2024-08-30

The following is a brief summary of the paper The structure and stability of persistence modules.

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Persistence Modules

Persistence Module

Given a poset $T\subseteq \mathbb{R}$, a $T$-persistence module $\mathbb{V}$ is an indexed family of vector spaces (over some field $k$) $(V_t{)}_{t\in T}$ with a family of linear maps $(v_s^t:V_s\to V_t\mid s\le t)$ such that $v_s^t\circ v_t^r=v_s^t$ for all $s\le t\le r$ and $v_s^s=1_{V_s}$.

We typically visualize persistence modules as a chain of vector spaces:

$$V_{t_1}\to V_{t_2}\to V_{t_3}\to \cdots$$

Given two $T$-persistence modules $\mathbb{U}$ and $\mathbb{V}$, we can construct a persistence module homomorphism between $\mathbb{U}$ and $\mathbb{V}$ to be a collection of linear maps

$$\Phi =({\varphi }_t:U_t\to V_t\mid t\in T)$$

such that the linear maps are preserved, i.e., the diagram below commutes for all $s\le t$.

Under such a homomorphism, persistence modules form a category. We denote the homset by $\text{Hom}(\mathbb{U},\mathbb{V})$.

A particularly useful persistence module is the interval module. Given posets $J\subseteq T\subseteq \mathbb{R}$ where $J$ is an interval, we define the $T$-persistence module ${\mathbb{I}}_T^J$ with spaces

$$I_t=\{\begin{array}{ll}k& \text{if }t\in J\\ 0& \text{otherwise}\end{array}$$

and maps

$$i_s^t=\{\begin{array}{ll}{1}_k& \text{if }s,t\in J\\ 0& \text{otherwise.}\end{array}$$

For example, if $a<b<c$ the $\{a,b,c\}$-interval module $\mathbb{I}[a,b]$ is

$$k\stackrel{1}{\to }k\stackrel{0}{\to }0.$$

Interval modules represent features that are present over an interval $J$ that are absent elsewhere. For example, in simplicial complex filtrations, the homology groups decompose into a direct sum of interval modules. Each interval module represents a class of cycles that are alive only in the interval $J$.

We can compose two or more persistence modules with the direct sum.

Direct Sum of Persistence Modules

Given two $T$-persistence modules $\mathbb{U},\mathbb{V}$, the direct sum $\mathbb{W}=\mathbb{U}\oplus \mathbb{V}$ is the $T$-persistence module with spaces

$$(W_t=U_t\oplus V_t\mid t\in T)$$

and linear maps

$$(w_s^t:u_s^t\oplus v_s^t\mid s\le t).$$

We often do this the opposite direction. Given a persistence module $\mathbb{V}$, we want to find interval modules $({\mathbb{I}}^{J_{\ell }}{)}_{\ell \in L}$ such that

$$\mathbb{V}=\underset{\ell \in L}{⨁}{\mathbb{I}}^{J_{\ell }}.$$

We call such a persistence module decomposable. Note that interval modules are indecomposable because the projection map $(u,v)\mapsto (u,0)$ is either trivial or an isomorphism, i.e., an interval module can only decompose as $\mathbb{I}=\mathbb{I}\oplus 0$ or $\mathbb{I}=0\oplus \mathbb{I}$.

Gabriel, Auslander, Ringel-Tachikawa, Webb

Let $\mathbb{V}$ be a persistence module over $T\subseteq \mathbb{R}$. Then $\mathbb{V}$ can be decomposed as a direct sum of interval modules if

1. $T$ is a finite set; or
2. $T$ is locally finite and each space $V_t$ is finite-dimensional.

This decomposition is unique up to ordering.

