Interleaving

Abstract

This chapter examines the interleaving relation between persistence modules, and the associated interleaving metric. Interleavings are approximate isomorphisms, and in the first instance may be defined by a pair of ‘shifted’ homomorphisms between the two persistence modules being compared. More abstractly, an interleaving can be thought of as a solution to a functor extension problem. The Interpolation Lemma is the main result of this chapter: it asserts that a pair of interleaved persistence modules can be interpolated by a 1-Lipschitz 1-parameter family. We give three different explicit constructions of the interpolation; two of them are the left and right Kan extensions (in the functor extension point of view), while the third mediates between the two.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Interleaving is a way of comparing two persistence modules. As with any category, two persistence modules ${\mathbb {U}}, {\mathbb {V}}$ are said to be isomorphic if there are maps

$$ \varPhi \in {\text {Hom}}({\mathbb {U}}, {\mathbb {V}}), \quad \varPsi \in {\text {Hom}}({\mathbb {V}}, {\mathbb {U}}), $$

such that

$$ \varPsi \varPhi = 1_{\mathbb {U}}, \quad \varPhi \varPsi = 1_{\mathbb {V}}. $$

This relation is too strong in situations where the persistence modules have been constructed from noisy or uncertain data. The natural response is to consider a weaker relation, $\delta$ -interleaving, where $\delta \ge 0$ quantifies the uncertainty.

In this section, we define the interleaving relation and study its elementary properties. We prove the nontrivial result (from [14]) that if two persistence modules are $\delta$-interleaved, then they are connected in the space of persistence modules by a path of length $\delta$. This ‘interpolation lemma’ is a crucial step in the proof of the stability theorem in Chap. 5.

4.1 Shifted Homomorphisms

The first step is to consider homomorphisms which shift the value of the persistence index. Let ${\mathbb {U}}, {\mathbb {V}}$ be persistence modules over ${\mathbf {R}}$, and let $\delta$ be any real number. A homomorphism of degree $\delta$ is a collection $\varPhi$ of linear maps

$$ \phi _t : U_t \rightarrow V_{t+\delta } $$

for all $t \in {\mathbf {R}}$, such that the diagram

commutes whenever $s \le t$.

We write

$$\begin{aligned} {\text {Hom}}^\delta ({\mathbb {U}}, {\mathbb {V}})&= \{ \text {homomorphisms } {\mathbb {U}}\rightarrow {\mathbb {V}}\text { of degree } \delta \}, \\ {\text {End}}^\delta ({\mathbb {V}})&= \{ \text {homomorphisms }{\mathbb {V}}\rightarrow {\mathbb {V}}\text { tof degree } \delta \}. \end{aligned}$$

Composition gives a map

$$ {\text {Hom}}^{\delta _2}({\mathbb {V}}, {\mathbb {W}}) \times {\text {Hom}}^{\delta _1}({\mathbb {U}}, {\mathbb {V}}) \rightarrow {\text {Hom}}^{\delta _1+\delta _2}({\mathbb {U}}, {\mathbb {W}}). $$

For $\delta \ge 0$, the most important degree-$\delta$ endomorphism is the shift map

$$ 1_{\mathbb {V}}^\delta \in {\text {End}}^\delta ({\mathbb {V}}), $$

which is the collection of maps $( v^t_{t+\delta } )$ from the persistence structure on ${\mathbb {V}}$. If $\varPhi$ is a homomorphism ${\mathbb {U}}\rightarrow {\mathbb {V}}$ of any degree, then by definition $\varPhi 1_{\mathbb {U}}^\delta = 1_{\mathbb {V}}^\delta \varPhi$ for all $\delta \ge 0$.

Remark 4.1

Here is another way to think of morphisms of non-zero degree. For any persistence module ${\mathbb {V}}$, and $\delta \in {\mathbf {R}}$, we let ${\mathbb {V}}[\delta ]$ denote the shifted module

$$\begin{aligned} (V[\delta ])_t = V_{t+\delta }, \quad (v[\delta ])^s_t = v^{s+\delta }_{t+\delta }. \end{aligned}$$

In other words, ${\mathbb {V}}[\delta ]$ is obtained from ${\mathbb {V}}$ by shifting all the information downwards by $\delta$. Then there are obvious identifications

$$ {\text {Hom}}^\delta ({\mathbb {U}},{\mathbb {V}}) = {\text {Hom}}({\mathbb {U}},{\mathbb {V}}[\delta ]) = {\text {Hom}}({\mathbb {U}}[a],{\mathbb {V}}[a+\delta ]) $$

for all $a \in {\mathbf {R}}$. To avoid excessive notation, we will use the same symbol for

$$ \varPhi = (\phi _t) : {\mathbb {U}}\rightarrow {\mathbb {V}}[\delta ] $$

as for its shifted equivalent

$$ \varPhi = (\phi _{t+a}) : {\mathbb {U}}[a] \rightarrow {\mathbb {V}}[a+\delta ] $$

since the constituent maps are the same.

