3 hold for any finite dimensional matrix $\Psi \in \mathbb {R}^{P \times Q}$ or linear operator $\Psi : \mathbb {R}^P\rightarrow L^2(\mathbb {T})$: The columns of $\Psi$ need not be the samples of a convolution operator.

5.2 Certificates Over a Discrete Grid

As in Sect. 2, we may compute the subdifferential of the $\ell ^1$ norm. For $m=m_{a,x}=\sum _{i=1}^N a_i\delta _{x_i}$ with support in $\mathcal {G}$:

$$\begin{aligned} \partial || m ||_{\text {TV},\mathcal {G}} = \left\{ \eta \in C(\mathbb {T}) \;;\; || \eta ||_{\infty ,\mathcal {G}} \leqslant 1, \forall \,i=1,\ldots ,N, \; \eta (x_i)={{\mathrm{sign}}}(a_i) \right\} . \end{aligned}$$

(29)

where

$$\begin{aligned} || \eta ||_{\infty ,\mathcal {G}}= \max \left\{ |\eta (t)| \;;\; t\in \mathcal {G} \right\} . \end{aligned}$$

We also introduce the corresponding dual problems:

Remark 2

Let us denote by $G$ the image by $\varPhi$ of all measures with support in $\mathcal {G}$. It may happen (for instance if the grid is too rough) that $y_0\notin G$, in which case $(\mathcal {P}_0^\mathcal {G}(y_0))$ is not feasible and $(\mathcal {D}_0^\mathcal {G}(y_0))$ has infinite value. But $(\mathcal {P}_\lambda ^\mathcal {G}(y_0))$ is then equivalent to $\mathcal {P}_\lambda ^\mathcal {G}(y_{0,G})$ where $y_0=y_{0,G}+y_{0,G^\perp }$ is an orthogonal decomposition. Problem $(\mathcal {P}_\lambda ^\mathcal {G}(y_0))$ is thus an approximation of $\mathcal {P}_0^\mathcal {G}(y_{0,G})$, and the relevant dual problems are $\mathcal {D}_\lambda ^\mathcal {G}(y_{0,G})$ and $\mathcal {D}_0^\mathcal {G}(y_{0,G})$. For the sake of simplicity, we shall assume from now on that $y_0\in G$, but the reader may keep in mind that this hypothesis can be withdrawn by replacing $y$ with $y_{0,G}$.

In view of Remark 2, we observe that problems $(\mathcal {D}_\lambda ^\mathcal {G}(y_0))$ and $(\mathcal {D}_0^\mathcal {G}(y_0))$ are in fact finite dimensional. Indeed, their constraints being invariant by addition of elements of $G^\perp$, we may consider their quotient with the space $G^\perp$. Therefore, the condition $p\in L^2(\mathbb {T})$ may be reduced to $p\in G$ where $G$ is a finite-dimensional space.

As a consequence, a solution to $(\mathcal {D}_0^\mathcal {G}(y_0))$ always exists, so that we may define the discrete minimal norm certificate:

$$\begin{aligned} \eta _0^\mathcal {G}=\varPhi ^* p_0^\mathcal {G},&\quad \text {where} \quad p_0^\mathcal {G}=\underset{p}{{{\mathrm{argmin}}}}\; \left\{ || p ||_2 \;;\; p \text{ is } \text{ a } \text{ solution } \text{ of } (\mathcal {D}_0^\mathcal {G}(y_0)) \right\} . \end{aligned}$$

(30)

The solutions of $(\mathcal {P}_\lambda ^\mathcal {G}(y_0))$ and $(\mathcal {D}_\lambda ^\mathcal {G}(y_0))$ [respectively $(\mathcal {P}_0^\mathcal {G}(y_0))$ and $(\mathcal {D}_0^\mathcal {G}(y_0))$] are related by the extremality conditions (11) [respectively (12)] where the total variation is replaced with its discrete counterpart $|| \cdot ||_{\text {TV},\mathcal {G}}$.

5.3 Noise Robustness

As in the continuous case (cf. Sect. 3), the support of the solutions of $\mathcal {P}_\lambda ^\mathcal {G}(y_0+w)$ for $\lambda \rightarrow 0^+$ and $|| w ||_2=O(\lambda )$ is governed by the minimal norm certificate. We introduce here the discrete counterpart of the extended support of a measure.

Definition 7

(Extended support) Let $m_0\in \mathcal {M}(\mathbb {T})$ such that $y_0=\varPhi (m_0)\in G$, and let $\eta ^\mathcal {G}_0$ be the discrete minimal norm certificate defined in (30). The extended support of $m_0$ relatively to $\mathcal {G}$ is defined as

$$\begin{aligned} {{\mathrm{Ext_\mathcal {G}}}}(m_0) = \left\{ t\in \mathcal {G} \;;\; |\eta _0^\mathcal {G}(t)|= 1 \right\} , \end{aligned}$$

(31)

and the extended signed support relatively to $\mathcal {G}$ as

$$\begin{aligned} {{\mathrm{Ext^\pm _\mathcal {G}}}}(m_0) = \left\{ (t,v) \in \mathcal {G}\times \{-1,+1\} \;;\; \eta _0^\mathcal {G}(t)= v \right\} . \end{aligned}$$

(32)

It is important to notice that the assumption $y_0\in G$ does not mean that the support of $m_0$ is included in $\mathcal {G}$, but that there exists a measure with support included in $\mathcal {G}$ which produces the same observation $y_0$. Therefore, the support of $m_0$ and its extended support may even be disjoint.

As in the continuous case, notice that $m_0$ is a solution of $(\mathcal {P}_0^\mathcal {G}(y_0))$ if and only if ${{\mathrm{Supp^{\pm }}}}(m_0) \subset {{\mathrm{Ext^\pm _\mathcal {G}}}}(m_0)$.