Persistence Diagrams

We often want to visualize the structure of persistence modules. For example, assume we have a persistence module composed of sublevelsets of a function $f:X\to \mathbb{R}$,

$${\mathbb{X}}_{\text{sub}}^f:X^{t_1}\to X^{t_2}\to X^{t_3}\to \cdots$$

where the maps are given by inclusion $X^{t_i}\hookrightarrow X^{t_j}$ when $i\le j$. If we apply the $p$-dimensional singular homology functor, we get a new persistence module of singular homology groups

$$H_p({\mathbb{X}}_{\text{sub}}^f):H_p(X^{t_1})\to H_p(X^{t_2})\to H_p(X^{t_3})\to \cdots .$$

One may show that each index $t_i$ corresponds to a critical point in the space $X$. So by plotting a point $(i,j)$ in the extended half-plane $\overline{\mathcal{H}}$ if a cycle is born at $t_i$ and dies at $t_j$, we get what is known as a persistence diagram. For example,

The above figure is an example of a decorated persistence diagram where the points are given a tick direction. In an undecorated persistence diagram the points are not given a tick direction. Note that we have two points at infinity. This implies that there is a 0-cycle born at time $a$ that never dies. Likewise there is a 1-cycle born at time $f$ that never dies.

Question: What exactly do the tick directions represent?

For persistence modules over a poset $T\subseteq \mathbb{R}$, it is important to distinguish the topology of the intervals (open, closed, half-open). We do so with decorated real numbers. Given that $p\le q$ we have the following four cases:

$$\begin{array}{r}(p^{-},q^{-})\text{ means }[p,q)\\ (p^{-},q^{+})\text{ means }[p,q]\\ (p^{+},q^{-})\text{ means }(p,q)\\ (p^{+},q^{+})\text{ means }(p,q]\end{array}$$

If the decoration is unknown (or we do not wish to specify), then we use the notation $(p^{\ast },q^{\ast })$ and any of the four options above are possibilities.

Assume that $\mathbb{V}$ is a decomposable persistence module over $T\subseteq \mathbb{R}$, i.e.,

$$\mathbb{V}=\underset{\ell \in L}{⨁}\mathbb{I}(p_{\ell }^{\ast },q_{\ell }^{\ast }).$$

Then we define the decorated persistence diagram of $\mathbb{V}$ to be the multiset

$$\text{Dgm}(\mathbb{V})=\{(p^{\ast },q^{\ast })\mid \ell \in L\}$$

and the undecorated persistence diagram of $\mathbb{V}$ to be the multiset

$$\text{dgm}(\mathbb{V})=\{(p,q)\mid \ell \in L\}\setminus \Delta$$

where $\Delta =\{(x,x)\in \overline{\mathcal{H}}\}$ is the diagonal in the extended half-plane.

Since the decomposition of persistence modules is unique up to order, these definitions are well-defined. In decorated persistence diagrams, the tick direction is drawn depending on the decoration of the interval. The possible tick directions are

Rectangle Measures

Let $\mathcal{D}\subset {\mathbb{R}}^2$. Define

$$\text{Rect}(\mathcal{D})=\{[a,b]\times [c,d]\subset \mathcal{D}\mid a<b,c<d\},$$

i.e., the set of closed rectangles contained in $\mathcal{D}$. A rectangle measure or $r$-measure on $\mathcal{D}$ is a function $\mu :\text{Rect}(\mathcal{D})\to \{0,1,2,\dots \}\cup \{\infty \}$ such that whenever $a<p<b\le c<q<d$ we get that

$$\begin{array}{rl}\mu ([a,b]\times [c,d])& ={\mu }_{\mathbb{V}}([a,p]\times [c,d])+{\mu }_{\mathbb{V}}([p,b]\times [c,d])\\ \mu ([a,b]\times [c,d])& ={\mu }_{\mathbb{V}}([a,b]\times [c,q])+{\mu }_{\mathbb{V}}([a,b]\times [q,d]).\end{array}$$

In other words, the function is additive with respect to horizontal and vertical splitting.