4.2 Interleaving

Let $\delta \ge 0$. Two persistence modules ${\mathbb {U}}, {\mathbb {V}}$ are said to be $\delta$ -interleaved if there are maps

$$ \varPhi \in {\text {Hom}}^\delta ({\mathbb {U}}, {\mathbb {V}}), \quad \varPsi \in {\text {Hom}}^\delta ({\mathbb {V}}, {\mathbb {U}}) $$

such that

$$ \varPsi \varPhi = 1_{\mathbb {U}}^{2\delta }, \quad \varPhi \varPsi = 1_{\mathbb {V}}^{2\delta }. $$

More expansively (with many more indices written out), there are maps

$$ \phi _t: U_t \rightarrow V_{t+\delta } \quad \text {and} \quad \psi _t: V_t \rightarrow U_{t+\delta } $$

defined for all t, such that the following diagrams

(4.1)

commute for all eligible parameter values; that is, for all $s \le t$.

Remark 4.2

Where possible, we will be concise rather than expansive.

Example 4.3

Let X be a topological space and let $f, g: X \rightarrow {\mathbf {R}}$. Suppose $\Vert f - g \Vert _\infty < \delta$. Then the persistence modules ${\text {H}}({\mathbb {X}}^f_\mathrm {sub})$, ${\text {H}}({\mathbb {X}}^g_\mathrm {sub})$ are $\delta$-interleaved. Indeed, there are inclusions

$$\begin{aligned} (X,f)^t&\subseteq (X,g)^{t+\delta } \\ (X,g)^t&\subseteq (X,f)^{t+\delta } \end{aligned}$$

for all t, which induce maps

$$\begin{aligned} \varPhi&: {\text {H}}({\mathbb {X}}^f_\mathrm {sub}) \rightarrow {\text {H}}({\mathbb {X}}^g_\mathrm {sub}) \\ \varPsi&: {\text {H}}({\mathbb {X}}^g_\mathrm {sub}) \rightarrow {\text {H}}({\mathbb {X}}^f_\mathrm {sub}) \end{aligned}$$

of degree $\delta$. Since all the maps are induced functorially from inclusion maps, the interleaving relations are automatically satisfied.

This is the situation for which the stability theorem of Cohen-Steiner, Edelsbrunner and Harer [19] was originally stated: if two functions f, g are close then the diagrams for their sublevelset persistent homology are close. Subsequently, stability has been formulated as a theorem about the diagrams of interleaved persistence modules [14, 15]. In the present work, we will come to view stability as a theorem about r-measures.

4.3 Interleaving (Continued)

An interleaving between two persistence modules can be thought of as a persistence module over a certain partially ordered set (poset). We develop this idea next.

Consider the standard partial order on the plane:

$$ (p_1, q_1) \le (p_2, q_2) \quad \Leftrightarrow \quad p_1 \le p_2 \;\text {and}\; q_1 \le q_2. $$

For any real number x, define the corresponding shifted diagonal in the plane:

$$ \varDelta _x = \{ (p,q) \mid q-p = 2x \} = \{ (t-x, t+x) \mid t \in {\mathbf {R}}\} $$

As a poset, this is isomorphic to the real line. We will use the specific isomorphism by which $t \in {\mathbf {R}}$ corresponds to $(t-x, t+x) \in \varDelta _x$. This gives a canonical identification between persistence modules over ${\mathbf {R}}$ and persistence modules over $\varDelta _x$.

Proposition 4.4

Let x, y be real numbers. Persistence modules ${\mathbb {U}}, {\mathbb {V}}$ are $|y-x|$-interleaved if and only if there is a persistence module ${\mathbb {W}}$ over $\varDelta _x \cup \varDelta _y$ such that ${\mathbb {W}}|_{\varDelta _x} = {\mathbb {U}}$ and ${\mathbb {W}}|_{\varDelta _y} = {\mathbb {V}}$. Here $\varDelta _x \cup \varDelta _y$ is regarded as a subposet of ${\mathbf {R}}^2$.

Proof

Assume $x < y$ without loss of generality. We claim that (i) the extra information carried by $(y-x)$-interleaving maps $\varPhi , \varPsi$ is equivalent to (ii) the extra information carried by ${\mathbb {W}}$. Let us describe both, more carefully:

   (i) In addition to ${\mathbb {U}}, {\mathbb {V}}$ we have a system of maps $\varPhi = (\phi _t)$, where

$$ \phi _t : U_t \rightarrow V_{t+y-x}, $$

and a system of maps $\varPsi = (\psi _t)$, where

$$ \psi _t : V_t \rightarrow U_{t+y-x}. $$

These are constrained by the relations (for all $\eta \ge 0$).

$$\begin{aligned} \varPhi 1_{\mathbb {U}}^\eta = 1_{\mathbb {V}}^\eta \varPhi , \quad \varPsi 1_{\mathbb {V}}^\eta = 1_{\mathbb {U}}^\eta \varPsi , \quad \varPsi \varPhi = 1_{\mathbb {U}}^{2y-2x}, \quad \varPhi \varPsi = 1_{\mathbb {V}}^{2y-2x}. \end{aligned}$$

(4.2)

There are no other constraints.

