Stable Signal Recovery from Phaseless Measurements

Abstract

The aim of this paper is to study the stability of the $\ell _1$ minimization for the compressive phase retrieval and to extend the instance-optimality in compressed sensing to the real phase retrieval setting. We first show that $m={\mathcal {O}}(k\log (N/k))$ measurements are enough to guarantee the $\ell _1$ minimization to recover k-sparse signals stably provided the measurement matrix A satisfies the strong RIP property. We second investigate the phaseless instance-optimality presenting a null space property of the measurement matrix A under which there exists a decoder $\Delta$ so that the phaseless instance-optimality holds. We use the result to study the phaseless instance-optimality for the $\ell _1$ norm. This builds a parallel for compressive phase retrieval with the classical compressive sensing.

1 Introduction

In this paper we consider the phase retrieval for sparse signals with noisy measurements, which arises in many different applications. Assume that

$$\begin{aligned} b_j:=|\left<a_j,x_0\right>|+e_j, \quad j=1,\ldots ,m \end{aligned}$$

where $x_0\in {\mathbb R}^N$, $a_j\in {\mathbb R}^N$ and $e_j\in {\mathbb R}$ is the noise. Our goal is to recover $x_0$ up to a unimodular scaling constant from $b:=(b_1,\ldots ,b_m)^\top$ with the assumption of $x_0$ being approximately k-sparse. This problem is referred to as the compressive phase retrieval problem [9].

The paper attempts to address two problems. Firstly we consider the stability of $\ell _1$ minimization for the compressive phase retrieval problem where the signal $x_0$ is approximately k-sparse, which is the $\ell _1$ minimization problem defined as follows:

$$\begin{aligned} \min \Vert x\Vert _1\quad \text{ subject } \text{ to } \quad \bigl \Vert |Ax|-|Ax_0|\bigr \Vert _2\le \epsilon , \end{aligned}$$

(1.1)

where $A:=[a_1,\ldots ,a_m]^\top$ and $|Ax_0|:=[|\left<a_1,x_0\right>|,\ldots ,|\left<a_m,x_0\right>|]^\top$. Secondly we investigate instance-optimality in the phase retrieval setting.

Note that in the classical compressive sensing setting the stable recovery of a k-sparse signal $x_0\in {\mathbb C}^N$ can be done using $m={\mathcal O}(k\log (N/k))$ measurements for several classes of measurement matrices A. A natural question is whether stable compressive phase retrieval can also be attained with $m={\mathcal O}(k\log (N/k))$ measurements. This has indeed proved to be the case in [6] if $x_0\in {\mathbb R}^N$ and A is a random real Gaussian matrix. In [8] a two-stage algorithm for compressive phase retrieval is proposed, which allows for very fast recovery of a sparse signal if the matrix A can be written as a product of a random matrix and another matrix (such as a random matrix) that allows for efficient phase retrieval. The authors proved that stable compressive phase retrieval can be achieved with $m={\mathcal O}(k\log (N/k))$ measurements for complex signals $x_0$ as well. In [10], the strong RIP (S-RIP) property is introduced and the authors show that one can use the $\ell _1$ minimization to recover sparse signals up to a global sign from the noiseless measurements $|Ax_0|$ provided A satisfies S-RIP. Naturally, one is interested in the performance of $\ell _1$ minimization for the compressive phase retrieval with noisy measurements. In this paper, we shall show that the $\ell _1$ minimization scheme given in (1.1) will recover a k-sparse signal stably from $m={\mathcal O}(k\log (N/k))$ measurements, provided that the measurement matrix A satisfies the strong RIP (S-RIP) property. This establishes an important parallel for compressive phase retrieval with the classical compressive sensing. Note that in [11] such a parallel in terms of the null space property was already established.

The notion of instance optimality was first introduced in [5]. We use $\Vert x\Vert _0$ to denote the number of non-zero elements in x . Given a norm $\Vert \cdot \Vert _X$ such as the $\ell _1$-norm and $x\in {\mathbb R}^N$, the best k-term approximation error is defined as

$$\begin{aligned} \sigma _k(x)_X\,\,:=\,\, \min _{z\in \Sigma _k}\Vert x-z\Vert _X, \end{aligned}$$

where

$$\begin{aligned} \Sigma _k:=\{x\in {\mathbb R}^N: \Vert x\Vert _0\le k\}. \end{aligned}$$

We use $\Delta : {\mathbb R}^m\mapsto {\mathbb R}^N$ to denote a decoder for reconstructing x. We say the pair $(A,\Delta )$ is instance optimal of order k with constant $C_0$ if

$$\begin{aligned} \Vert x-\Delta (Ax)\Vert _X\le C_0\sigma _k(x)_X \end{aligned}$$

(1.2)

holds for all $x\in {\mathbb R}^N$. In extending it to phase retrieval, our decoder will have the input $b=|Ax|$. A pair $(A,\Delta )$ is said to be phaseless instance optimal of order k with constant $C_0$ if

$$\begin{aligned} \min \Bigl \{\Vert x-\Delta (|Ax|)\Vert _X, \Vert x+\Delta (|Ax|)\Vert _X\Bigr \}\le C_0\sigma _k(x)_X \end{aligned}$$

(1.3)

holds for all $x\in {\mathbb R}^N$. We are interested in the following problem : Given $\Vert \cdot \Vert _X$ and $k<N$, what is the minimal value of m for which there exists $(A,\Delta )$ so that (1.3) holds?

The null space $\mathcal {N}(A):=\{x\in {\mathbb R}^N:Ax=0\}$ of A plays an important role in the analysis of the original instance optimality (1.2) (see [5]). Here we present a null space property for $\mathcal {N}(A)$, which is necessary and sufficient, for which there exists a decoder $\Delta$ so that (1.3) holds. We apply the result to investigate the instance optimality where X is the $\ell _1$ norm. Set

$$\begin{aligned} \Delta _1(|Ax|):\,= \,\mathop {\mathrm{argmin}}\limits _{z\in {\mathbb R}^N}\Bigl \{\Vert z\Vert _1: |Ax|=|Az|\Bigr \}. \end{aligned}$$

We show that the pair $(A,\Delta _1)$ satisfies (1.3) with X being the $\ell _1$-norm provided A satisfies the strong RIP property (see Definition 2.1). As shown in [10], the Gaussian random matrix $A\in {\mathbb R}^{m\times N}$ satisfies the strong RIP of order k for $m={\mathcal O}(k\log (N/k)$. Hence $m={\mathcal O}(k\log (N/k))$ measurements suffice to ensure the phaseless instance optimality (1.3) for the $\ell _1$-norm exactly as with the traditional instance optimality (1.2).

2 Auxiliary Results

In this section we provide some auxiliary results that will be used in later sections. For $x\in {\mathbb R}^N$ we use $\Vert x\Vert _p:=\Vert x\Vert _{\ell _p}$ to denote the p-norm of x for $0<p \le \infty$. The measurement matrix is given by $A:=[a_1,\ldots ,a_m]^T \in \mathbb {R}^{m\times N}$ as before. Given an index set $I\subset \{1,\ldots ,m\}$ we shall use $A_I$ to denote the sub-matrix of A where only rows with indices in I are kept, i.e.,

$$\begin{aligned} A_I:=[a_j:j\in I]^\top . \end{aligned}$$

The matrix A satisfies the Restricted Isometry Property (RIP) of order k if there exists a constant $\delta _k\in [0,1)$ such that for all k-sparse vectors $z\in \Sigma _k$ we have

$$\begin{aligned} (1-\delta _k)\Vert z\Vert _2^2\le \Vert Az\Vert _2^2\le (1+\delta _k)\Vert z\Vert _2^2. \end{aligned}$$

It was shown in [2] that one can use $\ell _1$-minimization to recover k-sparse signals provided that A satisfies the RIP of order t k and $\delta _{t k}<\sqrt{1-\frac{1}{t}}$ where $t>1$.

To investigate compressive phase retrieval, a stronger notion of RIP is given in [10]:

Definition 2.1

(S-RIP) We say the matrix $A=[a_1,\ldots ,a_m]^\top \in \mathbb {R}^{m\times N}$ has the Strong Restricted Isometry Property of order k with bounds $\theta _-,\ \theta _+\in (0, 2)$ if

$$\begin{aligned} \theta _-\Vert x\Vert _2^2\le \min _{ I\subseteq [m], |I|\ge m/2}\Vert A_Ix\Vert _2^2\le \max _{I\subseteq [m],|I|\ge m/2} \Vert A_Ix\Vert _2^2\le \theta _+\Vert x\Vert _2^2 \end{aligned}$$

(2.1)

holds for all k-sparse signals $x\in \mathbb {R}^N$, where $[m]:=\{1,\ldots ,m\}$. We say A has the Strong Lower Restricted Isometry Property of order k with bound $\theta _-$ if the lower bound in (2.1) holds. Similarly we say A has the Strong Upper Restricted Isometry Property of order k with bound $\theta _+$ if the upper bound in (2.1) holds.