Theorem 3

(Noise robustness, discrete case) Let $m_0\in \mathcal {M}(\mathbb {T})$ such that $y_0=\varPhi (m_0)\in G$. Then, there exists $\alpha >0, \lambda _0>0$, such that for $(\lambda ,w) \in D_{\alpha ,\lambda _0}$ (defined in (6)) any solution $\tilde{m}_{\lambda ,w}$ of $\mathcal {P}_\lambda ^\mathcal {G}(y_0+w)$ satisfies:

$$\begin{aligned} {{\mathrm{Supp^{\pm }}}}(\tilde{m}_{\lambda ,w})&\subset {{\mathrm{Ext^\pm _\mathcal {G}}}}(m_0). \end{aligned}$$

(33)

If, in addition, $\varPhi _{{{\mathrm{Ext_\mathcal {G}}}}(m_0)}$ has full rank and $m_0$ is a solution of $(\mathcal {P}_0^\mathcal {G}(y_0))$, then the solution $\tilde{m}_{\lambda ,w}$ is unique, $m_0$ is identifiable and choosing $\lambda = || w ||_2/\alpha$ ensures $|| \tilde{m}_{\lambda ,w}-m ||_{2,\mathcal {G}} = O(|| w ||)$, where

$$\begin{aligned} || \tilde{m}_{\lambda ,w}-m ||_{2,\mathcal {G}}^2 = \sum _{x\in \mathcal {G}} |m(\{x\})-\tilde{m}_{\lambda ,w}(\{x\})|^2. \end{aligned}$$

Proof

The proof is essentially the same as in the continuous case; therefore, we only sketch it. To simplify the notation, we write $J={{\mathrm{Ext_\mathcal {G}}}}(m_0)$. The solutions of $(\mathcal {D}_\lambda ^\mathcal {G}(y_0))$ converge to $p_0^\mathcal {G}\in L^2(\mathbb {T})$ for $\lambda \rightarrow 0^+$, where $\varPhi ^*p_0^\mathcal {G}=\eta _0^\mathcal {G}$ is the discrete minimal norm certificate.

By the triangle inequality

$$\begin{aligned} || \tilde{\eta }_\lambda - \eta _0^\mathcal {G} ||_{\infty ,\mathcal {G}}&\leqslant \underbrace{|| \tilde{\eta }_\lambda -\eta _\lambda ^\mathcal {G} ||_{\infty ,\mathcal {G}}}_{\leqslant C\frac{|| w ||_2}{\lambda } } + || \eta _\lambda ^\mathcal {G}-\eta _0^\mathcal {G} ||_{\infty ,\mathcal {G}} \end{aligned}$$

Thus, there exist two constants $\alpha >0$ and $\lambda _0>0$, such that for $\frac{|| w ||_2}{\lambda }\leqslant \alpha$ and $0<\lambda <\lambda _0$, $|\tilde{\eta }_\lambda (x)|<1$ for any $x\in \mathcal {G}\setminus J$. Then, the primal-dual extremality conditions imply that for any solution $\tilde{m}_{\lambda ,w}$ of $\mathcal {P}_\lambda ^\mathcal {G}(y_0+w)$, one has ${{\mathrm{Supp}}}(\tilde{m}_{\lambda ,w}) \subset J$ and equality of the signs.

Now, if $\varPhi _J$ has full rank, we can invert the extremality condition:

$$\begin{aligned} \frac{1}{\lambda }\varPhi _J^* \left( y_0 +w - \varPhi _J( {\tilde{m}_{\lambda ,w}\vert }_{J} ) \right)&= {\tilde{\eta }_\lambda \vert }_{J},\\ \text{ so } \text{ that } \quad {\tilde{m}_{\lambda ,w}\vert }_{J} = {m\vert }_{J} + \varPhi _J^+ w-&\lambda (\varPhi _J \varPhi _J^*)^{-1} { \tilde{\eta }_\lambda \vert }_{J}. \end{aligned}$$

Observing that $|| {\tilde{\eta }_\lambda \vert }_{J} ||_{\infty ,\mathcal {G}}\leqslant 1$, we obtain the $\ell _2$-robustness result. $\square$

Theorem 3 is analogous to Lemma 1 for the continuous problem. The discrete nature of the problem makes its conclusions more precise. Although the $\ell ^2$-robustness results are similar to those of Theorem 2, the focus here is a bit more general, in the sense that this theorem does not assert that the support of the recovered measures matches the support of the input measure $m_0$. In fact, if $m_0$ is a solution to $(\tilde{\mathcal {P}}_0^\mathcal {G}(y_0))$, ${{\mathrm{Supp^{\pm }}}}(m_0)\subset {{\mathrm{Ext^\pm _\mathcal {G}}}}(m_0)$, so that the recovered solutions to $\mathcal {P}_\lambda ^\mathcal {G}(y_0+w)$ have in general more spikes than $m_0$, and the spikes in ${{\mathrm{Ext_\mathcal {G}}}}(m_0)\setminus {{\mathrm{Supp}}}(m_0)$ must vanish as $\lambda \rightarrow 0, || w ||_2 \rightarrow 0$.

In order to get the exact recovery of the signed support for small noise, we may assume in addition that ${{\mathrm{Supp^{\pm }}}}(m_0)={{\mathrm{Ext^\pm _\mathcal {G}}}}(m_0)$ so as to obtain a result analogous to Theorem 2. Precisely, we obtain the following theorem which was initially proved by Fuchs [20]. First, we introduce a pre-certificate.

Definition 8

(Fuchs pre-certificate) Let $m_0 \in \mathcal {M}(\mathbb {T})$ such that ${{\mathrm{Supp}}}(m_0)\subset \mathcal {G}$. We define the Fuchs pre-certificate as

$$\begin{aligned} \eta _\mathrm{F}= \underset{\eta =\varPhi ^* p, p \in L^2}{{{\mathrm{argmin}}}}\; || p || \quad \text {subject to}\quad {\eta \vert }_{{{\mathrm{Supp}}}m_0} = {{\mathrm{sign}}}( {m_0\vert }_{{{\mathrm{Supp}}}m_0}). \end{aligned}$$

(34)

This pre-certificate, introduced in [20], is a certificate for $m_0$ if and only if $|| \eta _\mathrm{F} ||_{\infty ,\mathcal {G}} \leqslant 1$, in which case it is equal to the discrete minimal norm pre-certificate $\eta _0^\mathcal {G}$.