   (ii) In addition to ${\mathbb {U}}, {\mathbb {V}}$ the persistence module ${\mathbb {W}}$ carries maps between the two components $\varDelta _x, \varDelta _y$. These maps are constrained by the composition law

$$ w^R_T = w^S_T \circ w^R_S $$

for all $R,S,T \in \varDelta _x \cup \varDelta _y$ with $R \le S \le T$.

First, observe that we recover the maps $\phi _t, \psi _t$ as vertical maps from $\varDelta _x$ to $\varDelta _y$, and horizontal maps from $\varDelta _y$ to $\varDelta _x$, respectively (see Fig. 4.1):

$$\begin{aligned} \begin{array}{llll} U_t &{}= W_{(t-x,t+x)} &{}\rightarrow W_{(t-x,t+2y-x)} &{}= V_{t+y-x} \\ V_t &{}= W_{(t-y,t+y)} &{}\rightarrow W_{(t+y-2x,t+y)} &{}= U_{t+y-x} \end{array} \end{aligned}$$

Fig. 4.1

The maps $\varPhi , \varPsi$ recovered from the module ${\mathbb {W}}$ over $\varDelta _x \cup \varDelta _y$

Next, observe that the composition law implies all of the relations (4.2).

Finally, there is no additional information in ${\mathbb {W}}$, beyond the interleaving maps and relations. Indeed, all remaining maps $w^S_T$, where $S \le T$, can all be factored in the form:

$$\begin{aligned} \begin{array}{llll} w^S_T &{}= v^{s+{y-x}}_{t} \circ \phi _s &{}\quad \text {if }S \in \varDelta _x \text { and } T \in \varDelta _y, \\ w^S_T &{}= u^{s+{y-x}}_{t} \circ \psi _s &{}\quad \text {if } S \in \varDelta _y \text { and } T \in \varDelta _x. \end{array} \end{aligned}$$

Thus each map in ${\mathbb {W}}$ is an instance of one of

$$\begin{aligned} \begin{array}{lll} &{} 1_{\mathbb {U}}^\eta &{}\quad \text {from }\varDelta _x\text { to } \varDelta _x, \\ &{} 1_{\mathbb {V}}^\eta &{}\quad \text {from } \varDelta _y\text { to } \varDelta _y, \\ &{} 1_{\mathbb {V}}^\eta \varPhi &{}\quad \text {from } \varDelta _x\text { to } \varDelta _y, \\ &{} 1_{\mathbb {U}}^\eta \varPsi &{}\quad \text {from }\varDelta _y\text { to } \varDelta _x. \end{array} \end{aligned}$$

It is a simple matter to verify that the composition law is satisfied for each composable pair of maps. For instance:

$$ (1_{\mathbb {V}}^\eta \varPhi ) (1_{\mathbb {U}}^\zeta \varPsi ) = 1_{\mathbb {V}}^\eta \varPhi 1_{\mathbb {U}}^\zeta \varPsi = 1_{\mathbb {V}}^\eta 1_{\mathbb {V}}^\zeta \varPhi \varPsi = 1_{\mathbb {V}}^{\eta +\zeta } 1_{\mathbb {V}}^{2y-2x} = 1_{\mathbb {V}}^{\eta +\zeta +2y-2x} $$

This can be done using only the known relations, so there are no further constraints on the $w^S_T$. $\square$

Remark 4.5

This characterisation makes it clear (or, in another view, depends on the fact) that all composable combinations of the maps $u, v, \phi , \psi$ from a given domain to a given codomain must be equal: indeed, they must agree with the appropriate map $w^S_T$ of ${\mathbb {W}}$.

4.4 The Interpolation Lemma

In this section we prove a crucial result from [14]:

Lemma 4.6

(interpolation lemma) Suppose ${\mathbb {U}}$, ${\mathbb {V}}$ are a $\delta$ -interleaved pair of persistence modules. Then there exists a 1-parameter family of persistence modules $\left( {\mathbb {U}}_x \mid {x \in [0,\delta ]} \right)$ such that ${\mathbb {U}}_{0}, {\mathbb {U}}_{\delta }$ are equal to ${\mathbb {U}}, {\mathbb {V}}$ respectively, and ${\mathbb {U}}_x, {\mathbb {U}}_y$ are $|y-x|$-interleaved for all $x, y \in [0,\delta ]$. Moreover, if ${\mathbb {U}}, {\mathbb {V}}$ are q-tame then the $({\mathbb {U}}_x)$ may be assumed q-tame also.

We prove something sharper: given a specific pair of interleaving maps

$$ {\varPhi \in {\text {Hom}}^\delta ({\mathbb {U}},{\mathbb {V}})} \qquad \qquad {\varPsi \in {\text {Hom}}^\delta ({\mathbb {V}},{\mathbb {U}})} $$

we explicitly provide, for each $x < y$ in $[0,\delta ]$, a pair of interleaving maps

$$ \varPhi ^x_y \in {\text {Hom}}^{y-x}({\mathbb {U}}_x,{\mathbb {U}}_y) \qquad \qquad \varPsi ^y_x \in {\text {Hom}}^{y-x}({\mathbb {U}}_y,{\mathbb {U}}_x) $$

such that $\varPhi ^0_\delta = \varPhi$ and $\varPsi ^\delta _0 = \varPsi$, and moreover

$$ \varPhi ^y_z \varPhi ^x_y = \varPhi ^x_z \qquad \qquad \varPhi ^y_x \varPhi ^z_y = \varPhi ^z_x $$

for all $x< y < z$. In view of Proposition 4.4, this sharp form of the interpolation lemma can be restated as follows.

