Comparison between "Parametrized homology via zigzag persistence" and "Refined homology in the presence of a real-valued continuous function"

TL;DR

This paper demonstrates that certain persistence diagrams and measures from previous works are special cases of more general maps and measures, unifying different approaches in parametrized and refined homology.

Contribution

It shows that the persistence diagrams and measures in [5] are specific instances of the more general frameworks discussed in [2] and [1], unifying different homology approaches.

Findings

Persistence diagrams in [5] are special cases of maps in [2] and [1]

Measures in [5] are particular cases of those in [2] and [1]

Unification of parametrized and refined homology frameworks

Abstract

In this note one shows that the four persistence diagrams and measures defined in [5] are particular cases of the maps and the measures discussed in [2] and [1]

Equations

$$Dg{m}^{\wedge }(H\mathbb{X}),Dg{m}^{\vee }(H\mathbb{X}),Dg{m}^{\backslash \backslash }(H_r\mathbb{X}),Dg{m}^{//}(H\mathbb{X})$$

$${\mu }_{H_r}^c(R)=$$

$${\mu }_{H_r}^{c,o}(R)=$$

$${\mu }_{H_r}^o(R)=$$

$${\mu }_{H_r}^{o,c}(R)=$$

$$Dm{g}_r^c(a,b)=$$

$$Dm{g}_r^{c,o}(a,b)=$$

$$Dm{g}_r^o(a,b)=$$

$$Dm{g}_r^{o,c}(a,b)=$$

$${\mathbb{F}}_r(B):={\mathbb{F}}_r(b,c)/({\mathbb{F}}_r(a,c)+{\mathbb{F}}_r(b,d));$$

$$\dim {\mathbb{F}}_r(B)=F_r(b,c)+F_r(a,d)-F_r(a,c)-F_r(b,d).$$

$${\mathbb{T}}_r(B^{\prime }):={\mathbb{T}}_r(b,d)/j^{\prime }{\mathbb{T}}_r(a,d)+{\mathbb{T}}_r(b,c),$$

$$\dim {\mathbb{T}}_r(B^{\prime })=T_r(b,d)+T_r(a,c)-T_r(a,d)-T_r(b,c).$$

$${\mathbb{T}}_r(B^{\prime \prime }):={\mathbb{T}}_r(c,a)/j^{\prime \prime }{\mathbb{T}}_r(d,a)+{\mathbb{T}}_r(c,b)$$

$$\dim {\mathbb{T}}_r(B^{\prime \prime })=T_r(c,a)+T_r(d,b)-T_r(c,b)-T_r(d,a).$$

$$\dim H_r(f^{-1}(-\infty ,a])=\{\begin{array}{l}♯\{I\in {\mathcal{B}}_r^c\mid l(I)\le a\}\\ ♯\{I\in {\mathcal{B}}_{r-1}^o\mid I\subset (-\infty ,a)\}\\ ♯\{I\in {\mathcal{B}}_r^{co}\mid l(I)\le a<r(I)\}\end{array}$$

$$\dim {\mathbb{I}}_a(r)=\{\begin{array}{l}♯\{I\in {\mathcal{B}}_r^c\mid l(I)\le a\}\\ ♯\{I\in {\mathcal{B}}_{r-1}^o\mid I\subset (-\infty ,a)\}\end{array},$$

$$\dim H_r(f^{-1}([a,\infty ))=\{\begin{array}{l}♯\{I\in {\mathcal{B}}_r^c\mid r(I)\ge a\}\\ ♯\{I\in {\mathcal{B}}_{r-1}^o\mid I\subset (a,\infty )\}\\ ♯\{I\in {\mathcal{B}}_r^{oc}\mid l(I)<a\le r(I)\}\end{array}$$

$$\dim {\mathbb{I}}^a(r)=\{\begin{array}{l}♯\{I\in {\mathcal{B}}_r^c\mid r(I)\ge a\}\\ ♯\{I\in {\mathcal{B}}_{r-1}^o\mid I\subset (a,\infty )\}\end{array}.$$