The authors of [10] proved that Gaussian matrices with $m=\mathcal {O}(tk\log (N/k))$ satisfy S-RIP of order tk with high probability.

Theorem 2.1

([10]) Suppose that $t>1$ and $A=(a_{ij})\in \mathbb {R}^{m\times N}$ is a random Gaussian matrix with $m=\mathcal {O}(tk\log (N/k))$ and $a_{ij}\sim {\mathcal N}(0,\frac{1}{\sqrt{m}})$. Then there exist $\theta _-, \theta _+ \in (0,2)$ such that with probability $1-\exp (-cm/2)$ the matrix A satisfies the S-RIP of order tk with constants $\theta _-$ and $\theta _+$, where $c>0$ is an absolute constant and $\theta _-$, $\theta _+$ are independent of t.

The following is a very useful lemma for this study.

Lemma 2.1

Let $x_0\in \mathbb {R}^N$ and $\rho \ge 0$. Suppose that $A\in \mathbb {R}^{m\times N}$ is a measurement matrix satisfying the restricted isometry property with $\delta _{tk}\le \sqrt{\frac{t-1}{t}}$ for some $t>1$. Then for any

$$\begin{aligned} \hat{x}\in \Bigl \{x\in {\mathbb R}^N : \Vert x\Vert _1\le \Vert x_0\Vert _1+\rho , \, \Vert Ax-Ax_0\Vert _2\le \epsilon \Bigr \} \end{aligned}$$

we have

$$\begin{aligned} \Vert \hat{x}-x_0\Vert _2\le c_1\epsilon +c_2\frac{2\sigma _k(x_0)_1}{\sqrt{k}}+c_2\cdot \frac{\rho }{\sqrt{k}}, \end{aligned}$$

where $c_1=\frac{\sqrt{2(1+\delta )}}{1-\sqrt{t/(t-1)}\delta }$, $c_2=\frac{\sqrt{2}\delta +\sqrt{(\sqrt{t(t-1)}-\delta t)\delta }}{\sqrt{t(t-1)}-\delta t}+1.$

Remark 2.1

We build the proof of Lemma 2.1 following the ideas of Cai and Zhang [2]. The full proof is given in Appendix for completeness. It is well-known that an effective method to recover approximately-sparse signals $x_0$ in the traditional compressive sensing is to solve

$$\begin{aligned} x^\#:=\mathop {\mathrm{argmin}}\limits _{x}\{\Vert x\Vert _1 : \Vert Ax-Ax_0\Vert _2\le \epsilon \}. \end{aligned}$$

(2.2)

The definition of $x^\#$ shows that

$$\begin{aligned} \Vert x^\#\Vert _1\le \Vert x_0\Vert _1, \quad \Vert Ax^\#-Ax_0\Vert _2\le \epsilon , \end{aligned}$$

which implies that

$$\begin{aligned} \Vert x^\#-x_0\Vert _2\le C_1\epsilon +C_2\frac{\sigma _k(x_0)_1}{\sqrt{k}}, \end{aligned}$$

provided that A satisfies the RIP condition with $\delta _{tk}\le \sqrt{1-1/t}$ for $t>1$ (see [2]). However, in practice one prefers to design fast algorithms to find an approximation solution of (2.2), say $\hat{x}$. Thus it is possible to have $\Vert \hat{x}\Vert _1> \Vert x_0\Vert _1$. Lemma 2.1 gives an estimate of $\Vert \hat{x}-x_0\Vert _2$ for the case where $\Vert \hat{x}\Vert _1\le \Vert x_0\Vert _1+\rho$.

Remark 2.2

In [7], Han and Xu extend the definition of S-RIP by replacing the m / 2 in (2.1) by $\beta m$ where $0<\beta <1$. They also prove that, for any fixed $\beta \in (0,1)$, the $m\times N$ random Gaussian matrix satisfies S-RIP of order k with high probability provided $m=\mathcal {O}(k\log (N/k))$.

3 Stable Recovery of Real Phase Retrieval Problem

3.1 Stability Results

The following lemma shows that the map $\phi _A(x):=|Ax|$ is stable on $\Sigma _k$ modulo a unimodular constant provided A satisfies strong lower RIP of order 2k. Define the equivalent relation $\sim$ on ${\mathbb R}^N$ and ${\mathbb C}^N$ by the following: for any $x,y, x \sim y$ iff $x= cy$ for some unimodular scalar c, where x, y are in ${\mathbb R}^N$ or ${\mathbb C}^N$. For any subset Y of ${\mathbb R}^N$ or ${\mathbb C}^N$ the notation $Y/\sim$ denotes the equivalent classes of elements in Y under the equivalence. Note that there is a natural metric $D_\sim$ on ${\mathbb C}^N/\sim$ given by

$$\begin{aligned} D_\sim (x, y) = \min _{|c|=1} \Vert x-cy\Vert . \end{aligned}$$

Our primary focus in this paper will be on ${\mathbb R}^N$, and in this case $D_\sim (x,y) = \min \{\Vert x-y\Vert _2, \Vert x+y\Vert _2\}$.

Lemma 3.1

Let $A\in {\mathbb R}^{m\times N}$ satisfy the strong lower RIP of order 2k with constant $\theta _-$. Then for any $x, y \in \Sigma _k$ we have

$$\begin{aligned} \Vert |Ax|-|Ay|\Vert _2^2 \ge \theta _- \min (\Vert x-y\Vert _2^2, \Vert x+y\Vert _2^2). \end{aligned}$$

Proof

For any $x,y\in \Sigma _k$ we divide $\{1,\ldots ,m\}$ into two subsets:

$$\begin{aligned} T=\{j:~\mathrm{sign}(\langle {a_j,x} \rangle )=\mathrm{sign}(\langle {a_j,y} \rangle )\} \end{aligned}$$

and

$$\begin{aligned} T^c=\{j:~\mathrm{sign}(\langle {a_j,x} \rangle )=-\mathrm{sign}(\langle {a_j,y} \rangle )\}. \end{aligned}$$

Clearly one of T and $T^c$ will have cardinality at least m / 2. Without loss of generality we assume that T has cardinality no less than m / 2. Then

$$\begin{aligned} \Vert |Ax|-|Ay|\Vert ^2_2= & {} \Vert A_Tx-A_T y\Vert _2^2 + \Vert A_{T^c}x+A_{T^c}y\Vert _2^2\\\ge & {} \Vert A_Tx-A_T y\Vert _2^2 \\\ge & {} \theta _- \Vert x-y\Vert _2^2 \\\ge & {} \theta _- \min (\Vert x-y\Vert _2^2, \Vert x+y\Vert _2^2). \end{aligned}$$

$\square$

Remark 3.1

Note that the combination of Lemma 3.1 and Theorem 2.1 shows that for an $m\times N$ Gaussian matrix A with $m=O(k\log (N/k))$ one can guarantee the stability of the map $\phi _A(x):=|Ax|$ on $\Sigma _k/\sim$.

3.2 The Main Theorem

In this part, we will consider how many measurements are needed for the stable sparse phase retrieval by $\ell _1$-minimization via solving the following model:

$$\begin{aligned} \min \Vert x\Vert _1\,\,\, \text{ subject } \text{ to } \,\,\, \Vert |Ax|-|Ax_0|\Vert _2^2\le \epsilon ^2, \end{aligned}$$

(3.1)

where A is our measurement matrix and $x_0\in \mathbb {R}^N$ is a signal we wish to recover. The next theorem tells under what conditions the solution to (3.1) is stable.

Theorem 3.1

Assume that $A\in \mathbb {R}^{m\times N}$ satisfies the S-RIP of order tk with bounds $\theta _-, \theta _+ \in (0,2)$ such that

$$\begin{aligned} t\ge \max \Bigl \{\frac{1}{2\theta _--\theta _-^2},\frac{1}{2\theta _+-\theta _+^2}\Bigr \}. \end{aligned}$$

Then any solution $\hat{x}$ for (3.1) satisfies

$$\begin{aligned} \min \{\Vert \hat{x}-x_0\Vert _2,\Vert \hat{x}+x_0\Vert _2\}\le c_1\epsilon +c_2\frac{2\sigma _k(x_0)_1}{\sqrt{k}}, \end{aligned}$$

where $c_1$ and $c_2$ are constants defined in Lemma 2.1.

