The reflection distance between zigzag persistence modules

Abstract

By invoking the reflection functors introduced by Bernstein et al. (Russ Math Surv 28(2):17–32, 1973), in this paper we define a metric on the space of all zigzag modules of a given length, which we call the reflection distance. We show that the reflection distance between two given zigzag modules of the same length is an upper bound for the $\ell ^1$-bottleneck distance between their respective persistence diagrams.

Appendices

Appendices

Limits and colimits

We assume the reader is familiar with (small) categories, functors, and natural transformations (see Mac Lane 1971; Riehl 2017 for details). We denote by $\mathbf {Vect}$ the category of vector spaces over some fixed field $\mathbb {F}$.

Definition A1.1

Fix a small category $\mathcal {J}$.

• 1. A diagram of vector spaces of shape$\mathcal {J}$ is a functor $D:\mathcal {J}\rightarrow \mathbf Vect$.

• 2. A cone over a diagram $D:\mathcal {J}\rightarrow \mathbf Vect$ is a vector space N together with a collection of linear transformations $\lambda :=\{\lambda _J: N\rightarrow D(J)\ | \ J\in \text {Ob}(\mathcal {J})\}$, indexed by the objects of $\mathcal {J}$, such that for any morphism $f:A\rightarrow B$ in $\text {Hom}(\mathcal {J})$ we have $\lambda _B = D(f)\circ \lambda _A$. We denote such a cone by $(N,\lambda )$.

• 3. A cocone over a diagram $D:\mathcal {J}\rightarrow \mathbf Vect$ is a vector space M together with a collection of linear transformations $\gamma :=\{\gamma _J: D(J)\rightarrow M \ | \ J\in \text {Ob}(\mathcal {J})\}$, indexed by the objects of $\mathcal {J}$, such that for any morphism $f:A\rightarrow B$ in $\text {Hom}(\mathcal {J})$ we have $\gamma _A = \gamma _B\circ D(f)$. We denote such a cocone by $(M,\lambda )$.

Limits and colimits are then universal cones or cocones, respectively:

Definition A1.2

Let $D:\mathcal {J}\rightarrow \mathbf {Vect}$ be a diagram of vector spaces.

• 1. The limit of the diagram D is a cone $(L,\phi )$ over D such that for any other cone $(N,\lambda )$ over D, there exists a unique linear transformation $\psi :N\rightarrow L$ with $\lambda _A = \phi _A \circ \psi$ for all $A\in \text {Ob}(\mathcal {J})$. We denote the limit L by $\,\lim D$.

• 2. The colimit of the diagram D is a cocone $(C,\phi )$ over D such that for any other cocone $(M,\lambda )$ over D, there exists a unique linear transformation $\psi :M\rightarrow C$ with $\lambda _A = \phi _A \circ \psi$ for all $A\in \text {Ob}(\mathcal {J})$. We denote the colimit C by $\text {colim}(D)$.

Theorem A1.1

Let $\mathcal {J}$ be a small category and let $D^1,D^2$ be diagrams of vector spaces. If there exists a natural transformation $\eta :D^1\rightarrow D^2$ all of whose components are monomorphisms, then $(L^1,\eta _J\circ \lambda ^1_J)$ is a cone for $D^2$ and the unique morphism $\psi :L^1\rightarrow L^2$ satisfying $\eta _J\circ \lambda ^1_J = \lambda ^2_J\circ \psi$ for all $J\in \mathcal {J}$ is a monomorphism.

Proof

The proof can be found in more generality in Borceux et al. (1994 p. 89, Corollary 2.15.3). $\square$

Let $\mathbf {2}$ denote the discrete category with 2 objects. That is, $\text {Ob}(\mathbf {2}) = \{1,2\}$ and $\text {Hom}(1,2) = \emptyset$.

Definition A1.3

Let X, Y be objects in the category $\,\mathcal {C}$ and let $D:\mathbf {2}\rightarrow \mathcal {C}$ be the diagram given by $D(1) = X$ and $D(2) = Y$.

• 1. The product of B and C, denoted $B\times C$, is the limit of D, if it exists,

• 2. The coproduct of B and C, denoted $B\amalg C$, is the colimit of D, if it exists.

In $\mathbf {Vect}$, products and coproducts always exist and coincide.

Theorem A1.2

Let $\mathcal {J}$ be a small category and let $D^1, D^2$ be diagrams of vector spaces. Then $\lim (D^1\times D^2)\cong \lim (D^1)\times \lim (D^2)$ Dually, $\text {colim}(D^1\amalg D^2)\cong \text {colim}(D^1)\amalg \text {colim}(D^2)$. Here $D^1\times D^2$ and $D^1\amalg D^2$ denote the product and coproduct, respectively, of $D^1$ and $D^2$ in the functor category $\mathcal {C}^\mathcal {J}$.