If $\varPhi _{{{\mathrm{Supp}}}m_0}$ has full rank, then $\eta _\mathrm{F}$ can be computed by solving a linear system:

$$\begin{aligned} \eta _\mathrm{F}= \varPhi ^* \varPhi _{I}^{+,*} {{\mathrm{sign}}}( {m\vert }_{I}) \quad \text {where} \quad I={{\mathrm{Supp}}}m_0 \quad \text {and} \quad \varPhi _{I}^{+,*} = \varPhi _{I} (\varPhi _{I}^*\varPhi _{I})^{-1}. \end{aligned}$$

Corollary 3

(Exact support recovery, discrete case, [20]) Let $m_0 \in \mathcal {M}(\mathbb {T})$ such that ${{\mathrm{Supp}}}(m_0) \subset \mathcal {G}$, and that $\varPhi _{{{\mathrm{Supp}}}m_0}$ has full rank. If $| \eta _\mathrm{F}(t)|<1$ for all $t\in \mathcal {G}\setminus {{\mathrm{Supp}}}m_0$, then $m_0$ is identifiable for $\mathcal {G}$ and there exists $\alpha >0, \lambda _0>0$, such that for $(\lambda ,w) \in D_{\alpha ,\lambda _0}$ the solution $\tilde{m}_{\lambda ,w}$ of $\mathcal {P}_\lambda ^\mathcal {G}(y+w)$ is unique and satisfies ${{\mathrm{Supp^{\pm }}}}(\tilde{m})={{\mathrm{Supp^{\pm }}}}(m_0)$. Moreover,

$$\begin{aligned} {\tilde{m}_{\lambda ,w}\vert }_{I} = {m_0\vert }_{I} + \varPhi _{I}^+ w - \lambda (\varPhi _{I}\varPhi _{I}^*)^{-1} {{\mathrm{sign}}}( {m_0\vert }_{I}), \end{aligned}$$

(35)

where $I={{\mathrm{Supp}}}m_0$.

The condition $| \eta _\mathrm{F}(t)|<1$ for all $t\in \mathcal {G}\setminus {{\mathrm{Supp}}}m_0$ is often called the irrepresentability condition in the statistics literature, see [34]. This condition can be shown to be almost a necessary and sufficient condition to ensure exact recovery of the support of $m_0$. For instance, if $| \eta _\mathrm{F}(t)|>1$ for some $t\in \mathcal {G}\setminus {{\mathrm{Supp}}}m_0$, one can show that for all $\lambda > 0$ ${{\mathrm{Supp}}}(\tilde{m}_\lambda ) \ne {{\mathrm{Supp}}}m_0$ where $\tilde{m}_\lambda$ is any solution of $\mathcal {P}_\lambda ^\mathcal {G}(y_0)$, see [33]. In our framework, we see that this irrepresentability condition means that the pre-certificate $\eta _\mathrm{F}$ is indeed a certificate (so that it is equal to the minimal norm certificate) and that its saturation set is equal to the support of $m_0$.

For deconvolution problems, an important issue is that Corollary 3 is useless when studying the stability of the original infinite-dimensional problem $(\mathcal {P}_\lambda (y_0))$. Indeed, the pre-certificate (34) is not constrained to have vanishing derivatives, so that it generally takes some values strictly greater than 1 for a generic discrete input measure $m_0$. When the stepsize of the grid is small enough, such values are sampled and $|| \eta _\mathrm{F} ||_{\infty ,\mathcal {G}}$ necessarily becomes strictly larger than one. As detailed in Sect. 4, when shifting from the discrete grid setting to the continuous setting, the natural pre-certificate to consider is the vanishing derivative pre-certificate $\eta _\mathrm{V}$ defined in (27), and not the pre-certificate $\eta _\mathrm{F}$.

5.4 Structure of the Extended Support for Thin Grids

In the previous section, we have introduced the notion of extended signed support of a measure $m_0$ relatively to a grid $\mathcal {G}$, and we have proved that this set, ${{\mathrm{Ext^\pm _\mathcal {G}}}}{m_0}$, contains the signed supports of all the reconstructed measures for small noise. In this section, we focus on the structure of the extended support. We show that, if the support of $m_0$ belongs to the grid for a sufficiently small stepsize and if the non-degenerate source condition holds, the extended signed support consists in the signed support of $m_0$ and possibly one immediate neighbor with the same sign for each spike. Therefore, when the grid stepsize is small enough, the support of the measure is generally not stable for the discrete problem, but the support of the reconstructed measure is a close approximation of the original one.

From now on, for the sake of simplicity, we consider dyadic grids $\mathcal {G}_n= \left\{ \frac{j}{2^n} \;;\; 0\leqslant j \leqslant 2^n-1 \right\}$. The constraint sets in $\mathcal {D}_\lambda ^{\mathcal {G}_n}(y_0)$ and $(\mathcal {D}_\lambda (y_0))$ are denoted, respectively, by

$$\begin{aligned} C_n&= \left\{ p\in L^2(\mathbb {T}) \;;\; \left| (\varPhi ^*p)\left( \frac{j}{2^n}\right) \right| \leqslant 1,\ 0\leqslant j\leqslant 2^n \right\} , \end{aligned}$$

(36)

$$\begin{aligned} \text{ and } C&= \left\{ p\in L^2(\mathbb {T}) \;;\; \left\| \varPhi ^*p\right\| _{\infty }\leqslant 1 \right\} = \bigcap _{n\in \mathbb {N}} C_n. \end{aligned}$$

(37)

The structure of ${{\mathrm{Ext^\pm _\mathcal {G}}}}(m_0)$ for large $n$ is intimately related to the convergence of $p^{\mathcal {G}_n}_0$ to $p_0$. First, let us notice the following result, whose proof is given in “Appendix 3.”

Proposition 9

(Convergence for fixed $\lambda$) Let $m_0\in \mathcal {M}(\mathbb {T})$. Then, for any $\lambda >0$,

$$\begin{aligned} \lim _{n\rightarrow +\infty } p_\lambda ^{\mathcal {G}_n}&= p_\lambda \text{ for } \text{ the } L^2(\mathbb {T}) \text{(strong) } \text{ topology },\end{aligned}$$

(38)

$$\begin{aligned} \text{ and } \lim _{n\rightarrow +\infty } \eta _\lambda ^{\mathcal {G}_n}&= \eta _\lambda \text{ for } \text{ the } \text{ topology } \text{ of } \text{ the } \text{ uniform } \text{ convergence }. \end{aligned}$$

(39)

Moreover, if there exists a solution to the continuous dual problem $(\mathcal {D}_0(y_0))$,

$$\begin{aligned} \lim _{\lambda \rightarrow 0^+} \lim _{n\rightarrow +\infty } p_\lambda ^{\mathcal {G}_n}= p_0, \text{ and } \lim _{\lambda \rightarrow 0^+} \lim _{n\rightarrow +\infty } \eta _\lambda ^{\mathcal {G}_n}= \eta _0. \end{aligned}$$

(40)