$$\dim {\mathbb{F}}_r(a,b)=\{\begin{array}{l}♯\{I\in {\mathcal{B}}_r^c\mid l(I)\le a,r(I)\ge b\}\\ ♯\{I\in {\mathcal{B}}_{r-1}^o\mid b<l(I)<r(I)<a\}\end{array}$$

$${\delta }_r^f(a,b)=♯\{I=[a,b]\in {\mathcal{B}}_{r-1}^o\mid l(I)=b,r(I)=a\}$$

$$\dim {\mathbb{F}}_r(a,b)=♯\{I\in {\mathcal{B}}_r^c\mid l(I)\le a\le b\le r(I)\}$$

$${\delta }_r^f(a,b)=♯\{I=[a,b]\in {\mathcal{B}}_r^c\mid l(I)=a,r(I)=b\}$$

$$\dim {\mathbb{T}}_{a,b}(r)=♯\{I\in {\mathcal{B}}_r^{co}\mid l(I)\le a<r(I)\le b\}$$

$${\gamma }_r^f(a,b)=♯\{I=[a,b)\in {\mathcal{B}}_{r-1}^o\mid l(I)=a,r(I)=b\}$$

$$\dim {\mathbb{T}}^{a,b}(r)=♯\{I\in {\mathcal{B}}_r^{oc}\mid b\le l(I)<a\le r(I)\}$$

$${\gamma }_r^f(a,b)=♯\{I=[a,b]\in {\mathcal{B}}_{r-1}^o\mid l(I)=b,r(I)=a\}$$

Full text

1. 1 Introduction
2. 2 Definitions
3. 3 Equalities

1 Introduction

In a recent paper [5] published in this journal the authors have proposed for an $\mathbb{R}-$ space $\mathbb{X}$ (= a continuous map $f:X\to\mathbb{R}$), under reasonable topological hypotheses, four persistence diagrams

[TABLE]

regarded as the relevant invariants for parametrized homology of $\mathbb{X}.$ These persistence diagrams collect the four types of barcodes which can be associated to $f$ via zigzag persistence, are maps with discrete support from $\overline{R}^{2}_{+}=\{-\infty\leq a<b\leq+\infty\}$ to $\mathbb{Z}_{\geq 0}$ and can be interpreted as densities $d\mu$ of four integer valued measures $\mu=\mu^{\wedge}_{H_{r}(\mathbb{X})},$ $\mu^{\lor}_{H_{r}(\mathbb{X})},\mu^{\backslash\backslash}_{H_{r}(\mathbb{X})},\mu^{//}_{H_{r}(\mathbb{X})}$ for squares $R=[a,b]\times[c.d],$ $-\infty\leq a<b<c,d\leq\infty.$

In cite [2] sections 6 and 7 (cf. also [3]published in this journal) and with more details in the book [1] sections 5 and 6, under essentially the same hypotheses, two maps with discrete support, $\rho^{f}_{r}:\mathbb{R}^{2}\to\mathbb{Z}_{\geq 0},\gamma^{f}_{r}:\mathbb{R}^{2}\setminus\Delta\to\mathbb{Z}_{\geq 0}$ with $\Delta=\{(x,x)\in\mathbb{R}^{2}\}$, have been defined and studied. 111 In the above references the discussion refers mostly to $X$ compact but the conclusions remain the same without the need of any additional hypotheses when $X$ is locally compact and $f:X\to\mathbb{R}$ is a proper map. In this case the support of the maps $\delta_{r}^{f}$ and $\gamma^{f}_{r}$ is discrete; this is already used in the case of angle-valued maps. In the case $X$ is compact the support of the maps $\delta^{f}_{r}$ and $\gamma^{f}_{r}$ is finite, hence these maps are configurations of points with multiplicity, cf. [2], [1] and [3].