Theorem 4.7

(interpolation lemma, version 2) Any persistence module ${\mathbb {W}}$ over $\varDelta _{0} \cup \varDelta _{\delta }$ extends to a persistence module $\overline{{\mathbb {W}}}$ over the diagonal strip

$$ \varDelta _{[0,\delta ]} = \left\{ (p,q) \mid 0 \le q-p \le 2\delta \right\} \subset {\mathbf {R}}^2. $$

If ${\mathbb {W}}|_{\varDelta _0}, {\mathbb {W}}|_{\varDelta _\delta }$ are q-tame, then the extension may be chosen so that each $\overline{{\mathbb {W}}}|_{\varDelta _x}$ is q-tame.

Remark 4.8

The extension is by no means unique.

Let us clarify how Theorem 4.7 implies Lemma 4.6. If ${\mathbb {U}}$, ${\mathbb {V}}$ are $\delta$-interleaved, then there exists a persistence module ${\mathbb {W}}$ over $\varDelta _{0} \cup \varDelta _{\delta }$ such that ${\mathbb {W}}|_{\varDelta _{0}} = {\mathbb {U}}$ and ${\mathbb {W}}|_{\varDelta _{\delta }} = {\mathbb {V}}$. By Theorem 4.7, this extends to $\overline{{\mathbb {W}}}$ over the strip $\varDelta _{[0,\delta ]}$. If we define a 1-parameter family ${\mathbb {U}}_x = \overline{{\mathbb {W}}}|_{\varDelta _x}$, then ${\mathbb {U}}_x, {\mathbb {U}}_y$ are $|x-y|$-interleaved for all $x,y \in [0,\delta ]$.

For readers familiar with Kan extensions in category theory [43], here is a very short proof of the theorem. Let us regard the posets $\varDelta _{0} \cup \varDelta _{\delta }$ and $\varDelta _{[0,\delta ]}$ as categories (see Remark 2.1); then persistence modules over these posets are the same as functors to the category of vector spaces. The theorem asserts the existence of an extension $\overline{{\mathbb {W}}}$

for any functor ${\mathbb {W}}$. Peter Bubenik has pointed out to us that the Kan extension theorem immediately provides two such extensions, since the category $\mathrm {Vect}$ is both complete (which yields the ‘right-extension’) and co-complete (which yields the ‘left-extension’).

We proceed now to a detailed proof, for those who would still like one.

Proof

(Theorem  4.7 ) In order to express the proof more symmetrically, it is convenient to replace the interval $[0,\delta ]$ by the interval $[-1,1]$. This can be done by rescaling and translating the plane. Accordingly, suppose we are given a persistence module ${\mathbb {W}}$ over $\varDelta _{-1} \cup \varDelta _{1}$.

Our strategy is to construct two persistence modules over the strip $\varDelta _{[-1, 1]}$ and a module map between them. The image (or coimage) of this map is itself a persistence module over the strip, and will be the required extension.

By Proposition 4.4, ${\mathbb {W}}$ provides ${\mathbb {U}}= {\mathbb {W}}|_{\varDelta _{-1}}$ and ${\mathbb {V}}= {\mathbb {W}}|_{\varDelta _{1}}$, which we can view as persistence modules over ${\mathbf {R}}$ using the canonical parametrisation:

$$\begin{aligned} \begin{array}{ll} U_t&= W_{(t+1,t-1)} \quad \text {and}\quad V_t = W_{(t-1, t+1)} \end{array} \end{aligned}$$

and corresponding linear maps $u^s_t, v^s_t$. The module ${\mathbb {W}}$ also provides interleaving maps $\varPhi \in {\text {Hom}}^2({\mathbb {U}}, {\mathbb {V}})$ and $\varPsi \in {\text {Hom}}^2({\mathbb {V}}, {\mathbb {U}})$ of degree 2:

$$ \phi _t = w^{(t+1,t-1)}_{(t+1,t+3)} : U_t \rightarrow V_{t+2}, \quad \psi _t = w^{(t-1,t+1)}_{(t+3,t+1)} : V_t \rightarrow U_{t+2}, $$