We define the interior of $\mathcal{D}$ to be

$${\mathcal{D}}^{\circ }=\{(p,q)\mid \exists R\in \text{Rect}(\mathcal{D})\text{s.t.}(p,q)\in R^{\circ }\}$$

where $R^{\circ }$ is the interior of the rectangle $R$. In other words, the set of points in $\mathcal{D}$ such that there is an open rectangle contained inside $\mathcal{D}$ that contains said point.

For decorated points we use the $r$-interior

$${\mathcal{D}}^{■}=\{(p^{\ast },q^{\ast })\mid \exists R\in \text{Rect}(\mathcal{D})\text{s.t.}(p^{\ast },q^{\ast })\in R\}.$$

which behaves the same as the interior but we use the closed rectangle and require the tick to lie inside the rectangle as well.

Equivalence Theorem

Let $\mathcal{D}\subseteq {\mathbb{R}}^2$. There is a bijective correspondence between

• Finite $r$-measures on $\mathcal{D}$
• Locally finite multisets $A$ in ${\mathcal{D}}^{■}$

The measure $\mu$ corresponding to the multiset $A$ is related by the formula

$$\mu (R)=\text{card}(A{|}_R)=\sum _{(p^{\ast },q^{\ast })\in R}m((p^{\ast },q^{\ast }))$$

where $m:A\to \{0,1,2,\dots \}$ is the multiplicity function, for all rectangles $R\in \text{Rect}(\mathcal{D})$.

Therefore, one may view a finite $r$-measure of a rectangle $R$ as simply counting the number of points in $A$ that lie inside $R$.

One may extend the equivalence theorem to non-finite $r$-measures by restricting the $r$-interior ${\mathcal{D}}^{■}$ to only rectangles of finite size

$$\begin{array}{rl}{\mathcal{F}}^{\circ }(\mu )& =\{(p,q)\mid \exists R\in {\mathcal{D}}^{\circ }\text{ such that }\mu (R)<\infty \}\\ {\mathcal{F}}^{■}(\mu )& =\{(p^{\ast },q^{\ast })\mid \exists R\in {\mathcal{D}}^{■}\text{ such that }\mu (R)<\infty \}.\end{array}$$

and using horizontal or vertical splitting.

We define the decorated diagram of an $r$-measure $\mu$ to be the unique multiset $\text{Dgm}(\mu )$ that satisfies

$$\mu (R)=\text{card}(\text{Dgm}(\mu ){|}_R)$$

as described in the equivalence theorem. The undecorated diagram of $\mu$ then ignores the ticks and restricts to the interior, i.e.,

$$\text{dgm}(\mu )=\{(p,q)\mid (p^{\ast },q^{\ast })\in \text{Dgm}(\mu )\}\cap {\mathcal{F}}^{\circ }(\mu ).$$

The Persistence Measure

Let $\mathbb{V}$ be a $T$-persistence module where $T$ is a finite subset of $\mathbb{R}$, i.e.,

$$\mathbb{V}:V_{t_1}\to V_{t_2}\to \cdots \to V_{t_n}.$$

We can represent this module as a quiver

$$\bullet \text{—}\bullet \text{—}\cdots \text{—}\bullet$$

where a filled circle $\bullet$ indicates where the module has non-zero rank and the unfilled circle $\circ$ indicate where the module has rank 0. For example if $a<b<c$, the six possible interval modules over $\{a,b,c\}$ are

$$\begin{array}{rlrlrl}\mathbb{I}[a,a]& ={\bullet }_a\text{—}{\circ }_b\text{—}{\circ }_c& \mathbb{I}[a,b]& ={\bullet }_a\text{—}{\bullet }_b\text{—}{\circ }_c& \mathbb{I}[a,c]& ={\bullet }_a\text{—}{\bullet }_b\text{—}{\bullet }_c\\ \mathbb{I}[b,b]& ={\circ }_a\text{—}{\bullet }_b\text{—}{\circ }_c& \mathbb{I}[b,c]& ={\circ }_a\text{—}{\bullet }_b\text{—}{\bullet }_c& & \\ \mathbb{I}[c,c]& ={\circ }_a\text{—}{\circ }_b\text{—}{\bullet }_c& & & & \end{array}$$

We write

$$⟨[a_i,a_j]\mid \mathbb{V}⟩\text{ or }⟨\cdots \text{—}{\bullet }_{a_i}\text{—}\cdots \text{—}{\bullet }_{a_j}\text{—}\cdots \mid \mathbb{V}⟩$$

to indicate the number of copies of $\mathbb{I}[a_i,a_j]$ in the interval decomposition of $\mathbb{V}$.