Proof

Clearly any $\hat{x}\in {\mathbb R}^N$ satisfying (3.1) must have

$$\begin{aligned} \Vert \hat{x}\Vert _1\le \Vert x_0\Vert _1 \end{aligned}$$

(3.2)

and

$$\begin{aligned} \Vert |A\hat{x}|-|Ax_0|\Vert _2^2\le \epsilon ^2. \end{aligned}$$

(3.3)

Now the index set $\{1, 2, \dots , m\}$ is divisible into two subsets

$$\begin{aligned} T&=\{j:~ \mathrm{sign}(\langle {a_j,\hat{x}} \rangle )=\mathrm{sign}(\langle {a_j,x_0} \rangle )\}, \\ T^c&=\{j:~ \mathrm{sign}(\langle {a_j,\hat{x}} \rangle )=-\mathrm{sign}(\langle {a_j,x_0} \rangle )\}. \end{aligned}$$

Then (3.3) implies that

$$\begin{aligned} \Vert A_T\hat{x}-A_Tx_0\Vert _2^2+\Vert A_{T^c}\hat{x}+A_{T^c}x_0\Vert _2^2\le \epsilon ^2. \end{aligned}$$

(3.4)

Here either $|T|\ge m/2$ or $|T^c|\ge m/2$. Without loss of generality we assume that $|T|\ge m/2$. We use the fact

$$\begin{aligned} \Vert A_T\hat{x}-A_Tx_0\Vert _2^2\le \epsilon ^2. \end{aligned}$$

(3.5)

From (3.2) and (3.5) we obtain

$$\begin{aligned} \hat{x}\in \left\{ x\in {\mathbb R}^N: \Vert x\Vert _1\le \Vert x_0\Vert _1, \Vert A_Tx-A_Tx_0\Vert _2\le \epsilon \right\} . \end{aligned}$$

(3.6)

Recall that A satisfies S-RIP of order tk and constants $\theta _-, \ \theta _+$. Here

$$\begin{aligned} t\ge \max \{\frac{1}{2\theta _--\theta _-^2},\frac{1}{2\theta _+-\theta _+^2}\}>1. \end{aligned}$$

(3.7)

The definition of S-RIP implies that $A_T$ satisfies the RIP of order tk in which

$$\begin{aligned} \delta _{tk} \le \max \{1-\theta _-,\ \theta _+-1\}\le \sqrt{\frac{t-1}{t}} \end{aligned}$$

(3.8)

where the second inequality follows from (3.7). The combination of (3.6), (3.8) and Lemma 2.1 now implies

$$\begin{aligned} \Vert \hat{x}-x_0\Vert _2\le c_1\epsilon +c_2\frac{2\sigma _k(x_0)_1}{\sqrt{k}}, \end{aligned}$$

where $c_1$ and $c_2$ are defined in Lemma 2.1. If $|T^c|\ge \frac{m}{2}$ we get the corresponding result

$$\begin{aligned} \Vert \hat{x}+x_0\Vert _2\le c_1\epsilon +c_2\frac{2\sigma _k(x_0)_1}{\sqrt{k}}. \end{aligned}$$

The theorem is now proved. $\square$

This theorem demonstrates that, if the measurement matrix has the S-RIP, the real compressive phase retrieval problem can be solved stably by $\ell _1$-minimization.

4 Phase Retrieval and Best k-term Approximation

4.1 Instance Optimality from the Linear Measurements

We introduce some definitions and results in [5]. Recall that for a given encoder matrix $A\in {\mathbb R}^{m\times N}$ and a decoder $\Delta :{\mathbb R}^m\mapsto {\mathbb R}^N$, the pair $(A,\Delta )$ is said to have instance optimality of order k with constant $C_0$ with respect to the norm X if

$$\begin{aligned} \Vert x-\Delta (Ax)\Vert _X\le C_0\sigma _k(x)_X \end{aligned}$$

(4.1)

holds for all $x\in {\mathbb R}^N$. Set ${\mathcal N}(A):=\{\eta \in {\mathbb R}^N: A\eta =0\}$ to be the null space of A. The following theorem gives conditions under which the (4.1) holds.

Theorem 4.1

([5]) Let $A\in {\mathbb R}^{m\times N}$, $1 \le k \le N$ and $\Vert \cdot \Vert _X$ be a norm on ${\mathbb R}^N$. Then a sufficient condition for the existence of a decoder $\Delta$ satisfying (4.1) is

$$\begin{aligned} \Vert \eta \Vert _X\le \frac{C_0}{2}\sigma _{2k}(\eta )_X, \quad \forall \eta \in {\mathcal N}(A). \end{aligned}$$

(4.2)

A necessary condition for the existence of a decoder $\Delta$ satisfying (4.1) is

$$\begin{aligned} \Vert \eta \Vert _X\le C_0\sigma _{2k}(\eta )_X, \quad \forall \eta \in {\mathcal N}(A). \end{aligned}$$

(4.3)

For the norm $X=\ell _1$ it was established in [5] that instance optimality of order k can indeed be achieved, e.g. for a Gaussian matrix A, with $m=O(k\log (N/k))$. The authors also considered more generally taking different norms on both sides of (4.1). Following [5], we say the pair $(A,\Delta )$ has (p, q)-instance optimality of order k with constant $C_0$ if

$$\begin{aligned} \Vert x-\Delta (Ax)\Vert _{p}\le C_0 k^{\frac{1}{q}-\frac{1}{p}}\sigma _k(x)_{q}, \quad \forall x\in {\mathbb R}^N, \end{aligned}$$

(4.4)

with $1\le q\le p \le 2$. It was shown in [5] that the (p, q)-instance optimality of order k can be achieved at the cost of having $m=\mathcal {O}(k(N/k)^{2-2/q})\log (N/k)$ measurements.

4.2 Phaseless Instance Optimality

A natural question here is whether an analogous result to Theorem 4.1 exists for phaseless instance optimality defined in (1.3). We answer the question by presenting such a result in the case of real phase retrieval.

Recall that a pair $(A,\Delta )$ is said to be have the phaseless instance optimality of order k with constant $C_0$ for the norm $\Vert .\Vert _X$ if

$$\begin{aligned} \min \Bigl \{\Vert x-\Delta (|Ax|)\Vert _X, \Vert x+\Delta (|Ax|)\Vert _X\Bigr \}\le C_0\sigma _k(x)_X \end{aligned}$$

(4.5)

holds for all $x\in {\mathbb R}^N$.

Theorem 4.2

Let $A\in {\mathbb R}^{m\times N}$, $1 \le k \le N$ and $\Vert \cdot \Vert _X$ be a norm. Then a sufficient condition for the existence of a decoder $\Delta$ satisfying the phaseless instance optimality (4.5) is: For any $I\subseteq \{1,\ldots ,m\}$ and $\eta _1\in \mathcal {N}(A_I)$, $\eta _2\in \mathcal {N}(A_{I^c})$ we have

$$\begin{aligned} \min \{\Vert \eta _1\Vert _X , \Vert \eta _2\Vert _X \}\le \frac{C_0}{4}\sigma _k(\eta _1-\eta _2)_X+\frac{C_0}{4}\sigma _k(\eta _1+\eta _2)_X. \end{aligned}$$

(4.6)

A necessary condition for the existence of a decoder $\Delta$ satisfying (4.5) is: For any $I\subseteq \{1,\ldots ,m\}$ and $\eta _1\in \mathcal {N}(A_I)$, $\eta _2\in \mathcal {N}(A_{I^c})$ we have

$$\begin{aligned} \min \{\Vert \eta _1\Vert _X , \Vert \eta _2\Vert _X \}\le \frac{C_0}{2}\sigma _k(\eta _1-\eta _2)_X+\frac{C_0}{2}\sigma _k(\eta _1+\eta _2)_X. \end{aligned}$$

(4.7)

Proof

We first assume (4.6) holds, and show that there exists a decoder $\Delta$ satisfying the phaseless instance optimality (4.5). To this end, we define a decoder $\Delta$ as follows:

$$\begin{aligned} \Delta (|Ax_0|)=\mathop {\hbox {argmin}}_{|Ax|=|Ax_0|}\sigma _k(x)_X. \end{aligned}$$

Suppose $\hat{x}:=\Delta (|Ax_0|)$. We have $|A\hat{x}|=|Ax_0|$ and $\sigma _k(\hat{x})_X\le \sigma _k(x_0)_X$. Note that $\langle {a_j,\hat{x}} \rangle =\pm \langle {a_j,x_0} \rangle$. Let $I \subseteq \{1,\ldots ,m\}$ be defined by

$$\begin{aligned} I=\Bigl \{j:~\langle {a_j,\hat{x}} \rangle = \langle {a_j,x_0} \rangle \Bigr \}. \end{aligned}$$

Then

$$\begin{aligned} A_I(x_0-\hat{x})=0,\quad A_{I^c}(x_0+\hat{x})=0. \end{aligned}$$

Set

$$\begin{aligned} \eta _1&:=x_0-\hat{x}\in \mathcal {N}(A_I),\\ \eta _2&:=x_0+\hat{x}\in \mathcal {N}(A_{I^c}). \end{aligned}$$

A simple observation yields

$$\begin{aligned} \sigma _k(\eta _1-\eta _2)_X=2\sigma _k(\hat{x})_X\le 2\sigma _k(x_0)_X , \quad \sigma _k(\eta _1+\eta _2)_X=2\sigma _k(x_0)_X. \end{aligned}$$

(4.8)

Then (4.6) implies that

$$\begin{aligned} \min \{ \Vert \hat{x}-x_0\Vert _X, \Vert \hat{x}+x_0\Vert _X \}&= \min \{\Vert \eta _1\Vert _X , \Vert \eta _2\Vert _X \} \\&\le \frac{C_0}{4}\sigma _k(\eta _1-\eta _2)_X+\frac{C_0}{4}\sigma _k(\eta _1+\eta _2)_X \\&\le C_0\sigma _k(x_0)_X. \end{aligned}$$

Here the last equality is obtained by (4.8). This proves the sufficient condition.