From ${\mathbb {U}}, {\mathbb {V}}$ we construct four persistence modules over ${\mathbf {R}}^2$:

$$\begin{aligned} \begin{array}{llllll} \mathbb {A}&{} &{} \quad \text {defined by} \quad A_{(p,q)} &{}= U_{p-1} &{} \quad \text {and} \quad a^{(p,q)}_{(r,s)} &{}= u^{p-1}_{r-1} \\ \mathbb {B}&{} &{} \quad \text {defined by} \quad B_{(p,q)} &{}= V_{q-1} &{} \quad \text {and} \quad b^{(p,q)}_{(r,s)} &{}= v^{q-1}_{s-1} \\ \mathbb {C}&{} &{} \quad \text {defined by} \quad C_{(p,q)} &{}= U_{q+1} &{} \quad \text {and} \quad c^{(p,q)}_{(r,s)} &{}= u^{q+1}_{s+1} \\ \mathbb {D}&{} &{} \quad \text {defined by} \quad D_{(p,q)} &{}= V_{p+1} &{} \quad \text {and} \quad d^{(p,q)}_{(r,s)} &{}= v^{p+1}_{r+1} \end{array} \end{aligned}$$

Note that $\mathbb {A}, \mathbb {B}, \mathbb {C}, \mathbb {D}$ are the vertical and horizontal extensions of the modules ${\mathbb {W}}|_{\varDelta _{\pm 1}}$ to the whole plane. In particular, we have

$$\begin{aligned} \begin{array}{lllll}&\mathbb {A}|_{\varDelta _{-1}} = {\mathbb {U}},&\quad \mathbb {B}|_{\varDelta _{1}} = {\mathbb {V}},&\quad \mathbb {C}|_{\varDelta _{-1}} = {\mathbb {U}},&\quad \mathbb {D}|_{\varDelta _{1}} = {\mathbb {V}}, \end{array} \end{aligned}$$

with respect to our canonical parametrisations of each diagonal. Restricting each module to its ‘other’ diagonal, on the other hand, we have

$$\begin{aligned} \begin{array}{lllll}&\mathbb {A}|_{\varDelta _1} = {\mathbb {U}}[-2],&\quad \mathbb {B}|_{\varDelta _{-1}} = {\mathbb {V}}[-2],&\quad \mathbb {C}|_{\varDelta _1} = {\mathbb {U}}[2],&\quad \mathbb {D}|_{\varDelta _{-1}} = {\mathbb {V}}[2], \end{array} \end{aligned}$$

using the ‘shifted module’ notation of Remark 4.1.

Next, we construct four module maps:

$$\begin{aligned} \begin{array}{llll} \overline{1}_{\mathbb {U}}&{}: \mathbb {A}\rightarrow \mathbb {C}&{}\quad \text {defined at }(p,q)\text { to be} \quad u^{p-1}_{q+1} &{}: U_{p-1} \rightarrow U_{q+1} \\ \overline{\varPhi }&{}: \mathbb {A}\rightarrow \mathbb {D}&{}\quad \text {defined at } (p,q)\text { to be} \quad \phi _{p-1} &{}: U_{p-1} \rightarrow V_{p+1} \\ \overline{\varPsi }&{}: \mathbb {B}\rightarrow \mathbb {C}&{}\quad \text {defined at } (p,q)\text { to be} \quad \psi _{q-1} &{}: V_{q-1} \rightarrow U_{q+1} \\ \overline{1}_{\mathbb {V}}&{}: \mathbb {B}\rightarrow \mathbb {D}&{}\quad \text {defined at } (p,q)\text { to be } \quad v^{q-1}_{p+1} &{}: V_{q-1} \rightarrow V_{p+1} \end{array} \end{aligned}$$

The maps $\overline{\varPhi }, \overline{\varPsi }$ are defined over the whole plane, whereas $\overline{1}_{\mathbb {U}}$ is defined only where $p-1\le q+1$, and $\overline{1}_{\mathbb {V}}$ is defined only where $q-1 \le p+1$. To verify that the four definitions give module maps, it is enough to observe that the required commutation relations involve composable combinations of the maps $u, v, \phi , \psi$, which always agree by Remark 4.5.

Note that the intersection of the regions of definition, where all four maps are defined, is precisely the strip $\varDelta _{[-1,1]}$. Henceforth, we restrict $\mathbb {A}, \mathbb {B}, \mathbb {C}, \mathbb {D}$ and the four maps to that strip.

Define $\varOmega \in {\text {Hom}}(\mathbb {A}\oplus \mathbb {B}, \mathbb {C}\oplus \mathbb {D})$ by the 2-by-2 matrix

$$ \left[ \begin{array}{cc} \overline{1}_{\mathbb {U}}&{} \overline{\varPsi }\\ \overline{\varPhi }&{} \overline{1}_{\mathbb {V}}\end{array} \right] $$

of module maps. Our claim is that $\overline{{\mathbb {W}}}= {\mathrm {im}}(\varOmega )$ is the required extension. We may equivalently claim that $\overline{{\mathbb {W}}}= {\mathrm {coim}}(\varOmega ) = (\AA \oplus \mathsf {B}) / {\mathrm {ker}}(\varOmega )$ is the required extension.¹

Step 1. $\overline{{\mathbb {W}}}|_{\varDelta _{-1}}$ is isomorphic to ${\mathbb {U}}$.