Proof

Consider the product category $\mathbf {2}\times J$ and let $F:\mathbf {2}\times \mathcal {J}\rightarrow \mathcal {C}$ be the functor given by $F(i,j) = D^i(j)$ for $i = 1,2$ and $j\in \mathcal {J}$. By commutativity of limits,

$$\begin{aligned} \lim _\mathbf {2}\lim _\mathcal {J} F(i,j) = \lim _\mathcal {J}\lim _\mathbf {2}F(i,j) \end{aligned}$$

[see Riehl (2017 p. 111, Theorem 3.8.1)]. Now

$$\begin{aligned} \lim (D^1\times D^2)&= \lim _{\mathcal {J}}(D^1\times D^2) = \lim _{\mathcal {J}}\lim _\mathbf {2} F(i,j)\\&= \lim _{\mathbf {2}}\lim _\mathcal {J} F(i,j) = \lim _{\mathbf {2}}\lim D^i\\&= \lim (D^1)\times \lim (D^2). \end{aligned}$$

The second statement follows by a duality argument. $\square$

Matchings

The Matching Lemma is a useful tool for combining matchings in opposite directions into a single matching. For us, the lemma was crucial for proving our stability result, while Bjerkevik makes use of the lemma in proving his generalization of the algebraic stability theorem (Bjerkevik 2016). While it may be one of the earliest results in infinite matching theory (Aharoni 1991), we believe its applications to stability-type questions in persistence theory make it worth expounding upon here.

Lemma A2.1

(Matching lemma for finite matchings) Let S and T be sets and let $f:S\nrightarrow T$ and $g:T\nrightarrow S$ be finite matchings. Then there exists a matching $M:S\nrightarrow T$ such that

• 1. $\text {coim}(f)\subseteq \text {coim}(M)$,

• 2. $\text {coim}(g)\subseteq \text {im}(M)$,

• 3. if $M(s) = t$ then either $f(s) = t$ or $g(t) = s$.

Proof

The proof is inspired by a proof of the Cantor–Schröder–Bernstein theorem³ given in Schweizer (2000). Let $s\in \text {coim}(f)$, $t\in \text {coim}(g)$, and consider sequences of the form

$$\begin{aligned} \cdots {\mathop {\longrightarrow }\limits ^{g}}f^{-1}(g^{-1}(s)){\mathop {\longrightarrow }\limits ^{f}} g^{-1}(s){\mathop {\longrightarrow }\limits ^{g}}s{\mathop {\longrightarrow }\limits ^{f}} f(s) {\mathop {\longrightarrow }\limits ^{g}} g(f(s)) {\mathop {\longrightarrow }\limits ^{f}} \cdots \end{aligned}$$

and

$$\begin{aligned} \cdots {\mathop {\longrightarrow }\limits ^{f}}g^{-1}(f^{-1}(t)){\mathop {\longrightarrow }\limits ^{g}} f^{-1}(t){\mathop {\longrightarrow }\limits ^{f}}t{\mathop {\longrightarrow }\limits ^{g}} g(t) {\mathop {\longrightarrow }\limits ^{f}} f(g(t)) {\mathop {\longrightarrow }\limits ^{g}} \cdots , \end{aligned}$$

where we allow these sequences to terminate to the right or left when undefined. We refer to these sequences as the orbits of s or t. Since f and g are finite matchings, such sequences either terminate on both the left and right, or are infinite but periodic. By injectivity of f and g, every element of $\text {coim}(f)\sqcup \text {coim}(g)$ appears in exactly one orbit. Moreover, every orbit falls into one of the following five classes:

• 1. $s\rightarrow t\rightarrow \cdots \rightarrow s\rightarrow y$,

• 2. $s\rightarrow t\rightarrow \cdots \rightarrow s\rightarrow t\rightarrow x$,

• 3. $t\rightarrow s\rightarrow \cdots \rightarrow t\rightarrow x$,

• 4. $t\rightarrow s\rightarrow \cdots \rightarrow t\rightarrow s\rightarrow y$,

• 5.

  

where the s’s and t’s represent elements of $\text {coim}(f)$ and $\text {coim}(g)$, respectively, arrows represent either f or g, and y’s and x’s represent elements of $\text {im}(f){\setminus } \text {coim}(g)$ and $\text {im}(g){\setminus } \text {coim}(f)$, respectively. We define a matching $M:S\nrightarrow T$ as follows: for each $i = 1,\ldots ,5$ let

$$\begin{aligned} S_i^{\text {co}}:= \{s\in \text {coim}(f) \ | \ s \text { appears in an orbit of type } i\} \end{aligned}$$

and

$$\begin{aligned} T_i^{\text {co}}:= \{t\in \text {coim}(g) \ | \ t \text { appears in an orbit of type } i\}. \end{aligned}$$

Then and . We partition $S_1^{\text {co}}$ and $T_3^{\text {co}}$ further by defining

$$\begin{aligned} S_1^{\text {co}\backslash \text {p}} : = \{s\in S_1^{\text {co}} \ | \ f(s)\in \text {coim}(g)\}, \quad S_1^{\text {p}} : = \{s\in S_1^{\text {co}} \ | \ f(s)\not \in \text {coim}(g)\},\\ T_3^{\text {co}\backslash \text {p}} : = \{t\in T_3^{\text {co}} \ | \ g(t)\in \text {coim}(f)\}, \quad \text { and } \quad T_3^{\text {p}} : = \{t\in T_3^{\text {co}}\ | \ g(t)\not \in \text {coim}(f)\}, \end{aligned}$$

so that and . The image, coimage, and mapping of M is specified by the diagram

Note that $f(S_1^{\text {co}\backslash \text {p}}) = T_1^{\text {co}}$, $f(S_2^{\text {co}}) = T_2^{\text {co}}$, $g(T_3^{\text {co}\backslash \text {p}}) = S_3^{\text {co}}$, $g(T_4^{\text {co}}) = S_4^{\text {co}}$, and $f(S_5^{\text {co}}) = T_5^{\text {co}}$ so that M is surjective. Since f and g are injective, we see that M is a bijection between each of the parts specified and thus defines a matching. Moreover, and . Property (3) evidently holds by the definition of M. $\square$