These maps are derived from the vector space-valued maps $\hat{\rho}^{f}_{r}$ and $\hat{\gamma}^{f}_{r}$ with $\rho^{f}=\dim\hat{\gamma}^{f}_{r}$ and $\gamma^{f}_{r}=\dim\hat{\gamma}^{f}_{r}$ and are viewed as relevant topological invariants for a real-valued map $f.$ The support of $\rho_{r}$ consists of points in $\mathbb{R}^{2}$ which correspond to the closed $r-$bar codes and open $(r-1)-$ barcodes while the support of $\gamma_{r}$ of points which correspond to the closed-open and open-closed $r-$barcodes. In both [2] and [1] these two maps 222They become four when one treats separately the sigma algebras generated by the above diagonal squares and the below diagonal squares are defined in two ways, one being as ”densities” of the measures $\dim\mathbb{F}_{r}$ and $\dim\mathbb{T}_{r}$ on the sigma algebras associated to the same type of squares. A measure theoretic formulation of both $\delta^{f}_{r},\gamma^{f}_{r},$ and even more general of $\hat{\delta}^{f}_{r}$ and $\hat{\gamma}^{f}_{r},$ can be explicitly found in [1] subsection 9.2 and is implicit in [2] and [1] subsections 5.1 and 6.1.

The purpose of this note is to show that the persistence diagrams $Dmg^{\cdots}_{\cdots}$ and of the measures $\mu^{\cdots}_{\cdots}$ are the restrictions to the points and the rectangles above the diagonal resp. below diagonal after composing with the map $T(x,y)=(y,x)$ of the maps $\rho^{f}_{\cdots}$ and the measures $\dim\mathbb{F}_{\cdots},\dim\mathbb{T}_{\cdots}$ facts which might pass unnoticed in view of notational differences.

Precisely, one has:

Proposition 1.1

• •

(a): $Dmg^{\wedge}(H_{r}(\mathbb{X}))$ equals $\rho^{f}_{r}$ restricted to $\mathbb{R}^{2}_{+},$

• •

(b): $Dmg^{\lor}(H_{r}(\mathbb{X}))$ equals $\rho^{f}_{r-1}\cdot T$ restricted to $\mathbb{R}^{2}_{+},$

• •

(c): $Dmg^{\backslash\backslash}(H_{r}(\mathbb{X}))$ equals $\rho^{f}_{r}$ restricted to $\mathbb{R}^{2}_{+},$

• •

(d): $Dmg^{//}(H_{r}(\mathbb{X}))$ equals $\rho^{f}_{r-1}\cdot T$ restricted to $\mathbb{R}^{2}_{+}.$

$(R^{2}_{+}=\{a,b)\in\mathbb{R}^{2},a<b\}$) and

Proposition 1.2

• •

(A): $\mu^{\wedge}_{H_{r}}([a,b]\times[c,d))=\dim\mathbb{F}_{r}((a,b]\times[c,d)),$

• •

(B): $\mu^{\backslash\backslash}_{H_{r}}([a,b]\times[c,d])=\dim\mathbb{T}_{r}((a,b]\times(c,d]),$

• •

(C): $\mu^{//}_{H_{r-1}}([a,b]\times[c,d])=\dim\mathbb{F}_{r}((c,d]\times[a,b)),$

• •

(D): $\mu^{\lor}_{H_{r}}([a,b]\times[c,d])=\dim\mathbb{F}_{r}((c,d]\times[a,b)).$

Note that the stability results as stated in [5] follows in a straightforward manner from the stability results of [2] or [1] and the Alexander duality in [5] can be derived without effort from the Poincaré-duality in [2] or [1] in the same way the Alexander-duality can be derived from the Poincaré duality. (NOTE: After the posting of the first version of this note it was brought to my attention that the result on Alexander-duality as well as most of the arguments in [5] were contained in the thesis of one of the author, Sara Kalisnik, cf. http://www.matknjiz.si/doktorati/2013/Kalisnik-14521-4.pdf, and posted on arXiv cf. Sara Kalisnik , Alexander Duality for Parametrized Homology, arXiv:1303.1591.)