We next turn to the necessary condition. Let $\Delta$ be a decoder for which the phaseless instance optimality (4.5) holds. Let $I\subseteq \{1,\ldots ,m\}$. For any $\eta _1\in \mathcal {N}(A_I)$ and $\eta _2\in \mathcal {N}(A_{I^c})$ we have

$$\begin{aligned} |A(\eta _1+\eta _2)|=|A(\eta _1-\eta _2)|=|A(\eta _2-\eta _1)|. \end{aligned}$$

(4.9)

The instance optimality implies

$$\begin{aligned}&\min \Bigl \{\Vert \Delta (|A(\eta _1+\eta _2)|)+\eta _1+\eta _2\Vert _X,\Vert \Delta (|A(\eta _1+\eta _2)|)-(\eta _1+\eta _2)\Vert _X\Bigr \}\nonumber \\&\quad \le C_0\sigma _k(\eta _1+\eta _2)_X. \end{aligned}$$

(4.10)

Without loss of generality we may assume that

$$\begin{aligned} \Vert \Delta (|A(\eta _1+\eta _2)|)+\eta _1+\eta _2\Vert _X\,\,\le \,\,\Vert \Delta (|A(\eta _1+\eta _2)|)-(\eta _1+\eta _2)\Vert _X. \end{aligned}$$

Then (4.10) implies that

$$\begin{aligned} \Vert \Delta (|A(\eta _1+\eta _2)|)+\eta _1+\eta _2\Vert _X \le C_0\sigma _k(\eta _1+\eta _2)_X. \end{aligned}$$

(4.11)

By (4.9), we have

$$\begin{aligned} \Vert \Delta (|A(\eta _1+\eta _2)|)+\eta _1+\eta _2\Vert _X&=\Vert \Delta (|A(\eta _2-\eta _1)|)-(\eta _2-\eta _1)+2\eta _2\Vert _X \nonumber \\&\ge 2\Vert \eta _2\Vert _X-\Vert \Delta (|A(\eta _2-\eta _1)|)-(\eta _2-\eta _1)\Vert _X. \end{aligned}$$

(4.12)

Combining (4.11) and (4.12) yields

$$\begin{aligned} 2\Vert \eta _2\Vert _X\le C_0\sigma _k(\eta _1+\eta _2)_X+\Vert \Delta (|A(\eta _2-\eta _1)|)-(\eta _2-\eta _1)\Vert _X. \end{aligned}$$

(4.13)

At the same time, (4.9) also implies

$$\begin{aligned} \Vert \Delta (|A(\eta _1+\eta _2)|)+\eta _1+\eta _2\Vert _X&=\Vert \Delta (|A(\eta _2-\eta _1)|)+(\eta _2-\eta _1)+2\eta _1\Vert _X \nonumber \\&\ge 2\Vert \eta _1\Vert _X-\Vert \Delta (|A(\eta _2-\eta _1)|)+(\eta _2-\eta _1)\Vert _X. \end{aligned}$$

(4.14)

Putting (4.11) and (4.14) together, we obtain

$$\begin{aligned} 2\Vert \eta _1\Vert _X \le C_0\sigma _k(\eta _1+\eta _2)_X+\Vert \Delta (|A(\eta _2-\eta _1)|)+(\eta _2-\eta _1)\Vert _X. \end{aligned}$$

(4.15)

It follows from (4.13) and (4.15) that

$$\begin{aligned} \min \left\{ \Vert \eta _1\Vert _X, \Vert \eta _2\Vert _X\right\}&\le \frac{C_0}{2}\sigma _k(\eta _1+\eta _2)_X\\&\quad +\frac{1}{2}\min \{ \Vert \Delta (|A(\eta _2-\eta _1)|)-(\eta _2-\eta _1)\Vert _X,\Vert \Delta (|A(\eta _2-\eta _1)|)\\&\quad +(\eta _2-\eta _1)\Vert _X\}\le \frac{C_0}{2}\sigma _k(\eta _1+\eta _2)_X+\frac{C_0}{2}\sigma _k(\eta _1-\eta _2)_X. \end{aligned}$$

Here the last inequality is obtained by the instance optimality of $(A,\Delta )$. For the case where

$$\begin{aligned} \Vert \Delta (|A(\eta _1+\eta _2)|)-(\eta _1+\eta _2)\Vert _X\,\,\le \,\,\Vert \Delta (|A(\eta _1+\eta _2)|)+\eta _1+\eta _2\Vert _X, \end{aligned}$$

we obtain

$$\begin{aligned} \min \{\Vert \eta _1\Vert _X,\Vert \eta _2\Vert _X\}\le \frac{C_0}{2}\sigma _k(\eta _1+\eta _2)_X+\frac{C_0}{2}\sigma _k(\eta _1-\eta _2)_X \end{aligned}$$

via the same argument. The theorem is now proved. $\square$

We next present a null space property for phaseless instance optimality, which allows us to establish parallel results for sparse phase retrieval.

Definition 4.1

We say a matrix $A \in {\mathbb R}^{m\times N}$ satisfies the strong null space property (S-NSP) of order k with constant C if for any index set $I\subseteq \{1,\ldots ,m\}$ with $|I|\ge m/2$ and $\eta \in {\mathcal N}(A_I)$ we have

$$\begin{aligned} \Vert \eta \Vert _X\le C\cdot \sigma _{k}(\eta )_X. \end{aligned}$$

Theorem 4.3

Assume that a matrix $A\in \mathbb {R}^{m\times N}$ has the strong null space property of order 2k with constant $C_0/2$. Then there must exist a decoder $\Delta$ having the phaseless instance optimality (1.3) with constant $C_0$. In particular, one such decoder is

$$\begin{aligned} \Delta (|Ax_0|)=\mathop {\hbox {argmin}}_{|Ax|=|Ax_0|}\sigma _k(x)_X. \end{aligned}$$

Proof

Assume that $I\subseteq \{1,\ldots ,m\}$. For any $\eta _1\in \mathcal {N}(A_I)$ and $\eta _2\in \mathcal {N}(A_{I^c})$ we must have either $\Vert \eta _1\Vert _X\le \frac{C_0}{2}\sigma _{2k}(\eta _1)_X$ or $\Vert \eta _2\Vert _X\le \frac{C_0}{2}\sigma _{2k}(\eta _2)_X$ by the strong null space property. If $\Vert \eta _1\Vert _X\le \frac{C_0}{2}\sigma _{2k}(\eta _1)_X$ then

$$\begin{aligned} \Vert \eta _1\Vert _X\le \frac{C_0}{2}\sigma _{2k}(\eta _1)_X\le \frac{C_0}{4}\sigma _k(\eta _1-\eta _2)_X+\frac{C_0}{4}\sigma _k(\eta _1+\eta _2)_X. \end{aligned}$$

Similarly if $\Vert \eta _2\Vert _X\le \frac{C_0}{2}\sigma _{2k}(\eta _2)_X$ we will have

$$\begin{aligned} \Vert \eta _2\Vert _X\le \frac{C_0}{2}\sigma _{2k}(\eta _2)_X\le \frac{C_0}{4}\sigma _k(\eta _1-\eta _2)_X+\frac{C_0}{4}\sigma _k(\eta _1+\eta _2)_X. \end{aligned}$$

It follows that

$$\begin{aligned} \min \{\Vert \eta _1\Vert _X , \Vert \eta _2\Vert _X \} \le \frac{C_0}{4}\sigma _k(\eta _1-\eta _2)_X+\frac{C_0}{4}\sigma _k(\eta _1+\eta _2)_X. \end{aligned}$$

(4.16)

Theorem 4.2 now implies that the required decoder $\Delta$ exists. Furthermore, by the proof of the sufficiency part of Theorem 4.2,

$$\begin{aligned} \Delta (|Ax_0|)=\mathop {\hbox {argmin}}_{|Ax|=|Ax_0|}\sigma _k(x)_X \end{aligned}$$

is one such decoder. $\square$

4.3 The Case $X=\ell _1$

We will now apply Theorem 4.3 to the $\ell _1$-norm case. The following lemma establishes a relation between S-RIP and S-NSP for the $\ell _1$-norm.