For a persistence module $\mathbb{V}$, we define the persistence measure of $\mathbb{V}$ to be the $r$-measure

$${\mu }_{\mathbb{V}}(R)=⟨{\circ }_a\text{—}{\bullet }_b\text{—}{\bullet }_c\text{—}{\circ }_d\mid \mathbb{V}⟩$$

for rectangles $R=[a,b]\times [c,d]$ with $a<b\le c<d$. It was shown by Gunnar Carlsson and Vin de Silva that

$${\mu }_{\mathbb{V}}(R)=\text{dim}\left[\frac{\text{Im}(v_b^c)\cap \text{Ker}(v_c^d)}{\text{Im}(v_a^c)\cap \text{Ker}(v_c^d)}\right]$$

where $R=[a,b]\times [c,d]$. Moreover, if the spaces $V_a,V_b,V_c,V_d$ are finite-dimensional (or at least $r_b^c<\infty$), then

$${\mu }_{\mathbb{V}}(R)=r_b^c-r_a^c-r_b^d+r_a^d$$

where $r_s^t=\text{rank}(v_s^t:V_s\to V_t)$.

For a decomposable persistence module over $\mathbb{R}$, one may show that the persistence measure corresponds to the decorated diagram as in the equivalence theorem, i.e.,

$${\mu }_{\mathbb{V}}(R)=\text{card}(\text{Dgm}(\mathbb{V}){|}_R).$$

Hence, one may say that

$$\text{Dgm}(\mathbb{V})=\text{Dgm}({\mu }_{\mathbb{V}}).$$

Tameness

We say a persistence module $\mathbb{V}$ is finite if it is a finite direct sum of interval modules. We say it is locally finite if and only if

1. each $V_t$ is finite-dimensional
2. there is a locally finite set $S\subset \mathbb{R}$ such that $v_b^c:V_b\to V_c$ is an isomorphism for every pair $b<c$ with $[b,c]\cap S=\varnothing$.

There are four notions of tameness.

1. We say $\mathbb{V}$ is $q$-tame if ${\mu }_{\mathbb{V}}(Q)<\infty$ for every quadrant $Q$ not touching the diagonal, i.e., for all $b<c$

$$\text{rank}(V_b\to V_c)=⟨{\bullet }_b\text{—}{\bullet }_c\mid \mathbb{V}⟩<\infty .$$

2. We say $\mathbb{V}$ is $h$-tame if ${\mu }_{\mathbb{V}}(H)<\infty$ for every horizontally infinite strip $H$ not touching the diagonal, i.e., for all $b<c<d$

$$⟨{\bullet }_b\text{—}{\bullet }_c\text{—}{\circ }_d\mid \mathbb{V}⟩<\infty$$

3. We say $\mathbb{V}$ is $v$-tame if ${\mu }_{\mathbb{V}}(V)<\infty$ for every vertically infinite strip $V$ not touching the diagonal, i.e., for all $a<b<c$

$$⟨{\circ }_a\text{—}{\bullet }_b\text{—}{\bullet }_c\mid \mathbb{V}⟩<\infty$$

4. We say that $\mathbb{V}$ is $r$-tame if ${\mu }_{\mathbb{V}}(R)<\infty$ for every finite rectangle $R$ not touching the diagonal, i.e., for all $a<b<c<d$

$$⟨{\circ }_a\text{—}{\bullet }_b\text{—}{\bullet }_c\text{—}{\circ }_d\mid \mathbb{V}⟩<\infty$$

The tameness conditions have the following inclusion diagram:

We will be particularly interested in $q$-tame modules.