2 Definitions

2.1 CSKM-definitions

For simplicity in writing one shortens the notations in [5] by replacing the notation : $\wedge,$ $\lor,$ $\backslash\backslash,$ $//$ by $c,$ $o,$ $co,$ $oc$ abreviatioins of closed, open, closed-open, open-closed and $\mu^{\cdots}_{H_{r}\mathbb{X}}$ by $\mu^{\cdots}_{r}.$

Consider

$\mathcal{B}^{c}_{r}=$ the multi-set of closed $r-$bar codes,

$\mathcal{B}^{o}_{r}=$ the multi-set of open $r-$ bar codes,

$\mathcal{B}^{c,o}_{r}=$ the multi-set of closed-open $r-$bar codes,

$\mathcal{B}^{o,c}_{r}=$ the multi-set of open-closed $r-$ bar codes.

The definitions of barcodes $\mathcal{B}^{\cdots}_{r}$ in [5] (called in [5] ”intervals” and / or ”decorated pairs”) are based on the initial presentation of zigzag persistence introduced by Carlsson, de-Silva, Morozov in 2009 . A reformulation of these definitions in terms of ”death” and of ”observability” is provided in [1] subsection 9.1.1.

If for a barcode $I$ one denotes by $l(I)$ resp. $r(I)$ the left end resp. the right end, then a careful reading of the definitions in [5] shows that for a box $R=[a,b]\times[c,d]$ with $-\infty\leq a<b<c<d\leq\infty$ one has:

[TABLE]

and then

[TABLE]

2.2 BH-definitions

Denote by :

$\mathbb{I}_{a}(r)=\mathrm{i}mg(H_{r}(f^{-1}((-\infty,a]))\to H_{r}(X)),$

$\mathbb{I}^{a}(r)=\mathrm{i}mg(H_{r}(f^{-1}([a,\infty)))\to H_{r}(X)),$

$\mathbb{F}_{r}(a,b)=\mathbb{I}_{a}(r)\cap\mathbb{I}^{b}(r),$ $F_{r}(a,b):=\dim\mathbb{F}_{r}(a,b),$

$\mathbb{T}_{r}(a,b):=\ker(H_{r}(f^{-1}((-\infty,a])\to H_{r}(f^{-1}((-\infty,b]))$ when $a<b,$

$\mathbb{T}_{r}(a,b):=\ker(H_{r}(f^{-1}([a,\infty))\to f^{-1}([b,\infty))$ when $a>b,$

$T_{r}(a,b)=\dim\mathbb{T}_{r}(a,b).$

For a box $B=(a,b]\times[c,d),\ a<b,c<d$ one defines

[TABLE]

and one observes

[TABLE]

For a box above diagonal $B^{\prime}=(a,b]\times(c,d],a<b\leq c<d$ one defines

[TABLE]

with $j^{\prime}:\mathbb{T}_{r}{(a,d)}\to\mathbb{T}_{r}{(b,d)}$ the obviously induced linear map, and one observes

[TABLE]

For a box below diagonal $B^{\prime\prime}=[c,d)\times[a,b),a<b\leq c<d$ one defines

[TABLE]

with $j^{\prime\prime}:\mathbb{T}_{r}({d,a})\to\mathbb{T}_{r}(c,a)$ the obviously induced linear map, and one observes

[TABLE]

Recall from [4] Theorem 3.2 and Proposition 5.3 or from [2] Proposition 4.1 333a real valued map can be regarded as an angle valued map or from [1] Proposition 4.3 the following equalities:

• •

[TABLE]

and then

[TABLE]

• •

[TABLE]

and then

[TABLE]

As a consequence we have

for $a>b$

[TABLE]

[TABLE] 2. 2.

for $a\leq b$

[TABLE]

[TABLE] 3. 3.

for $a<b$

[TABLE]

[TABLE] 4. 4.

for $a>b$

[TABLE]

[TABLE]

3 Equalities

For $R=[a,b]\times[c,d]$ with $a<b<c<d$

(1)+ (8) +(3) imply Proposition1.2 (A) and (2)+ (9) imply Proposition1.1(a),

(1)+ (6) +(3) imply Proposition 1.2(B) and (2)+ (7) imply Proposition1.1(b),

(1)+ (10) +(4) imply Proposition1.2(C) and (2)+ (11) imply Proposition1.1(c),

(1)+ (12) +(4) imply Proposition1.2(D) and (2)+ (13) imply Proposition1.1(d).