Lemma 4.1

Let a, b, k be integers. Assume that $A\in \mathbb {R}^{m\times N}$ satisfies the S-RIP of order $(a+b)k$ with constants $\theta _-, \ \theta _+\in (0, 2)$. Then A satisfies the S-NSP of order ak under the $\ell _1$-norm with constant

$$\begin{aligned} C_0=1+\sqrt{\frac{a(1+\delta )}{b(1-\delta )}}, \end{aligned}$$

where $\delta$ is the restricted isometry constant and $\delta :=\max \{1-\theta _-,\theta _+-1\}<1$.

We remark that the above lemma is the analogous to the following lemma providing a relationship between RIP and NSP, which was shown in [5]:

Lemma 4.2

([5, Lemma 4.1]) Let $a=l/k$, $b=l'/k$ where $l,l'\ge k$ are integers. Assume that $A\in \mathbb {R}^{m\times N}$ satisfies the RIP of order $(a+b)k$ with $\delta =\delta _{(a+b)k}<1$. Then A satisfies the null space property under the $\ell _1$-norm of order ak with constant $C_0=1+\frac{\sqrt{a(1+\delta )}}{\sqrt{b(1-\delta )}}$.

Proof

By the definition of S-RIP, for any index set $I\subseteq \{1,\ldots ,m\}$ with $|I|\ge m/2$, the matrix $A_I\in {\mathbb R}^{|I|\times N}$ satisfies the RIP of order $(a+b)k$ with constant $\delta _{(a+b)k}=\delta :=\max \{1-\theta _-,\theta _+-1\}< 1$. It follows from Lemma 4.2 that

$$\begin{aligned} \Vert \eta \Vert _1\le \left( 1+\sqrt{\frac{a(1+\delta )}{b(1-\delta )}}\,\right) \sigma _{ak}(\eta )_1 \end{aligned}$$

for all $\eta \, \in \mathcal {N}(A_I)$. This proves the lemma. $\square$

Set $a=2$ and $b=1$ in Lemma 4.1 we infer that if A satisfies the S-RIP of order 3k with constants $\theta _-, \ \theta _+\in (0, 2)$, then A satisfies the S-NSP of order 2k under the $\ell _1$-norm with constant $C_0=1+\sqrt{\frac{2(1+\delta )}{1-\delta }}$. Hence by Theorem 4.3, there must exist a decoder that has the instance optimality under the $\ell _1$-norm with constant $2C_0$. According to Theorem 2.1, by taking $m=O(k\log (N/k))$ a Gaussian random matrix A satisfies S-RIP of order 3k with high probability. Hence, there exists a decoder $\Delta$ so that the pair $(A, \Delta )$ has the the $\ell _1$-norm phaseless instance optimality at the cost of $m=O(k\log (N/k))$ measurements, as with the traditional instance optimality.

We are now ready to prove the following theorem on phaseless instance optimality under the $\ell _1$-norm.

Theorem 4.4

Let $A \in {\mathbb R}^{m\times N}$ satisfy the S-RIP of order tk with constants $0<\theta _-<1< \theta _+<2$, where

$$\begin{aligned} t\ge \max \left\{ \frac{2}{\theta _-}, \frac{2}{2-\theta _+} \right\} >2. \end{aligned}$$

Let

$$\begin{aligned} \Delta (|Ax_0|)=\mathop {\hbox {argmin}}_{x\in {\mathbb R}^N}\left\{ \Vert x\Vert _1: |Ax|=|Ax_0|\right\} . \end{aligned}$$

(4.17)

Then $(A,\Delta )$ has the $\ell _1$-norm phaseless instance optimality with constant $C=\frac{2C_0}{2-C_0}$, where $C_0=1+\sqrt{\frac{1+\delta }{(t-1)(1-\delta )}}$ and as before

$$\begin{aligned} \delta :=\max \{1-\theta _-,\theta _+-1\}\le 1-\frac{2}{t}. \end{aligned}$$

Proof of Lemma 4.1

Let $x_0 \in {\mathbb R}^N$ and set $\hat{x} =\Delta (|Ax_0|)$. Then by definition

$$\begin{aligned} \Vert \hat{x}\Vert _1\le \Vert x_0\Vert _1\quad \text {and}\quad |A\hat{x}|=|Ax_0|. \end{aligned}$$

Denote by $I\subseteq \{1,\ldots ,m\}$ the set of indices

$$\begin{aligned} I=\left\{ j: \langle {a_j,\hat{x}} \rangle =\langle {a_j,x_0} \rangle \right\} , \end{aligned}$$

and thus $\langle {a_j,\hat{x}} \rangle =-\langle {a_j,x_0} \rangle$ for $j \in I^c$. It follows that

$$\begin{aligned} A_I(\hat{x}-x_0)=0 \quad \text{ and } \quad A_{I^c}(\hat{x}+x_0)=0. \end{aligned}$$

Set

$$\begin{aligned} \eta :=\hat{x}-x_0\in \mathcal {N}(A_I). \end{aligned}$$

We know that A satisfies the S-RIP of order tk with constants $\theta _-,\ \theta _+$ where

$$\begin{aligned} t\ge \max \left\{ \frac{2}{\theta _-}, \frac{2}{2-\theta _+} \right\} >2. \end{aligned}$$

For the case $|I|\ge m/2$, $A_I$ satisfies the RIP of order tk with RIP constant

$$\begin{aligned} \delta =\delta _{tk}:=\max \{1-\theta _-, \theta _+-1\}\le 1-\frac{2}{t}< 1. \end{aligned}$$

Take $a:=1,\ b:=t-1$ in Lemma 4.1. Then A satisfies the $\ell _1$-norm S-NSP of order k with constant

$$\begin{aligned} C_0=1+\sqrt{\frac{1+\delta }{(t-1)(1-\delta )}}<2. \end{aligned}$$

This yields

$$\begin{aligned} \Vert \eta \Vert _1\le C_0\Vert \eta _{T^c}\Vert _1, \end{aligned}$$

(4.18)

where T is the index set for the k largest coefficients of $x_0$ in magnitude. Hence $\Vert \eta _T\Vert _1\le (C_0-1)\Vert \eta _{T^c}\Vert _1$. Since $\Vert \hat{x}\Vert _1\le \Vert x_0\Vert _1$ we have

$$\begin{aligned} \Vert x_0\Vert _1\ge \Vert \hat{x}\Vert _1&=\Vert x_0+\eta \Vert _1 =\Vert x_{0,T}+x_{0,T^c}+\eta _T+\eta _{T^c}\Vert _1\\&\ge \Vert x_{0,T}\Vert _1-\Vert x_{0,T^c}\Vert _1+\Vert \eta _{T^c}\Vert _1-\Vert \eta _T\Vert _1. \end{aligned}$$

It follows that

$$\begin{aligned} \Vert \eta _{T^c}\Vert _1\le \Vert \eta _T\Vert _1+2\sigma _k(x_0)_1\le (C_0-1)\Vert \eta _{T^c}\Vert _1+2\sigma _k(x_0)_1 \end{aligned}$$

and thus

$$\begin{aligned} \Vert \eta _{T^c}\Vert _1\le \frac{2}{2-C_0}\sigma _k(x_0)_1. \end{aligned}$$

Now (4.18) yields

$$\begin{aligned} \Vert \eta \Vert _1\le C_0\Vert \eta _{T^c}\Vert _1\le \frac{2C_0}{2-C_0}\sigma _k(x_0)_1, \end{aligned}$$

which implies

$$\begin{aligned} \Vert \hat{x}-x_0\Vert _1\le C_0\Vert \eta _{T^c}\Vert _1\le \frac{2C_0}{2-C_0}\sigma _k(x_0)_1. \end{aligned}$$

For the case $|I^c|\ge m/2$ identical argument yields

$$\begin{aligned} \Vert \hat{x}+x_0\Vert _1\le C_0\Vert \eta _{T^c}\Vert _1\le \frac{2C_0}{2-C_0}\sigma _k(x_0)_1. \end{aligned}$$

The theorem is now proved. $\square$

By Theorem 2.1, an $m\times N$ random Gaussian matrix with $m=\mathcal {O}(tk\log (N/k))$ satisfies the S-RIP of order tk with high probability. We therefore conclude that the $\ell _1$-norm phaseless instance optimality of order k can be achieved at the cost of $m=\mathcal {O}(tk\log (N/k))$ measurements.

4.4 Mixed-Norm phaseless Instance Optimality

We now consider mixed-norm phaseless instance optimality. Let $1\le q\le p\le 2$ and $s=1/q-1/p$. We seek estimates of the form

$$\begin{aligned} \min \{\Vert x-\Delta (|Ax|)\Vert _{p}, \Vert x+\Delta (|Ax|)\Vert _{p}\} \le C_0 k^{-s}\sigma _k(x)_{q} \end{aligned}$$

(4.19)

for all $x\in {\mathbb R}^N$. We shall prove both necessary and sufficient conditions for mixed-norm phaseless instance optimality.