Proof

On the diagonal $\varDelta _{-1}$ we have

$$ (\mathbb {A}\oplus \mathbb {B})|_{\varDelta _{-1}} = {\mathbb {U}}\oplus {\mathbb {V}}[-2], \quad (\mathbb {C}\oplus \mathbb {D})|_{\varDelta _{-1}} = {\mathbb {U}}\oplus {\mathbb {V}}[2], $$

and the homomorphism $\varOmega |_{\varDelta _{-1}}$ takes the matrix form

$$ \left[ \begin{array}{cc} 1_{\mathbb {U}}&{} \varPsi \\ \varPhi &{} 1^4_{\mathbb {V}}\end{array} \right] . $$

Since $1^4_{\mathbb {V}}= \varPhi \varPsi$, this factorises as

where

$$ \varOmega _1 = [ \begin{array}{cc} 1_{\mathbb {U}}&\varPsi \end{array}] \quad \text {and}\quad \varOmega _2 = \left[ \begin{array}{c} 1_{\mathbb {U}}\\ \varPhi \end{array} \right] $$

in matrix form. Thanks to the $1_{\mathbb {U}}$ entries, we see that $\varOmega _1$ is surjective and $\varOmega _2$ is injective. This implies that $\varOmega _1, \varOmega _2$ induce isomorphisms

as required. $\square$

Step 2. $\overline{{\mathbb {W}}}|_{\varDelta _{1}}$ is isomorphic to ${\mathbb {V}}$.

Proof

On the diagonal $\varDelta _{1}$ we have

$$ (\mathbb {A}\oplus \mathbb {B})_{\varDelta _1} = {\mathbb {U}}[-2] \oplus {\mathbb {V}}, \quad (\mathbb {C}\oplus \mathbb {D})_{\varDelta _1} = {\mathbb {U}}[2] \oplus {\mathbb {V}}, $$

and the homomorphism $\varOmega |_{\varDelta _1}$ takes the matrix form

$$ \left[ \begin{array}{cc} 1^4_{\mathbb {U}}&{} \varPsi \\ \varPhi &{} 1_{\mathbb {V}}\end{array} \right] . $$

Since $1^4_{\mathbb {U}}= \varPsi \varPhi$, this factorises as

where

$$ \varOmega _3 = \left[ \begin{array}{cc} \varPhi&1_{\mathbb {V}}\end{array} \right] \quad \text {and}\quad \varOmega _4 = \left[ \begin{array}{c} \varPsi \\ 1_{\mathbb {V}}\end{array} \right] $$

in matrix form. Thanks to the $1_{\mathbb {V}}$ entries, we see that $\varOmega _3$ is surjective and $\varOmega _4$ is injective. This implies that $\varOmega _3, \varOmega _4$ induce isomorphisms

as required. $\square$

Step 3. The cross-maps of $\overline{{\mathbb {W}}}$ between $\varDelta _{-1}$ and $\varDelta _1$ correspond to $\varPhi$ and $\varPsi$ under the isomorphisms of Steps 1 and 2.

Proof

The cross maps for $\overline{{\mathbb {W}}}$ are induced by the cross maps for $\mathbb {A}\oplus \mathbb {B}$ (if we view $\overline{{\mathbb {W}}}$ as a coimage) and equally by the cross maps for $\mathbb {C}\oplus \mathbb {D}$ (if we view $\overline{{\mathbb {W}}}$ as an image). The vertical cross-map for $\mathbb {A}\oplus \mathbb {B}$ is a map

of degree 2 which we can identify as

Alternatively, the vertical cross-map for $\mathbb {C}\oplus \mathbb {D}$ is a map

of degree 2 which we can identify as

The following diagram shows the vertical cross-maps for $\mathbb {A}\oplus \mathbb {B}$ (on the left) and $\mathbb {C}\oplus \mathbb {D}$ (on the right), the factorisations of Steps 1 and 2, and the map $\varPhi$.

It is enough to show that either square commutes. And indeed

$$ \varOmega _3 (1_{\mathbb {U}}\oplus 1^4_{\mathbb {V}}) = \left[ \begin{array}{cc} \varPhi&1_{\mathbb {V}}\end{array}\right] \left[ \begin{array}{cc} 1_{\mathbb {U}}&{} 0 \\ 0 &{} 1^4_{\mathbb {V}}\end{array}\right] = [ \begin{array}{cc} \varPhi&1^4_{\mathbb {V}}\end{array} ] = [ \varPhi ] [ \begin{array}{cc} 1_{\mathbb {U}}&\varPsi \end{array}] = \varPhi \varOmega _1 $$

for the left square, and

$$ (1^4_{\mathbb {U}}\oplus 1_{\mathbb {V}}) \varOmega _2 = \left[ \begin{array}{cc} 1^4_{\mathbb {U}}&{} 0 \\ 0 &{} 1_{\mathbb {V}}\end{array}\right] \left[ \begin{array}{c} 1_{\mathbb {U}}\\ \varPhi \end{array}\right] = \left[ \begin{array}{c} 1^4_{\mathbb {U}}\\ \varPhi \end{array}\right] = \left[ \begin{array}{c} \varPsi \\ 1_{\mathbb {V}}\end{array}\right] [\varPhi ] = \varOmega _4 \varPhi $$

for the right square. Thus the induced vertical cross-map corresponds to $\varPhi$.