Interleaving

We say that two persistence modules $\mathbb{U},\mathbb{V}$ are isomorphic if there are homomorphisms

$$\Phi \in \text{Hom}(\mathbb{U},\mathbb{V}),\Psi \in \text{Hom}(\mathbb{V},\mathbb{U})$$

such that

$$\Psi \Phi =1_{\mathbb{U}}\text{ and }\Phi \Psi =1_{\mathbb{V}}.$$

That is the diagram

commutes for all indices $a\le b$.

Such a relation in practice is too strong. Instead we use a weaker relation that allows a level of uncertainty. Let $\mathbb{U},\mathbb{V}$ be persistence modules over $T\subset \mathbb{R}$ and $\delta \ge 0$. A homomorphism of degree $\delta$ is a collection of linear maps

$$\Phi =({\varphi }_t:U_t\to V_{t+\delta }\mid t\in T)$$

such that the diagram

commutes for all $a\le b$. The set of all homomorphisms of degree $\delta$ between $\mathbb{U}$ and $\mathbb{V}$ is denoted by ${\text{Hom}}^{\delta }(\mathbb{U},\mathbb{V})$. Note that the composition of a homomorphism of degree ${\delta }_1$ and a homomorphism of degree ${\delta }_2$ gives a homomorphism of degree ${\delta }_1+{\delta }_2$.

For $\delta \ge 0$ the simplest homomorphism of degree $\delta$ is called the shift map, $1_{\mathbb{V}}^{\delta }\in {\text{Hom}}^{\delta }(\mathbb{V},\mathbb{V})$, which is given by the maps

$$1_{\mathbb{V}}^{\delta }=(v_t^{t+\delta }:V_t\to V_{t+\delta }\mid t\in T)$$

from the underlying persistence diagram structure.

We say that two persistence modules $\mathbb{U},\mathbb{V}$ are $\delta$-interleaved if there are two homomorphisms

$$\Phi \in {\text{Hom}}^{\delta }(\mathbb{U},\mathbb{V}),\Psi \in {\text{Hom}}^{\delta }(\mathbb{V},\mathbb{U})$$

such that

$$\Psi \Phi =1_{\mathbb{U}}^{2\delta },\Phi \Psi =1_{\mathbb{V}}^{2\delta }.$$

In other words, for all $a\le b$ the four diagrams below commute:

Given two $\delta$-interleaved persistence modules, we can continuously interpolate between the two.

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 $({\mathbb{W}}_x\mid x\in [0,\delta ])$ such that ${\mathbb{W}}_0,{\mathbb{W}}_{\delta }$ are equal to $\mathbb{U},\mathbb{V}$ respectively, and ${\mathbb{W}}_x,{\mathbb{W}}_y$ are $|y-x|$-interleaved.

More specifically, if $\Phi \in {\text{Hom}}^{\delta }(\mathbb{U},\mathbb{V})$ and $\Psi \in {\text{Hom}}^{\delta }(\mathbb{V},\mathbb{U})$ are the interleaving maps, then for each $x<y$ in $[0,\delta ]$ we get a pair of interleaving maps

$${\Phi }_x^y\in {\text{Hom}}^{y-x}({\mathbb{W}}_x,{\mathbb{W}}_y),{\Psi }_y^x\in {\text{Hom}}^{y-x}({\mathbb{W}}_y,{\mathbb{W}}_x)$$

such that ${\Phi }_0^{\delta }=\Phi$, ${\Psi }_{\delta }^0=\Psi$ and

$${\Phi }_x^y={\Phi }_z^y{\Phi }_x^z\text{ and }{\Psi }_y^x={\Psi }_z^x{\Psi }_y^z$$

for all $x<z<y$. That is, we get a persistence module over ${\mathbb{R}}^2$.