Theorem 4.5

Let $A\in {\mathbb R}^{ m\times N}$ and $1\le q\le p\le 2$. Set $s=1/q-1/p$. Then a decoder $\Delta$ satisfying the mixed norm phaseless instance optimality (4.19) with constant $C_0$ exists if: for any index set $I\subseteq \{1,\ldots ,m\}$ and any $\eta _1\in \mathcal {N}(A_I)$, $\eta _2\in \mathcal {N}(A_{I^c})$ we have

$$\begin{aligned} \min \{\Vert \eta _1\Vert _p , \Vert \eta _2\Vert _p \} \le \frac{C_0}{4}k^{-s}\Bigl (\sigma _k(\eta _1-\eta _2)_q+\sigma _k(\eta _1+\eta _2)_q\Bigr ). \end{aligned}$$

(4.20)

Conversely, assume a decoder $\Delta$ satisfying the mixed norm phaseless instance optimality (4.19) exists. Then for any index set $I\subseteq \{1,\ldots ,m\}$ and any $\eta _1\in \mathcal {N}(A_I)$, $\eta _2\in \mathcal {N}(A_{I^c})$ we have

$$\begin{aligned} \min \{\Vert \eta _1\Vert _p , \Vert \eta _2\Vert _p \} \le \frac{C_0}{2}k^{-s}\Bigl (\sigma _k(\eta _1-\eta _2)_q+\sigma _k(\eta _1+\eta _2)_q\Bigr ). \end{aligned}$$

Proof of Lemma 4.1

The proof is virtually identical to the proof of Theorem 4.2. We shall omit the details here in the interest of brevity. $\square$

Definition 4.2

(Mixed-Norm Strong Null Space Property) We say that A has the mixed strong null space property in norms $(\ell _p,\ell _q)$ of order k with constant C if for any index set $I\subseteq \{1,\ldots ,m\}$ with $|I|\ge m/2$ the matrix $A_I\in {\mathbb R}^{|I|\times N}$ satisfies

$$\begin{aligned} \Vert \eta \Vert _p\le Ck^{-s}\sigma _k(\eta )_q \end{aligned}$$

for all $\eta \in \mathcal {N}(A_I)$, where $s = 1/q-1/p$.

The above is an extension of the standard definition of the mixed null space property of order k in norms $(\ell _p,\ell _q)$ (see [5]) for a matrix A, which requires

$$\begin{aligned} \Vert \eta \Vert _p\le Ck^{-s}\sigma _k(\eta )_q \end{aligned}$$

for all $\eta \in \mathcal {N}(A)$. We have the following straightforward generalization of Theorem 4.3.

Theorem 4.6

Assume that $A\in \mathbb {R}^{m\times N}$ has the mixed strong null space property of order 2k in norms $(\ell _p,\ell _q)$ with constant $C_0/2$, where $1\le q\le p\le 2$. Then there exists a decoder $\Delta$ such that the mixed-norm phaseless instance optimality (4.19) holds with constant $C_0$.

We establish relationships between mixed-norm strong null space property and the S-RIP. First we present the following lemma that was proved in [5].

Lemma 4.3

([5, Lemma 8.2]) Let $k\ge 1$ and $\tilde{k} = \lceil {k(\frac{N}{k})^{2-2/q}}\rceil$. Assume that $A\in {\mathbb R}^{m\times N}$ satisfies the RIP of order $2k+\tilde{k}$ with $\delta :=\delta _{2k+\tilde{k}}<1$. Then A satisfies the mixed null space property in norms $(\ell _p,\ell _q)$ of order 2k with constant $C_0=2^{1/p+1/2}\sqrt{\frac{1+\delta }{1-\delta }}+2^{1/p-1/q}$.

Proposition 4.1

Let $k\ge 1$ and $\tilde{k} = \lceil {k(\frac{N}{k})^{2-2/q}}\rceil$. Assume that $A\in {\mathbb R}^{m\times N}$ satisfies the S-RIP of order $2k+\tilde{k}$ with constants $0<\theta _- <1 <\theta _+<2$. Then A satisfies the mixed strong null space property in norms $(\ell _p, \ell _q)$ of order 2k with constant $C_0=2^{1/p+1/2}\sqrt{\frac{1+\delta }{1-\delta }}+2^{1/p-1/q}$, where $\delta$ is the RIP constant and $\delta :=\delta _{2k+\tilde{k}}= \max \{1-\theta _-, \theta _+-1\}$.

Proof of Lemma 4.1

By definition for any index set $I\subseteq \{1,\ldots ,m\}$ with $|I|\ge m/2$, the matrix $A_I\in {\mathbb R}^{|I|\times N}$ satisfies RIP of order $2k+\tilde{k}$ with constant $C_0=2^{1/p+1/2}\sqrt{\frac{1+\delta }{1-\delta }}+2^{1/p-1/q}$, where $\delta$ is the RIP constant and $\delta :=\delta _{2k+\tilde{k}}= \max \{1-\theta _-, \theta _+-1\}$. By Lemma 4.3, we know that $A_I$ satisfies the mixed null space property in norms $(\ell _p,\ell _q)$ of order 2k with constant $C_0=2^{1/p+1/2}\sqrt{\frac{1+\delta }{1-\delta }}+2^{1/p-1/q}$, in other words for any $\eta \in \mathcal {N}(A_I)$,

$$\begin{aligned} \Vert \eta \Vert _p\le Ck^{-s}\sigma _{2k}(\eta )_q. \end{aligned}$$

So A satisfies the mixed strong null space property. $\square$

Corollary 4.1

Let $k\ge 1$ and $\tilde{k} = k(\frac{N}{k})^{2-2/q}$. Assume that $A\in {\mathbb R}^{m\times N}$ satisfies the S-RIP of order $2k+\tilde{k}$ with constants $0<\theta _- <1 <\theta _+<2$. Let $\delta :=\delta _{2k+\tilde{k}} =\max \{1-\theta _-, \theta _+-1\}<1$. Define the decoder $\Delta$ for A by

$$\begin{aligned} \Delta (|Ax_0|)=\mathop {\hbox {argmin}}_{|Ax|=|Ax_0|}\sigma _k(x)_q. \end{aligned}$$

(4.21)

Then (4.19) holds with constant $2C_0$, where $C_0=2^{1/p+1/2}\sqrt{\frac{1+\delta }{1-\delta }}+2^{1/p-1/q}$.

Proof of Lemma 4.1

By the Proposition 4.1, the matrix A satisfies the mixed strong null space property in $(\ell _p,\ell _q)$ of order 2k with constant $C_0=2^{1/p+1/2}\sqrt{\frac{1+\delta }{1-\delta }}+2^{1/p-1/q}$. The corollary now follows immediately from Theorem 4.6. $\square$

Remark 4.1

Combining Theorem 2.1 and Corollary 4.1, the mixed phaseless instance optimality of order k in norms $(\ell _p,\ell _q)$ can be achieved for the price of $\mathcal {O}(k(N/k)^{2-2/q}\log (N/k))$ measurements, just as with the traditional mixed $(\ell _p,\ell _q)$-norm instance optimality. Theorem 3.1 implies that the $\ell _1$ decoder satisfies the $(p,q)=(2,1)$ mixed-norm phaseless instance optimality at the price of $\mathcal {O}(k\log (N/k))$ measurements.

5 Appendix: Proof of Lemma 2.1

We will first need the following two Lemmas to prove Lemma 2.1.

Lemma 5.1

(Sparse Representation of a Polytope [2, 12]) Let $s\ge 1$ and $\alpha >0$. Set

$$\begin{aligned} T(\alpha ,s):=\Bigl \{u\in \mathbb {R}^n: \Vert u\Vert _\infty \le \alpha ,\ \Vert u\Vert _1\le s\alpha \Bigr \}. \end{aligned}$$

For any $v\in \mathbb {R}^n$ let

$$\begin{aligned} U (\alpha ,s,v):=\Bigl \{u\in \mathbb {R}^n:\hbox {supp}(u)\subseteq \hbox {supp}(v),\Vert u\Vert _0\le s,\Vert u\Vert _1=\Vert v\Vert _1,\Vert u\Vert _\infty \le \alpha \Bigr \}. \end{aligned}$$

Then $v\in T(\alpha ,s)$ if and only if v is in the convex hull of $U (\alpha ,s,v)$, i.e. v can be expressed as a convex combination of some $u_1, \dots , u_N$ in $U (\alpha ,s,v)$.

Lemma 5.2

([1, Lemma 5.3]) Assume that $a_1\ge a_2\ge \cdots \ge a_m\ge 0$. Let $r \le m$ and $\lambda \ge 0$ such that $\sum _{i=1}^{r}a_i + \lambda \ge \sum _{i=r+1}^{m}a_i$. Then for all $\alpha \ge 1$ we have

$$\begin{aligned} \sum _{j=r+1}^{m}a_j^\alpha \le r \left( \root \alpha \of {\frac{\sum _{i=1}^{r}a_i^\alpha }{r}}+\frac{\lambda }{r}\right) ^\alpha . \end{aligned}$$

(5.1)

In particular for $\lambda =0$ we have

$$\begin{aligned} \sum _{j=r+1}^{m}a_j^\alpha \le \sum _{i=1}^{r}a_i^\alpha . \end{aligned}$$

We are now ready to prove Lemma 2.1.