Proposition 9 simply states that the projection onto convex sets $C_n$ which converge (in the sense of set convergence) to $C$ converges to the projection onto $C$. However, the case $\lambda =0$ is not as straightforward, and for instance, one cannot easily swap the limits in (40). In fact, given any decreasing sequence of polyhedra $C_n$, it is not true in general that the minimal norm solution of $\sup _{p\in C_n} \langle y_0,\,p\rangle$ should converge to the minimal norm solution of $\sup _{p\in C} \langle y_0,\,p\rangle$ where $C=\bigcap _{n\in \mathbb {N}}C_n$. As a consequence, it is not clear to us whether this convergence always holds for polyhedra of the form

$$\begin{aligned} C_n = \left\{ p\in L^2 \;;\; || \varPhi ^*p ||_{\infty ,\mathcal {G}_n}\leqslant 1 \right\} . \end{aligned}$$

However, when the spikes locations belong to the grid for $n$ large enough, the convergence of the minimal norm certificates holds. In the case of dyadic grids, this is equivalent to $m_0\in \mathcal {M}(\mathbb {T})$ being a discrete dyadic measure, i.e., such that for some $n_0\in \mathbb {N}$:

$$\begin{aligned} m = \sum _{i=1}^N a_{i} \delta _{x_{i}}, \quad \text {with} \quad x_i=\frac{j_i}{2^{n_0}} \text{ and } 0\leqslant j_i \leqslant 2^{n_0}-1. \end{aligned}$$

(41)

The proofs given below make use of a remark given in [8]: If a solution of the continuous problem $(\mathcal {P}_0(y_0))$ has support in the grid $\mathcal {G}$, then it is also a solution of the discrete problem $(\mathcal {P}_0^\mathcal {G}(y_0))$.

Proposition 10

(Convergence for dyadic measures) Let $m_0\in \mathcal {M}(\mathbb {T})$ be a discrete dyadic measure [see (41)], and assume that the (possibly degenerate) source condition holds. Then,

$$\begin{aligned} \lim _{n\rightarrow +\infty } p_{0}^{\mathcal {G}_n}&=p_0 \text{ for } \text{ the } L^2 \text{(strong) } \text{ topology },\end{aligned}$$

(42)

$$\begin{aligned} \text{ and } \lim _{n\rightarrow +\infty } \eta _{0}^{\mathcal {G}_n(i)}&=\eta _0^{(i)} \text{ for } 0\leqslant i \leqslant 2, \text{ in } \text{ the } \text{ sense } \text{ of } \text{ the } \text{ uniform } \text{ convergence, } \end{aligned}$$

(43)

where $\eta _0 = \varPhi ^* p_0$ (respectively $\eta _{0}^{\mathcal {G}_n}=\varPhi ^* p_{0}^{\mathcal {G}_n}$) denotes the corresponding minimal norm certificate.

Proof

First, following [8], we observe that, since $(\varPhi ^*p_0)(x_i)={{\mathrm{sign}}}(a_i)$ and $|| \varPhi ^*p_0 ||_{\infty } \leqslant 1$ (a fortiori $|\varPhi ^*p_0\left( \frac{j}{2^n}\right) |\leqslant 1$ for $1\leqslant j\leqslant 2^n-1$), $\varPhi ^*p_0$ is also a dual certificate for $(\mathcal {P}_0^{\mathcal {G}_n})$ provided $n\geqslant n_0$. As a consequence $|| p_{0}^{\mathcal {G}_n} ||_2\leqslant || p_0 ||_2$.

The sequence $(p_{0}^{\mathcal {G}_n})_{n\in \mathbb {N}}$ being bounded in $L^2(\mathbb {T})$, we may extract a subsequence (still denoted by $p_{0}^{\mathcal {G}_n}$) which weakly converges to some $\tilde{p}\in L^2(\mathbb {T})$, and

$$\begin{aligned} || \tilde{p} ||_2 \leqslant \liminf _{n\rightarrow +\infty } || p_{0}^{\mathcal {G}_n} ||_2 \leqslant \limsup _{n\rightarrow +\infty } || p_{0}^{\mathcal {G}_n} ||_2 \leqslant || p_0 ||_2. \end{aligned}$$

(44)

Moreover, by optimality of $p_{0}^{\mathcal {G}_n}$ for the discrete problem, for each $p\in C\subset C_n$, $\langle y_0,\,p_{0}^{\mathcal {G}_n}\rangle \geqslant \langle y_0,\,p\rangle$ so that in the limit $\langle y_0,\,\tilde{p}\rangle \geqslant \langle y_0,\,p\rangle$. Observing that $\tilde{p}\in C=\bigcap _{n\in \mathbb {N}}{C_n}$ (since each $C_n$ is weakly closed), we conclude that $\tilde{p}=p_0$. Since the limit does not depend on the extracted subsequence, we conclude that the whole sequence $(p_0^{\mathcal {G}_n})_{n\in \mathbb {N}}$ converges to $p_0$, and equality in (44) implies that the convergence is strong.

The consequence regarding $\eta _{0}^{\mathcal {G}_n}$ is straightforward. $\square$

We may now describe the structure of the extended support for dyadic measures which satisfy the non-degenerate source condition.

Proposition 11

(Extended support) Let $m_0=\sum _{i=1}^N a_i \delta _{x_i}$ be a discrete dyadic measure which satisfies the non-degenerate source condition. Then, for $n$ large enough, there exists $\varepsilon ^n \in \{+1,-1\}^N$ such that:

$$\begin{aligned} {{\mathrm{Supp^{\pm }}}}(m_0) \subset {{\mathrm{Ext^\pm }}}_{\mathcal {G}_n}(m_0) \subset {{\mathrm{Supp^{\pm }}}}(m_0) \cup \left( {{\mathrm{Supp^{\pm }}}}(m_0) + \frac{\varepsilon ^n}{2^n} \right) , \end{aligned}$$

(45)

where ${{\mathrm{Supp^{\pm }}}}(m_0) + \frac{\varepsilon ^n}{2^n}= \left\{ (x_i+\frac{\varepsilon ^n_i}{2^n},\eta _{0}^{\mathcal {G}_n}(x_i)) \;;\; 1\leqslant i\leqslant N \right\}$.