A similar argument using the diagram

shows that the induced horizontal cross-map corresponds to $\varPsi$. $\square$

This completes the construction of the extension $\overline{{\mathbb {W}}}$. Now we verify the last assertion of theorem. Suppose that ${\mathbb {U}}, {\mathbb {V}}$ are q-tame, meaning that their non-identity structure maps have finite rank. On any diagonal $\varDelta _x$, the restricted modules $\mathbb {A}|_{\varDelta _x}$, $\mathbb {B}|_{\varDelta _x}$ are shifted copies of ${\mathbb {U}},{\mathbb {V}}$ so they are q-tame. It follows that the direct sum $\mathbb {A}\oplus \mathbb {B}|_{\varDelta _x}$ and its homomorphic image ${\mathbb {U}}_x = \varOmega (\mathbb {A}\oplus \mathbb {B})|_{\varDelta _x}$ are q-tame.

This completes the proof of Theorem 4.7. $\square$

We point out that Step 3 isn’t necessary to deduce the interpolation Lemma 4.6. It is sufficient to show that ${\mathbb {U}}= {\mathbb {U}}_{-1}$ and ${\mathbb {V}}= {\mathbb {U}}_{1}$ for some 1-parameter family of persistence modules such that each pair ${\mathbb {U}}_x, {\mathbb {U}}_y$ admits an $|x-y|$-interleaving. This already follows from Steps 1 and 2. We do not need to know that the 2-interleaving of ${\mathbb {U}},{\mathbb {V}}$ induced from $\overline{{\mathbb {W}}}$ is equal to the original 2-interleaving.

4.5 The Interpolation Lemma (Continued)

In this optional section, we study the interpolation lemma in greater depth. The results are not used elsewhere. Given two modules ${\mathbb {U}}, {\mathbb {V}}$ and a $\delta$-interleaving between them, there are at least three natural ways to construct an interpolation. We describe these constructions and some relationships between them.

As in the proof of Theorem 4.7, we may suppose that $\delta = 2$ and that ${\mathbb {U}}, {\mathbb {V}}$ and their interleaving are represented as a module over $\varDelta _{-1} \cup \varDelta _{1}$ in the plane, which we wish to extend to a module over $\varDelta _{[-1,1]}$.

It will be helpful to introduce some temporary notation. Let ${\mathbb {V}}$ be a persistence module over ${\mathbf {R}}$. Then ${\mathbb {V}}^{\textsc {p}}$, ${\mathbb {V}}^{\textsc {q}}$ are the persistence modules over ${\mathbf {R}}^2$ defined by

$$ (V^\textsc {p})_{(p,q)} = V_p, \quad (V^\textsc {q})_{(p,q)} = V_q, $$

and the canonical linear maps.

Now consider the sequence

(4.3)

of modules over $\varDelta _{[-1,1]}$ with maps

$$ \varOmega ' = \left[ \begin{array}{rr} \overline{1}_{\mathbb {U}}&{} -\overline{\varPsi }\\ -\overline{\varPhi }&{} \overline{1}_{\mathbb {V}}\end{array} \right] , \quad \varOmega = \left[ \begin{array}{cc} \overline{1}_{\mathbb {U}}&{} \overline{\varPsi }\\ \overline{\varPhi }&{} \overline{1}_{\mathbb {V}}\end{array} \right] , \quad \varOmega '' = \left[ \begin{array}{rr} \overline{1}_{\mathbb {U}}&{} -\overline{\varPsi }\\ -\overline{\varPhi }&{} \overline{1}_{\mathbb {V}}\end{array} \right] $$

defined analogously to $\varOmega$ from the proof of Theorem 4.7.

Notice that $\varOmega$, $\varOmega '$ and $\varOmega ''$ are essentially the same map. Certainly $\varOmega ', \varOmega ''$ are formally identical, up to a translation $\tau$ of the strip. In fact, each of the modules in the sequence is related to the next by an isomorphism $\sigma$ which changes the sign of the ${\mathbb {V}}$-term and transforms indices by $(p,q) \mapsto (q+2,p+2)$. We have $\tau = \sigma ^2$, and conjugacies $\varOmega = \sigma \varOmega ' \sigma ^{-1}$ and $\varOmega '' = \sigma \varOmega \sigma ^{-1}$.

Proposition 4.9

Each of the three modules

$$ {\mathrm {coker}}(\varOmega '), \quad {\mathrm {coim}}(\varOmega ) = {\mathrm {im}}(\varOmega ), \quad {\mathrm {ker}}(\varOmega '') $$

over $\varDelta _{[-1,1]}$ defines an interpolating family between ${\mathbb {U}},{\mathbb {V}}$.

Proof

We already know this for ${\mathrm {coim}}(\varOmega ) = {\mathrm {im}}(\varOmega )$ from the proof of Theorem 4.7. Now we outline the proof that ${\mathrm {coker}}(\varOmega ')$ and ${\mathrm {ker}}(\varOmega '')$ restrict on $\varDelta _{-1}$ to modules isomorphic to ${\mathbb {U}}$.