Proof

Set $h:=\hat{x}-x_0$. Let $T_0$ denote the set of the largest k coefficients of $x_0$ in magnitude. Then

$$\begin{aligned} \Vert x_0\Vert _1+\rho&\ge \Vert \hat{x}\Vert _1 =\Vert x_0+h\Vert _1 \\&=\Vert x_{0,T_0}+h_{T_0}+x_{0,T_0^c}+h_{T_0^c}\Vert _1\\&\ge \Vert x_{0,T_0}\Vert _1-\Vert h_{T_0}\Vert _1-\Vert x_{0,T_0^c}\Vert _1+\Vert h_{T_0^c}\Vert _1. \end{aligned}$$

It follows that

$$\begin{aligned} \Vert h_{T_0^c}\Vert _1&\le \Vert h_{T_0}\Vert _1+2\Vert x_{0,T_0^c}\Vert _1+\rho \\&=\Vert h_{T_0}\Vert _1+2\sigma _k(x_0)_1+\rho . \end{aligned}$$

Suppose that $S_0$ is the index set of the k largest entries in absolute value of h . Then we can get

$$\begin{aligned} \Vert h_{S_0^c}\Vert _1\le \Vert h_{T_0^c}\Vert _1&\le \Vert h_{T_0}\Vert _1+2\sigma _k(x_0)_1+\rho \\&\le \Vert h_{S_0}\Vert _1+2\sigma _k(x_0)_1+\rho . \end{aligned}$$

Set

$$\begin{aligned} \alpha :=\frac{\Vert h_{S_0}\Vert _1+2\sigma _k(x_0)_1+\rho }{k}. \end{aligned}$$

We divide $h_{S_0^c }$ into two parts $h_{S_0^c }=h^{(1)}+h^{(2)}$, where

$$\begin{aligned} h^{(1)}:=h_{S_0^c}\cdot I_{\{i:\,|h_{S_0^c}(i)|>\alpha /(t-1)\}}, \quad h^{(2)}:=h_{S_0^c}\cdot I_{\{i:\,|h_{S_0^c}(i)|\le \alpha /(t-1)\}} . \end{aligned}$$

A simple observation is that $\Vert h^{(1)}\Vert _1\le \Vert h_{S_0^c}\Vert _1\le \alpha k$. Set

$$\begin{aligned} \ell := |\mathrm{supp}(h^{(1)})|=\Vert h^{(1)}\Vert _0 . \end{aligned}$$

Since all non-zero entries of $h^{(1)}$ have magnitude larger than $\alpha /(t-1)$, we have

$$\begin{aligned} \alpha k\ge \Vert h^{(1)}\Vert _1=\sum _{i\in \mathrm{supp}(h^{(1)})}|h^{(1)}(i)| \ge \sum _{i\in \mathrm{supp}(h^{(1)})}\frac{\alpha }{t-1}=\frac{\alpha \ell }{t-1}, \end{aligned}$$

which implies $\ell \le (t-1)k$. Thus we have:

$$\begin{aligned} \big \langle A(h_{S_0}+h^{(1)}), Ah\big \rangle \le \Vert A(h_{S_0}+h^{(1)})\Vert _2\cdot \Vert Ah\Vert _2 \le \sqrt{1+\delta }\cdot \Vert h_{S_0}+h^{(1)}\Vert _2\cdot \epsilon .\nonumber \\ \end{aligned}$$

(5.2)

Here we apply the facts that $\Vert h_{S_0}+h^{(1)}\Vert _0=\ell +k\le tk$ and A satisfies the RIP of order tk with $\delta :=\delta _{tk}^A$. We shall assume at first that tk as an integer. Note that $\Vert h^{(2)}\Vert _\infty \le \frac{\alpha }{t-1}$ and

$$\begin{aligned} \Vert h^{(2)}\Vert _1 =\Vert h_{S_0^c}\Vert _1-\Vert h^{(1)}\Vert _1 \le k\alpha - \frac{ \alpha \ell }{t-1}=(k(t-1)-\ell )\frac{\alpha }{t-1}. \end{aligned}$$

(5.3)

We take $s:=k(t-1)-\ell$ in Lemma 5.1 and obtain that $h^{(2)}$ is a weighted mean

$$\begin{aligned} h^{(2)}=\sum _{i=1}^{N}\lambda _iu_i,\quad \quad 0\le \lambda _i\le 1, \quad \sum _{i=1}^N\lambda _i=1 \end{aligned}$$

where $\Vert u_i\Vert _0\le k(t-1)-\ell , \Vert u_i\Vert _1=\Vert h^{(2)}\Vert _1$, $\Vert u_i\Vert _\infty \le \alpha /(t-1)$ and $\hbox {supp}(u_i)\subseteq \hbox {supp}(h^{(2)})$. Hence

$$\begin{aligned} \Vert u_i\Vert _2\le \sqrt{\Vert u_i\Vert _0}\cdot \Vert u_i\Vert _\infty&=\sqrt{k(t-1)-\ell }\cdot \Vert u_i\Vert _\infty \\&\le \sqrt{k(t-1)}\cdot \Vert u_i\Vert _\infty \\&\le \alpha \sqrt{k/(t-1)}. \end{aligned}$$

Now for $0\le \mu \le 1$ and $d\ge 0$, which will be chosen later, set

$$\begin{aligned} \beta _j:=h_{S_0}+h^{(1)}+\mu \cdot u_j, \quad j=1,\ldots ,N. \end{aligned}$$

Then for fixed $i\in [1,N]$

$$\begin{aligned} \sum _{j=1}^{N}\lambda _j\beta _j-d\beta _i&=h_{S_0}+h^{(1)}+\mu \cdot h^{(2)}-d\beta _i\\&=(1-\mu -d)(h_{S_0}+h^{(1)})-d\mu u_i+\mu h. \end{aligned}$$

Recall that $\alpha =\frac{\Vert h_{S_0}\Vert _1+2\sigma _k(x_0)_1+\rho }{k}$. Thus

$$\begin{aligned} \Vert u_i\Vert _2&\le \sqrt{k/(t-1)}\alpha \\\nonumber&\le \frac{\Vert h_{S_0}\Vert _2}{\sqrt{t-1}}+\frac{2\sigma _k(x_0)_1+\rho }{\sqrt{k(t-1)}}\\\nonumber&\le \frac{\Vert h_{S_0}+h^{(1)}\Vert _2}{\sqrt{t-1}}+\frac{2\sigma _k(x_0)_1+\rho }{\sqrt{k(t-1)}}\\\nonumber&=\frac{z+R}{\sqrt{t-1}},\nonumber \end{aligned}$$

(5.4)

where $z:=\Vert h_{S_0}+h^{(1)}\Vert _2$ and $R:=\frac{2\sigma _k(x_0)_1+\rho }{\sqrt{k}}$. It is easy to check the following identity:

$$\begin{aligned} (2d-1)&\sum _{1\le i<j\le N}\lambda _i\lambda _j\Vert A(\beta _i-\beta _j)\Vert _2^2 \nonumber \\&=\sum _{i=1}^{N}\lambda _i\Bigl \Vert A(\sum _{j=1}^{N}\lambda _j\beta _j-d\beta _i)\Bigr \Vert _2^2 - \sum _{i=1}^{N}\lambda _i(1-d)^2\Vert A\beta _i\Vert _2^2, \end{aligned}$$

(5.5)

provided that $\sum _{i=1}^N\lambda _i=1$. Choose $d=1/2$ in (5.5) we then have

$$\begin{aligned} \sum _{i=1}^{N}\lambda _i\Bigl \Vert A\Bigl ( (\frac{1}{2}-\mu )(h_{S_0}+h^{(1)})-\frac{\mu }{2}u_i+\mu h\Bigr )\Bigr \Vert _2^2-\sum _{i=1}^{N}\frac{\lambda _i}{4}\Vert A\beta _i\Vert _2^2 = 0. \end{aligned}$$

Note that for $d=1/2$,

$$\begin{aligned}&\Bigl \Vert A\Bigl ( (\frac{1}{2}-\mu )(h_{S_0}+h^{(1)})-\frac{\mu }{2}u_i+\mu h\Bigr )\Bigr \Vert _2^2\\&\quad = \Bigl \Vert A\Bigl ( (\frac{1}{2}-\mu )(h_{S_0}+h^{(1)})-\frac{\mu }{2}u_i\Bigr ) \Bigr \Vert _2^2\\&\quad \quad +2\Bigl \langle A\Bigl ((\frac{1}{2}-\mu )(h_{S_0}+h^{(1)})-\frac{\mu }{2}u_i\Bigr ), \mu Ah\Bigr \rangle +\mu ^2\Vert Ah\Vert _2^2. \end{aligned}$$