Corollary 4

Under the hypotheses of Proposition 11, for $n$ large enough, there exist two constants $\alpha (n)>0$ and $\lambda _0(n)>0$ such that, for $\frac{|| w ||_2}{\lambda }<\alpha (n)$ and $0<\lambda <\lambda _{0}(n)$, any solution $\tilde{m}_\lambda ^{\mathcal {G}_n}$ of $(\mathcal {P}_\lambda ^{\mathcal {G}_n})$ has support in $\{ x_i,\ 1\leqslant i\leqslant N\} \cup \{x_i+\frac{\varepsilon ^n_i}{2^n}, \ 1\leqslant i\leqslant N \}$, with signs $\eta _{0}^{\mathcal {G}_n}(x_i)$, $1\leqslant i\leqslant N$.

Proof of Proposition 11

We describe the points where the value of $\eta _0^{\mathcal {G}_n}$ may be $\pm 1$. By the non-degenerate source condition, there exists $\varepsilon >0$ small enough such that the intervals $(x_{0,i}-\varepsilon , x_{0,i}+\varepsilon )$, $1\leqslant i\leqslant N$, do not intersect and that for all $t\in \bigcup _{i=1}^N (x_{i}-\varepsilon , x_{i}+\varepsilon )$, $|\eta _0(t)|\geqslant C>0$ and $|\eta _0''(t)|\geqslant C>0$. Moreover, $\sup _{K_\varepsilon } |\eta _0| <1$ with $K_\varepsilon = \mathbb {T}\setminus \bigcup _{i=1}^N (x_{i}-\varepsilon , x_{i}+\varepsilon )$.

Therefore, by Proposition 10, for $n$ large enough:

• $|\eta _{0}^{\mathcal {G}_n}(t)|\geqslant \frac{C}{2}>0$ for $t\in (x_{i,0}-\varepsilon ,x_{i,0}+\varepsilon )$,

• $|(\eta _{0}^{\mathcal {G}_n})''(t)|\geqslant \frac{C}{2}>0$ for $t\in (x_{i}-\varepsilon ,x_{i}+\varepsilon )$,

• $\sup _{K_\varepsilon } |\eta _{0}^{\mathcal {G}_n}| <1$,

and in each interval $(x_{i,0}-\varepsilon ,x_{i}+\varepsilon )$, $\eta _{0}^{\mathcal {G}_n}$ has the same sign as $\eta _{0}$ and it is strictly concave (respectively strictly convex) if $\eta _0(x_i)=1$ (respectively $-1$).

Assume for instance that $\eta _0(x_i)=1$. The extremality conditions between $p_0$ and $m_0$ for $(\mathcal {P}_0(y))$ also imply that $m_0$ is a solution of $(\mathcal {P}_0^{\mathcal {G}_n}(y_0))$. Then, the extremality conditions between $p_{0}^{\mathcal {G}_n}$ and $m_0$ imply that $\eta _{0}^{\mathcal {G}_n}(x_{i})=1$ as well. By the strict concavity of $\eta _{0}^{\mathcal {G}_n}$, there is at most one other point $t^\star \in (x_{i}-\varepsilon ,x_{i}+\varepsilon )$ such that $\eta _{0}^{\mathcal {G}_n}(t^\star )=1$, and since $\eta _{0}^{\mathcal {G}_n}(x_{i}\pm \frac{1}{2^n})\leqslant 1$, $|t^\star -x_i|\leqslant \frac{1}{2^n}$. Such a point $t^\star$ contributes to the extended support of $m$ if and only if it belongs to the grid (i.e., $t^\star =x_i\pm \frac{1}{2^n}$).

The argument for $\eta _0(x_i)=-1$ is similar. This concludes the proof. $\square$

Corollary 4 highlights the difference between the continuous and the discretized problems. In the first case, any small noise would induce a slight perturbation of the spikes locations and amplitudes, but their number would stay the same. In the second case, the spikes cannot “move,” so that new spikes may appear, but only at one of the immediate neighbors of the original ones.

For non-dyadic measures, we may show using Proposition 9 that for small, fixed $\lambda >0$, and $n$ large enough, there is at most one pair of spikes (located at consecutive points of the grid) in the neighborhood of each original spike. From our numerical experiments described below (in the case of the ideal low-pass filter), we conjecture that, in the case where there are indeed two spikes, they surround the location of the original spike.

5.5 Application to the Ideal Low-pass Filter

To conclude this section, we compare the different (pre-)certificates involved in the above discussion, whether on the discrete grid or in the continuous domain. Then, we illustrate the convergence of the sets $(C_n)_{n\in \mathbb {N}}$ toward $C$.

Certificates Figure 5 illustrates the results of Sect. 5.4. The numerical values are $f_c=6$, $n=7$, and the distance between the two opposite spikes is $\frac{0.6}{f_c}$. The continuous minimal norm certificate $\eta _0$ is shown: It satisfies $|\eta _0(t)|\leqslant 1$ for all $t\in \mathbb {T}$ and $\eta _0(x_i)={{\mathrm{sign}}}m_0(\{x_i\})$ for $1\leqslant i\leqslant N$. The discrete minimal norm certificate $\eta _0^{\mathcal {G}_n}$ satisfies $|\eta _0^{\mathcal {G}_n}(t)|\leqslant 1$ for all $t$ in the grid, and $\eta (u)={{\mathrm{sign}}}m_0(\{x_i\})$ for all $u\in {{\mathrm{Ext}}}_{\mathcal {G}_n}$ in the neighborhood of $x_i$. For a dyadic measure, such points are $x_i$ and possibly one of its immediate neighbors. For non-dyadic measures, we conjecture that such points are the two immediate neighbors of $x_i$.

The Fuchs pre-certificate $\eta _\mathrm{F}$ is also shown. Some points $t$ of the grid do not satisfy $| \eta _\mathrm{F}(t)|\leqslant 1$; hence, the Fuchs pre-certificate is not a certificate and the support is not stable. This was already clear from the fact that ${{\mathrm{Supp}}}(m_0) \subsetneq {{\mathrm{Ext}}}_{\mathcal {G}_n}(m_0)$.