On the diagonal $\varDelta _{-1}$ the sequence (4.3) restricts to:

and we have factorisations

$$ \varOmega ' \; \text {or} \; \varOmega '' = \left[ \begin{array}{rr} 1^4_{\mathbb {U}}&{} -\varPsi \\ -\varPhi &{} 1_{\mathbb {V}}\end{array} \right] = \left[ \begin{array}{r} -\varPsi \\ 1_{\mathbb {V}}\end{array} \right] \left[ \begin{array}{rr} -\varPhi&1_{\mathbb {V}}\end{array} \right] = \varOmega '_1 \varOmega '_2 \; \text {or} \; \varOmega ''_1 \varOmega ''_2. $$

These reveal that ${\mathrm {im}}(\varOmega ') = {\mathrm {im}}(\varOmega '_1)$ is a complementary submodule to ${\mathbb {U}}\oplus 0$ in ${\mathbb {U}}\oplus {\mathbb {V}}[t-2]$, and that ${\mathrm {ker}}(\varOmega '') = {\mathrm {ker}}(\varOmega ''_2)$ is a complementary submodule to $0 \oplus {\mathbb {V}}[t+2]$ in ${\mathbb {U}}\oplus {\mathbb {V}}[t+2]$. It follows that ${\mathrm {coker}}(\varOmega ')$ and ${\mathrm {ker}}(\varOmega '')$ are each isomorphic to ${\mathbb {U}}$.

By a symmetric argument, the restriction of each module to $\varDelta _1$ is isomorphic to ${\mathbb {V}}$. This completes the proof that ${\mathrm {coker}}(\varOmega ')$ and ${\mathrm {ker}}(\varOmega '')$ interpolate between ${\mathbb {U}}$ and ${\mathbb {V}}$. $\square$

Which of the three constructions should we prefer? It turns out that ${\mathrm {coker}}(\varOmega ')$ and ${\mathrm {ker}}(\varOmega '')$ are respectively isomorphic to the Kan left- and right-extensions, so these are natural from the category theoretic point of view. Now observe that $\varOmega \varOmega ' = 0$ and $\varOmega '' \varOmega = 0$, meaning that (4.3) is a chain complex. It follows that there is a natural projection and a natural inclusion

$$ {\mathrm {coker}}(\varOmega ') \twoheadrightarrow {\mathrm {coim}}(\varOmega ) = {\mathrm {im}}(\varOmega ) \hookrightarrow {\mathrm {ker}}(\varOmega '') $$

by which we see that ${\mathrm {coim}}(\varOmega ) = {\mathrm {im}}(\varOmega )$ is isomorphic to the image of the composite map ${\mathrm {coker}}(\varOmega ') \rightarrow {\mathrm {ker}}(\varOmega '')$. In this sense, it is intermediate between the left- and right-extensions; and structurally it is the ‘smallest’ of the three, being a quotient of one and a subobject of the other.

The surplus information carried by the two Kan extensions may be measured as the kernel of the projection and the cokernel of the inclusion. These are precisely the homology at the second and third terms of (4.3). It follows from the conjugacies described above that the two homology modules are isomorphic upon translating the strip by 2 and interchanging p and q (i.e. reversing the interpolation parameter).

We can use the ‘vineyard’ technique of [23] to visualise the 1-parameter family of persistence modules produced by each of the three constructions. We obtained the vineyards by sketching the supports of the eight module summands in (4.3) and using the sketches to partition the interpolation parameter range $[-1,1]$ into suitable intervals for case splitting. It is perhaps easier done than described, so we invite readers to conduct their own calculations and confirm that our vineyards are correct. As further corroboration, one verifies that the homology modules are isomorphic in the sense described above.

In Fig. 4.2, we consider the canonical 2-interleaving between interval modules $\mathbf {k}{[0,4)}$ and $\mathbf {k}{[1,6)}$. The thick black lines show how the points of the persistence diagram travel in the plane as we proceed along the interpolating family, for each of the three constructions. Each point travels with speed 1 and traverses a path of length 2 (in the ${\mathrm {d}^\infty }$-metric). The cokernel interpolation has an extra ‘ghost’ summand which emerges from the diagonal at (3, 3) at the beginning of the interpolation, and is reabsorbed by the diagonal at (2, 2) at the end.

Fig. 4.2

Vineyards of the cokernel (left), image (middle), and kernel (right) interpolations for the 2-interleaving between $\mathbf {k}{[0,4)}$ and $\mathbf {k}{[1,6)}$

In Fig. 4.3 we repeat the exercise using the canonical 3-interleaving between $\mathbf {k}{[0,4)}$ and $\mathbf {k}{[1,6)}$. The thick black paths now have length three, and the kernel and cokernel interpolations both produce ‘ghosts’ at the diagonal.

Fig. 4.3

Vineyards of the cokernel (left), image (middle), and kernel (right) interpolations for the 3-interleaving between $\mathbf {k}{[0,4)}$ and $\mathbf {k}{[1,6)}$