It follows from $\sum _{i=1}^N \lambda _i =1$ and $h^{(2)}=\sum _{i=1}^{N}\lambda _iu_i$ that

$$\begin{aligned}&\sum _{i=1}^{N}\lambda _i\Bigl \Vert A\Bigl ( (\frac{1}{2}-\mu )(h_{S_0}+h^{(1)})-\frac{\mu }{2}u_i+\mu h\Bigr )\Bigr \Vert _2^2\nonumber \\&= \sum _i\lambda _i\Bigl \Vert A\Bigl ( (\frac{1}{2}-\mu )(h_{S_0}+h^{(1)})-\frac{\mu }{2}u_i\Bigr ) \Bigr \Vert _2^2\nonumber \\&\quad +2\Bigl \langle A\Bigl ((\frac{1}{2}-\mu )(h_{S_0}+h^{(1)})-\frac{\mu }{2}h^{(2)}\Bigr ), \mu Ah\Bigr \rangle +\mu ^2\Vert Ah\Vert _2^2 \nonumber \\&= \sum _i\lambda _i\Bigl \Vert A\Bigl ( (\frac{1}{2}-\mu )(h_{S_0}+h^{(1)})-\frac{\mu }{2}u_i\Bigr ) \Bigr \Vert _2^2\nonumber \\&\quad + \mu (1-\mu )\Bigl \langle A(h_{S_0}+h^{(1)}),Ah\Bigr \rangle -\sum _{i=1}^{N}\frac{\lambda _i}{4}\Vert A\beta _i\Vert _2^2. \end{aligned}$$

(5.6)

Set $\mu =\sqrt{t(t-1)}-(t-1)$. We next estimate the three terms in (5.6). Noting that $\Vert h_{S_0}\Vert _0\le k$, $\Vert h^{(1)}\Vert _0\le \ell$ and $\Vert u_i\Vert _0\le s =k(t-1)-\ell$, we obtain

$$\begin{aligned} \Vert \beta _i\Vert _0\le \Vert h_{S_0}\Vert _0 + \Vert h^{(1)}\Vert _0+ \Vert u_i\Vert _0\le t\cdot k \end{aligned}$$

and $\Vert (\frac{1}{2}-\mu )(h_{S_0}+h^{(1)})-\frac{\mu }{2}u_i\Vert _0\le t\cdot k$. Since A satisfies the RIP of order $t\cdot k$ with $\delta$, we have

$$\begin{aligned}&\Bigl \Vert A\Bigl ( (\frac{1}{2}-\mu )(h_{S_0}+h^{(1)})-\frac{\mu }{2}u_i\Bigr ) \Bigr \Vert _2^2 \\&\qquad \le (1+\delta )\Vert (\frac{1}{2}-\mu )(h_{S_0}+h^{(1)})-\frac{\mu }{2}u_i\Vert _2^2\\&\quad = (1+\delta )\Bigl ((\frac{1}{2}-\mu )^2 \Vert (h_{S_0}+h^{(1)})\Vert _2^2+\frac{\mu ^2}{4}\Vert u_i\Vert _2^2\Bigr )\\&\quad = (1+\delta )\Bigl ((\frac{1}{2}-\mu )^2 z^2+\frac{\mu ^2}{4}\Vert u_i\Vert _2^2\Bigr ) \end{aligned}$$

and

$$\begin{aligned} \Vert A\beta _i\Vert _2^2\ge & {} (1-\delta )\Vert \beta _i\Vert _2^2 =(1-\delta )( \Vert h_{S_0}+h^{(1)}\Vert _2^2+\mu ^2\cdot \Vert u_i\Vert _2^2)\\= & {} (1-\delta )(z^2+\mu ^2\cdot \Vert u_i\Vert _2^2). \end{aligned}$$

Combining the result above with (5.2) and (5.4) we get

$$\begin{aligned}&0\le (1+\delta )\sum _{i=1}^{N}\lambda _i\Bigl ((\frac{1}{2}-\mu )^2z^2+\frac{\mu ^2}{4}\Vert u_i\Vert _2^2\Bigr ) +\mu (1-\mu )\sqrt{1+\delta }\cdot z\cdot \epsilon \\&\qquad -(1-\delta )\sum _{i=1}^{N}\frac{\lambda _i}{4}(z^2+\mu ^2\Vert u_i\Vert _2^2)\\&\quad =\sum _{i=1}^{N}\lambda _i\Bigl (\Bigl ((1+\delta )(\frac{1}{2}-\mu )^2-\frac{1-\delta }{4} \Bigr )z^2+\frac{\delta }{2}\mu ^2\Vert u_i\Vert _2^2 \Bigr )+\mu (1-\mu )\sqrt{1+\delta }\cdot z\cdot \epsilon \\&\quad \le \sum _{i=1}^{N}\lambda _i\Bigl (\Bigl ((1+\delta )(\frac{1}{2}-\mu )^2-\frac{1-\delta }{4} \Bigr )z^2+\frac{\delta }{2}\mu ^2\frac{(z+R)^2}{t-1} \Bigr )\\&\qquad +\mu (1-\mu )\sqrt{1+\delta }\cdot z\cdot \epsilon \\&\quad =\Bigl ((\mu ^2-\mu )+\delta \Bigl ( \frac{1}{2}-\mu +(1+\frac{1}{2(t-1)})\mu ^2\Bigr ) \Bigr )z^2\\&\qquad +\Bigl ( \mu (1-\mu )\sqrt{1+\delta }\cdot \epsilon +\frac{\delta \mu ^2R}{t-1}\Bigr )z+\frac{\delta \mu ^2R^2}{2(t-1)} \\&\quad =-t\Bigl ((2t-1)-2\sqrt{t(t-1)} \Bigr ) (\sqrt{\frac{t-1}{t}}-\delta )z^2\\&\qquad +\Bigl ( \mu ^2\sqrt{\frac{t}{t-1}}\sqrt{1+\delta }\cdot \epsilon +\frac{\delta \mu ^2R}{t-1}\Bigr )z+\frac{\delta \mu ^2R^2}{2(t-1)}\\&\quad =\frac{\mu ^2}{t-1}\Bigl (-t(\sqrt{\frac{t-1}{t}}-\delta )z^2+(\sqrt{t(t-1)(1+\delta )}\epsilon +\delta R)z+\frac{\delta R^2}{2} \Bigr ), \end{aligned}$$

which is a quadratic inequality for z. We know $\delta <\sqrt{(t-1)/t}$. So by solving the above inequality we get

$$\begin{aligned} z&\le \frac{(\sqrt{t(t-1)(1+\delta )}\epsilon +\delta R)+\left( (\sqrt{t(t-1)(1+\delta )}\epsilon +\delta R)^2+2t(\sqrt{(t-1)/t}-\delta )\delta R^2 \right) ^{1/2} }{2t(\sqrt{(t-1/t)}-\delta ) }\\&\le \frac{\sqrt{t(t-1)(1+\delta )}}{t(\sqrt{(t-1)/t}-\delta )}\epsilon +\frac{2\delta +\sqrt{2t(\sqrt{(t-1)/t}-\delta )\delta }}{2t(\sqrt{(t-1)/t}-\delta )}R. \end{aligned}$$

Finally, noting that $\Vert h_{S_0^c}\Vert _1\le \Vert h_{S_0}\Vert _1+R\sqrt{k}$, in the Lemma 5.2, if we set $m=N$, $r=k$, $\lambda =R\sqrt{k}\ge 0$ and $\alpha =2$ then $\Vert h_{S_0^c}\Vert _2\le \Vert h_{S_0}\Vert _2+R$. Hence

$$\begin{aligned} \Vert h\Vert _2&=\sqrt{\Vert h_{S_0}\Vert _2^2+\Vert h_{S_0^c}\Vert _2^2}\\&\le \sqrt{\Vert h_{S_0}\Vert _2^2+(\Vert h_{S_0}\Vert _2+R)^2}\\&\le \sqrt{2\Vert h_{S_0}\Vert _2^2}+R\le \sqrt{2}z+R\\&\le \frac{\sqrt{2(1+\delta )}}{1-\sqrt{t/(t-1)}\delta }\epsilon +\left( \frac{\sqrt{2}\delta +\sqrt{t(\sqrt{(t-1)/t}-\delta )\delta }}{t(\sqrt{(t-1)/t}-\delta )}+1 \right) R. \end{aligned}$$

Substitute R into this inequality and the conclusion follows.

For the case where $t\cdot k$ is not an integer, we set $t^*:=\lceil tk\rceil / k$, then $t^*>t$ and $\delta _{t^*k}=\delta _{tk}<\sqrt{\frac{t-1}{t}}<\sqrt{\frac{t^*-1}{t^*}}$. We can then prove the result by working on $\delta _{t^*k}$. $\square$