Fig. 5

Comparison of certificates for a dyadic (left) and a non-dyadic measure (right). The second row is a zoom of the first one near the left spike. The (continuous) minimum norm certificate $\eta _0$ (in continuous red line) is everywhere bounded by 1. The (discrete) minimum norm certificate $\eta _0^{\mathcal {G}_n}$ (in dashed blue line) is bounded by 1 at the grid points. The Fuchs pre-certificate $\eta _\mathrm{F}$ (dash-dot green line) is above 1 at some points of the grid: The Fuchs criterion is not satisfied (Color figure online)

Figure 6 focuses on the reconstructed amplitudes $\tilde{a}_i$ using $\mathcal {P}_\lambda (y_0)$ as $\lambda \rightarrow 0$. Each curve represents a path $\lambda \mapsto \tilde{a}_i$. Note that for the problem on a finite grid, such paths are piecewise affine. In the dyadic case (left part of the figure), the amplitude at $x_i$ (continuous line) and at the next point of the grid (dashed line) are shown. As $\lambda \rightarrow 0$, the spike at the neighbor vanishes and the result tends to the original identifiable measure. In the non- dyadic case (right part of the figure), the amplitude at the two immediate neighbors of $x_i$ are shown (continuous and dashed lines). Here, ${{\mathrm{Supp}}}m_0 \not \subset \mathcal {G}$ so that $m_0$ is not identifiable for the discrete problem. For each spike, the amplitudes of the two neighbors converge to some nonzero value. The limit measure as $\lambda \rightarrow 0$ is the solution of $\mathcal {P}_0(y_{0G})$.

Fig. 6

Display of the solution path (as a function of $\lambda$) for the measure displayed on Fig. 5. Left Amplitudes of the coefficients at $x_i$ (continuous line) and at the next point of the grid (dashed line) as $\lambda$ varies. Right: idem for the two immediate neighbors of $x_i$. Some other spikes (gray continuous line) appear and vanish before the last segments, as $\lambda \rightarrow 0$

Set convergence Now, we interpret the convergence of the discrete problems through the convergence of the corresponding constraint set for the dual problem. Writing $\varPhi ^* p(x)=\int p(t)\varphi (x-t) \mathrm {d}t= \langle p,\,\varphi _x\rangle _{L^2}$ with $\varphi _x : t\mapsto \varphi (x-t)$, we observe that:

$$\begin{aligned} C_n&= \left\{ p\in {{\mathrm{Im}}}\varPhi \;;\; \left| \varPhi ^*p \left( \frac{j}{2^n}\right) \right| \leqslant 1, \ 0\leqslant j \leqslant 2^n-1 \right\} \end{aligned}$$

(46)

$$\begin{aligned}&= \left\{ p\in {{\mathrm{Im}}}\varPhi \;;\; |\langle p,\,\varphi _{\frac{j}{2^n}}\rangle _{L^2}| \leqslant 1, \ 0 \leqslant j \leqslant 2^n-1 \right\} . \end{aligned}$$

(47)

As a consequence, $C_n$ is the polar set of the convex hull of $\left\{ \pm \varphi _{\frac{j}{2^n}} \;;\; 0\leqslant j \leqslant 2^n-1 \right\}$.

In the case of the Dirichlet kernel, the vector space ${{\mathrm{Im}}}\varPhi$ is the space of trigonometric polynomials with degree less than or equal to $f_c$. An orthonormal basis of ${{\mathrm{Im}}}\varPhi$ is given by: $(c_0,c_1,\ldots c_{f_c}, s_1, \ldots s_{f_c})$ where $c_0 \equiv 1$, $c_k: t\mapsto \sqrt{2}\cos (2\pi kt)$ and $s_k: t\mapsto \sqrt{2}\sin (2\pi kt)$ for $1\leqslant k\leqslant f_c$.

Moreover,

$$\begin{aligned} \varphi (x-t)&= \frac{1}{2f_c+1}\left( 1+\sum _{k=1}^{f_c} 2\cos (2\pi k (x-t))\right) \\&= \frac{1}{2f_c+1}\left( 1+2\sum _{k=1}^{f_c} \left( \cos (2\pi kx) \cos (2\pi kt) +\sin (2\pi kx)\sin (2\pi kt) \right) \right) \end{aligned}$$

so that we may write:

$$\begin{aligned} \varphi _x=\frac{1}{2f_c+1}\left( c_0+\sqrt{2}\sum _{k=1}^{f_c}\left( \cos (2\pi kx) c_k + \sin (2\pi kx)s_k \right) \right) . \end{aligned}$$

For $f_c=1$, we obtain $\varphi _x=\frac{1}{3}\left( c_0+\sqrt{2}\left( \cos (2\pi x) c_1 + \sin (2\pi x)s_1 \right) \right)$, and the vectors $\varphi _x$ lie on a circle. The convex hull of $\left\{ \pm \varphi _{\frac{j}{2^n}} \;;\; 0\leqslant j \leqslant 2^n-1 \right\}$ is thus a cylinder, and its polar set $C_n$ is displayed in Fig. 7 for $n=3$, 4, and 7.

Fig. 7

Top The convex set $C_n$ for $f_c=1$, and $n=3$, 4 or 7 (from left to right). Bottom: same convex sets, the red spheres indicate the (rescaled) vectors $\varphi _{\frac{j}{2^n}}$ (Color figure online)

Problem $(\mathcal {D}_\lambda ^{\mathcal {G}_n}(y_0+w))$ corresponds to the projection of $\frac{y_0+w}{\lambda }$ onto the polytope $C_n$. Each face of $C_n$ corresponds to a possible signed support of the solutions $\tilde{m}_{\lambda ,w}$. The large, flat faces of $C_n$ yield stability to the support of $\tilde{m}_{\lambda ,w}$ for small noise $w$, as described by Theorem 3. As $n\rightarrow +\infty$, these faces converge into a piecewise smooth manifold and the support of $\tilde{m}_{\lambda ,w}$ is allowed to vary smoothly in $\mathbb {T}$, according to Theorem 2.

6 Conclusion

In this paper, we have given a precise statement about the support recovery property of sparse spikes deconvolution with total variation regularization. This support recovery is governed by the non-degeneracy of a minimal norm certificate. This hypothesis can be checked by computing a vanishing derivative pre-certificate, which can be computed in closed form. We have shown that under this non-degeneracy hypothesis, one recovers the same number of spikes and that these spikes converge to the original ones when $\lambda$ and $|| w ||/\lambda$ are small enough. While previous stability results [1, 7, 19] hold for an arbitrary noise level and make use of any non-degenerate certificate, they are formulated in terms of local averages of the recovered measure and do not describe precisely the support. In contrast, our result which requires a specific certificate to be non-degenerate and a regime where $\lambda$ and $|| w ||/\lambda$ are small enough provides exact support stability. These settings and results are thus not comparable and provide complementary information about the performance of total variation regularization.