Let $\mathbb{U},\mathbb{V}$ be persistence modules. We define the interleaving distance of $\mathbb{U}$ and $\mathbb{V}$ to be

$$\begin{array}{rl}{d}_i(\mathbb{U},\mathbb{V})& =\inf \{\delta \mid \text{U and V are δ-interleaved}\}\\ & =\min \{\delta \mid \text{U and V are (δ+ϵ)-interleaved for all ϵ>0}\}.\end{array}$$

If no such $\delta$-interleaving exists, we say that $d_i(\mathbb{U},\mathbb{V})=+\infty$.

The interleaving distance forms a pseudometric. It is symmetric and it satisfies the triangle inequality. However, there are non-isomorphic modules with an interleaving distance of zero.

The Bottleneck Distance

The bottleneck distance between two persistence modules is measured using their persistence diagrams. That is, given persistence modules $\mathbb{U}$ and $\mathbb{V}$, the bottleneck distance looks at all possible pairings in $\text{dgm}(\mathbb{U})$ and $\text{dgm}(\mathbb{V})$ and finds their distance. For unmatched pairs, we take half the distance from the diagonal.

We define a partial matching between multisets $A$ and $B$ to be a collection $M\subset A\times B$ of ordered pairs such that

1. For every $\alpha \in A$ there is at most one $\beta \in B$ such that $(\alpha ,\beta )\in M$.
2. For every $\beta \in B$ there is at most one $\alpha \in A$ such that $(\alpha ,\beta )\in M$.

Given a partial matching $M$, we say that it is a $\delta$-matching if

1. If $(\alpha ,\beta )\in M$, then $d^{\infty }(\alpha ,\beta )\le \delta$
2. If $\alpha \in A$ is unmatched, then $d^{\infty }(\alpha ,\Delta )\le \delta$
3. If $\beta \in B$ is unmatched, then $d^{\infty }(\beta ,\Delta )\le \delta$.

Here we are using the ${\ell }^{\infty }$ metric on ${\mathbb{R}}^2$,

$$d^{\infty }((p,q),(r,s))=\max (|p-r|,|q-s|).$$

In the case that one or more of the points are at infinity, we use the following distances:

$$\begin{array}{rl}{d}^{\infty }((-\infty ,q),(-\infty ,s))& =|q-s|\\ d^{\infty }((p,\infty ),(r,\infty ))& =|p-r|\\ d^{\infty }((-\infty ,\infty ),(-\infty ,\infty ))& =0\\ d^{\infty }((p,q),(-\infty ,s))=\infty & \\ d^{\infty }((p,q),(r,\infty ))=\infty & \end{array}$$

For the unmatched points, we take half the ${\ell }^{\infty }$ distance down to the diagonal. That is,

$$d^{\infty }((p,q),\Delta )=\frac{1}{2}(q-p).$$

The bottleneck distance is then taken to be the smallest possible $\delta$-matching, i.e., given multisets $A$ and $B$,

$$d_b(A,B)=\inf \{\delta \mid \text{there is a δ-matching between A and B}\}.$$

Like the interleaving distance, the bottleneck distance satisfies the triangle inequality.

The Isometry Theorem

In the case that $\mathbb{U}$ and $\mathbb{V}$ are $q$-tame, the interleaving and bottleneck distances correspond.

Isometry Theorem

If $\mathbb{U},\mathbb{V}$ are $q$-tame persistence modules, then

$$d_i(\mathbb{U},\mathbb{V})=d_b(\text{dgm}(\mathbb{U}),\text{dgm}(\mathbb{V})).$$

This theorem is the combination of two parts: the stability theorem

$$d_i(\mathbb{U},\mathbb{V})\ge d_b(\text{dgm}(\mathbb{U}),\text{dgm}(\mathbb{V}))$$

and the converse stability theorem

$$d_i(\mathbb{U},\mathbb{V})\le d_b(\text{dgm}(\mathbb{U}),\text{dgm}(\mathbb{V})).$$