Developing a similar framework for the discrete $\ell ^1$ setting, we have also improved upon existing results about stability of the support by introducing the notion of extended support of a measure. Our study highlights the difference between the continuous and the discrete case: When the size of the grid is small enough, the stable recovery of the support is generally not possible in the discrete framework. Yet, in the non-degenerate case, the reconstructed support at small noise is a slight modification of the original one: Each original spike yields at most one pair of consecutive spikes which surround it.

Finally, let us note that the proposed method extends to non-stationary filtering operators and to arbitrary dimensions.

Appendices

Appendix 1: Auxiliary results

For the convenience of the reader, we give here the proofs of several auxiliary results which are needed in the discussion.

Proposition 12

(Subdifferential of the total variation) Let us endow $\mathcal {M}(\mathbb {T})$ with the weak-* topology and $C(\mathbb {T})$ with the weak topology. Then, for any $m\in \mathcal {M}(\mathbb {T})$, we have:

$$\begin{aligned} \partial || m ||_{\text {TV}} = \left\{ \eta \in C(\mathbb {T}) \;;\; || \eta ||_{\infty } \leqslant 1 \quad \text {and} \quad \int \eta \, \mathrm {d}m =|| m ||_{\text {TV}} \right\} . \end{aligned}$$

Proof

Let $A= \left\{ \eta \in C(\mathbb {T}) \;;\; \forall m\in \mathcal {M}(\mathbb {T}), \ \langle \eta ,\,m\rangle \leqslant || m ||_{\text {TV}} \right\}$. It is clear that $B_\infty (0,1)\subset A$, where $B_\infty (0,1)$ is the $L^\infty (\mathbb {T})$ closed unit ball. Conversely, we observe that $A\subset B_\infty (0,1)$ by considering the Dirac masses $(\pm \delta _t)_{t\in \mathbb {T}}$.

Let us write $J(m):= || m ||_{\text {TV}}$. The function $J:\mathcal {M}(\mathbb {T})\rightarrow \mathbb {R}\cup \{+\infty \}$ is convex, proper, lower semi-continuous (for the weak-* topology), positively homogeneous and:

$$\begin{aligned} J^*(\eta )&=\sup _{m\in \mathcal {M}(\mathbb {T})} \sup _{t>0} \left( \langle \eta ,\,t m\rangle -J(t m) \right) \\&=\sup _{t>0} t\left( \sup _{m\in \mathcal {M}(\mathbb {T})} \langle \eta ,\,m\rangle -J(m) \right) \\&=\left\{ \begin{array}{ll}0 &{}\quad \text{ if } \eta \in A, \\ +\infty &{}\quad \text{ otherwise. } \end{array}\right. \end{aligned}$$

By Proposition I.5.1 in [17], for any $\eta \in C(\mathbb {T})$:

$$\begin{aligned} \eta \in \partial J(m) \Longleftrightarrow \langle \eta ,\,m\rangle = J(m) +J^*(\eta ), \end{aligned}$$

which is equivalent to $|| \eta ||_{\infty }\leqslant 1$ and $\int \eta \mathrm {d}m = || m ||_{\text {TV}}$. $\square$

Proposition 13

There exists a solution to $(\mathcal {P}_0(y_0))$ and the strong duality holds between $(\mathcal {P}_0(y_0))$ and $(\mathcal {D}_0(y_0))$, i.e.,

$$\begin{aligned} \underset{\varPhi (m)=y_0}{\min }\; || m ||_{\text {TV}} = \underset{|| \varPhi ^* p ||_{\infty }\leqslant 1}{\sup }\; \langle y_0,\,p\rangle . \end{aligned}$$

(48)

Moreover, if a solution $p^\star$ to $(\mathcal {D}_0(y_0))$ exists,

$$\begin{aligned} \varPhi ^* p^\star \in \partial {|| m^\star ||_{\text {TV}}} \end{aligned}$$

(49)

where $m^\star$ is any solution to $(\mathcal {P}_0(y_0))$. Conversely, if (49) holds, then $m^\star$ and $p^\star$ are solutions of, respectively, $(\mathcal {P}_0(y_0))$ and $(\mathcal {D}_0(y_0))$.

Proof

We apply [17, Theorem II.4.1] to $(\mathcal {D}_0(y_0))$ (and not to $(\mathcal {P}_0(y_0))$ as would be natural) rewritten as

$$\begin{aligned} \inf _{|| \varPhi ^* p ||_{\infty } \leqslant 1} \langle -y_0,\,p\rangle , \end{aligned}$$

The infimum is finite since for any admissible $p$, $\langle -y_0,\,p\rangle =\langle m_0,\,\varPhi p\rangle \geqslant - || m_0 ||_{\text {TV}}$. Let $V=L^2(\mathbb {T})$, $Y=C(\mathbb {T})$ (endowed with the strong topology), $Y^*=\mathcal {M}(\mathbb {T})$, $F(u)=\langle -y_0,\,u\rangle$ for $u\in V$, $G(\psi )=\iota _{|| \cdot ||_{\infty }\leqslant 1}(\psi )$ for $\psi \in Y$ and $\Lambda =\varPhi ^*$. It is clear that $F$ and $G$ are proper convex lower semi-continuous functions. Eventually, $F$ is finite at 0, G is finite and continuous at $0=\Lambda 0$. Hence the result. $\square$

Appendix 2: Proof of Proposition 6

Assume that for some $(u,v)\in \mathbb {R}^N\times \mathbb {R}^N$, $\varGamma _x (u,v)=0$. Then,

$$\begin{aligned} \forall \,t\in \mathbb {T}, \quad 0&=\sum _{j=1}^N \left( u_j\varphi (t-x_j)+v_j \varphi '(t-x_j)\right) \\&=\sum _{k=-f_c}^{f_c} \left( \sum _{j=1}^N (u_j + 2ik\pi v_j)e^{-2ik\pi x_j} \right) e^{2ik\pi t} \end{aligned}$$

We deduce that

$$\begin{aligned} \forall \,k\in \{-f_c,\ldots f_c \}, \quad \sum _{j=1}^N (u_j + k \tilde{v}_j)r_j^k =0 \quad \text {where} \quad \left\{ \begin{array}{l} r_j=e^{-2i\pi x_j},\\ \tilde{v}_j=2i\pi v_j. \end{array} \right. \end{aligned}$$

It is therefore sufficient to prove that the columns of the following matrix are linearly independent

$$\begin{aligned} \begin{pmatrix} r_1^{-f_c} &{} \ldots &{}r_N^{-f_c} &{} (-f_c)r_1^{-f_c}&{} \ldots &{}(-f_c)r_N^{-f_c} \\ \vdots &{} &{}\vdots &{} \vdots &{} &{}\vdots \\ r_1^{k} &{} \ldots &{}r_N^{k} &{} kr_1^{k}&{} \ldots &{}k r_N^{k} \\ \vdots &{} &{}\vdots &{} \vdots &{} &{}\vdots \\ r_1^{f_c} &{} \ldots &{}r_N^{f_c} &{} (f_c)r_1^{f_c}&{} \ldots &{}(f_c)r_N^{f_c} \\ \end{pmatrix}. \end{aligned}$$

If $N<f_c$, we complete the family $\{r_1, \ldots r_N\}$ in a family $\{r_0,r_1,\ldots r_{f_c}\}\subset \mathbb {S}^1$ such that the $r_i$’s are pairwise distinct. We obtain a square matrix $M$ by inserting the corresponding columns

$$\begin{aligned} M = \begin{pmatrix} r_1^{-f_c} &{} \ldots &{}r_{f_c}^{-f_c} &{} r_0^{-f_c} &{}(-f_c)r_1^{-f_c}&{} \ldots &{}(-f_c)r_{f_c}^{-f_c} \\ \vdots &{} &{}\vdots &{}\vdots &{} \vdots &{} &{}\vdots \\ r_1^{k} &{} \ldots &{}r_{f_c}^{k} &{} r_0^k &{}kr_1^{k}&{} \ldots &{}k r_{f_c}^{k} \\ \vdots &{} &{}\vdots &{}\vdots &{} \vdots &{} &{}\vdots \\ r_1^{f_c} &{} \ldots &{}r_{f_c}^{f_c} &{} r_0^{f_c}&{}(f_c)r_1^{f_c}&{} \ldots &{}(f_c)r_{f_c}^{f_c} \\ \end{pmatrix}. \end{aligned}$$

We claim that $M$ is invertible. Indeed, if there exists $\alpha \in \mathbb {C}^{(2f_c+1)}$ such that $M^T \alpha =0$, then the rational function $F(z) = \sum _{k=-f_c}^{f_c} \alpha _k z^{k}$ satisfies:

$$\begin{aligned} F(r_j)&=0 \quad \text {and} \quad F'(r_j)=0 \ \text{ for } 1\leqslant j \leqslant f_c,\\ F(r_0)&=0. \end{aligned}$$

Hence, $F$ has at least $2f_c+1$ roots in $\mathbb {S}^1$, counting the multiplicities. This imposes that $F=0$, thus $\alpha =0$, and $M$ is invertible. The result is proved.

Appendix 3: Proof of Proposition 9

Let us denote by $P_{C_n}(x)$ the projection of $x\in L^2(\mathbb {T})$ onto $C_n$. We have:

$$\begin{aligned} \left\| P_{C_n}\left( \frac{y_0}{\lambda }\right) -P_{C_n}(0)\right\| _2 \leqslant \left\| \frac{y_0}{\lambda } - 0\right\| _2, \end{aligned}$$

so that the sequence $p_{\lambda }^{\mathcal {G}_n}=P_{C_n}(\frac{y_0}{\lambda })$ is bounded in $L^2(\mathbb {T})$, and we may extract a subsequence $p_{\lambda }^{\mathcal {G}_n'}$ which weakly converges to some $p_\lambda ^\star \in L^2(\mathbb {T})$. Since $C_{n'}$ is (weakly) closed for all $n'$, $p_\lambda ^\star \in \bigcap _{n'} C_{n'}=C$.

Moreover, by the characterization of the projection onto convex sets:

$$\begin{aligned} \forall z\in C\subset C_n', \ \left\langle \frac{y_0}{\lambda } - p_{\lambda }^{\mathcal {G}_n'},z\right\rangle - \left\langle \frac{y_0}{\lambda },p_{\lambda }^{\mathcal {G}_n'}\right\rangle +|| p_{\lambda }^{\mathcal {G}_n'} ||_2^2&\leqslant 0.\\ \text{ Passing } \text{ to } \text{ the } \text{ limit } n'\rightarrow +\infty , \ \left\langle \frac{y_0}{\lambda } - p_{\lambda }^\star ,z\right\rangle - \left\langle \frac{y_0}{\lambda },p_{\lambda }^\star \right\rangle + \liminf _{n'} || p_{\lambda }^{\mathcal {G}_n'} ||_2^2&\leqslant 0,\\ \left\langle \frac{y_0}{\lambda } - p_{\lambda }^\star ,z\right\rangle - \left\langle \frac{y_0}{\lambda },p_{\lambda }^\star \right\rangle + || p_{\lambda }^\star ||_2^2&\leqslant 0,\\ \left\langle \frac{y_0}{\lambda } - p_\lambda ^\star ,z - p_\lambda ^\star \right\rangle&\leqslant 0.\\ \end{aligned}$$

Thus, $p_\lambda ^\star$ is the orthogonal projection of $\frac{y_0}{\lambda }$ on $C$: $p_\lambda ^\star =P_{C}\left( \frac{y_0}{\lambda }\right) =p_\lambda$. Since this is true for any subsequence, the whole sequence $p_{\lambda }^{\mathcal {G}_n}$ weakly converges to $p_\lambda$.

Moreover, by lower semi-continuity and the inclusion $C\subset C_n$ we have:

$$\begin{aligned} \left\| \frac{y_0}{\lambda }-p_\lambda \right\| _2&\leqslant \liminf _{n\rightarrow +\infty } \left\| \frac{y_0}{\lambda }-p_{\lambda }^{\mathcal {G}_n}\right\| _2\leqslant \limsup _{n\rightarrow +\infty }\left\| \frac{y_0}{\lambda }-p_{\lambda }^{\mathcal {G}_n}\right\| _2 \leqslant \left\| \frac{y_0}{\lambda }-p_{\lambda }\right\| _2, \end{aligned}$$

so that $\frac{y_0}{\lambda }-p_{\lambda }^{\mathcal {G}_n}$ converges strongly to $\frac{y_0}{\lambda }-p_{\lambda }$, hence the strong convergence of $p_{\lambda }^{\mathcal {G}_n}$ to $p_\lambda$.

The rest of the statement follows from Proposition 1.