A Distance Between Filtered Spaces Via Tripods

TL;DR

This paper introduces a new intrinsic distance measure between filtered spaces that enhances stability analysis and constructs explicit geodesics, improving understanding of filtrations on different finite spaces.

Contribution

It extends stability results to filtrations on different spaces without interleaving and constructs explicit geodesics to demonstrate the intrinsic nature of the distance.

Findings

Established a new intrinsic distance measure for filtered spaces.

Constructed explicit geodesics between filtered spaces.

Strengthened stability results for filtrations.

Abstract

We present a simplified treatment of stability of filtrations on finite spaces. Interestingly, we can lift the stability result for combinatorial filtrations from [CSEM06] to the case when two filtrations live on different spaces without directly invoking the concept of interleaving. We then prove that this distance is intrinsic by constructing explicit geodesics between any pair of filtered spaces. Finally we use this construction to obtain a strengthening of the stability result.

Equations

$$L_{\varepsilon}(\mathbf{X}):=\big{\{}\sigma\subseteq X|\,F_{X}(\sigma)\leq\varepsilon\big{\}}.$$

$$L(\mathbf{X}):=\big{\{}L_{\varepsilon}(\mathbf{X})\subset L_{\varepsilon^{\prime}}(\mathbf{X})\}_{\varepsilon\leq\varepsilon^{\prime}}$$

$$\mathbb{V}_{k}(\mathbf{X}):=\big{\{}V_{\varepsilon}(\mathbf{X})\stackrel{{\scriptstyle v_{\varepsilon,\varepsilon^{\prime}}}}{{\longrightarrow}}V_{{\varepsilon^{\prime}}}(\mathbf{X})\big{\}}_{\varepsilon\leq\varepsilon^{\prime}},$$

$$D=\{(b_{\alpha },d_{\alpha }),0\le b_{\alpha }\le d_{\alpha },\alpha \in A\}$$

$$\mathbf{X}\longmapsto L(\mathbf{X})\longmapsto\mathbb{V}_{k}(\mathbf{X})\longmapsto I\big{(}\mathbb{V}_{k}(\mathbf{X})\big{)}.$$

$$J(\phi ):=\max \left(\underset{\beta \in B_1}{\max }\Vert P_{\beta }-Q_{\phi (\beta )}{\Vert }_{\infty },\underset{\alpha \in A_1\backslash B_1}{\max }\frac{1}{2}\mathrm{pers}(P_{\alpha }),\underset{\alpha \in A_2\backslash B_2}{\max }\frac{1}{2}\mathrm{pers}(P_{\alpha })\right).$$

$$d_{\mathcal{D}}(D_1,D_2):=\underset{(B_1,B_2,\phi )}{\min }J(\phi ),$$

$$d_{\mathcal{D}}({\mathrm{D}}_k(F),{\mathrm{D}}_k(G))\le \underset{\sigma \in \mathrm{pow}(X)}{\max }|F(\sigma )-G(\sigma )|,$$

$$Z\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$d_{\mathcal{F}}\big{(}\mathbf{X},\mathbf{Y}\big{)}:=\\ \inf\left\{\max_{\tau\in\mathrm{pow}(Z)}\big{|}\varphi^{\ast}_{X}F_{X}(\tau)-\varphi^{\ast}_{Y}F_{Y}(\tau)\big{|};\,\varphi_{X}:Z\twoheadrightarrow X,\,\varphi_{Y}:Z\twoheadrightarrow Y\,\,\mbox{parametrizations}\right\}.$$

$$d_{\mathcal{F}}(\mathbf{X},\mathbf{Y})<{\eta }_1\text{mbox}and{d}_{\mathcal{F}}(\mathbf{Y},\mathbf{W})<{\eta }_2.$$

$$\Vert F_X\circ {\psi }_X-F_Y\circ {\psi }_Y{\Vert }_{{\ell }^{\infty }(\mathrm{pow}(Z_1))}<{\eta }_1$$

$$\Vert F_Y\circ {\zeta }_Y-F_W\circ {\zeta }_W{\Vert }_{{\ell }^{\infty }(\mathrm{pow}(Z_2))}<{\eta }_2.$$

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$$d_{\mathcal{F}}(\mathbf{X},\mathbf{W})$$

$$\le \Vert F_X\circ {\varphi }_X-F_Y\circ {\varphi }_Y{\Vert }_{{\ell }^{\infty }(\mathrm{pow}(Z))}+\Vert F_Y\circ {\varphi }_Y-F_W\circ {\varphi }_W{\Vert }_{{\ell }^{\infty }(\mathrm{pow}(Z))}$$

$$=\Vert F_X\circ {\psi }_X-F_Y\circ {\psi }_Y{\Vert }_{{\ell }^{\infty }(\mathrm{pow}(Z_1))}+\Vert F_Y\circ {\zeta }_Y-F_W\circ {\zeta }_W{\Vert }_{{\ell }^{\infty }(\mathrm{pow}(Z_2))}$$

$$\le {\eta }_1+{\eta }_2.$$

$$d_{\mathcal{D}}({\mathrm{D}}_k(\mathbf{X}),{\mathrm{D}}_k(\mathbf{Y}))\le d_{\mathcal{F}}(\mathbf{X},\mathbf{Y}).$$

$$d_{\mathcal{D}}({\mathrm{D}}_k({\phi }_X^{\ast }{F}_X),{\mathrm{D}}_k({\phi }_Y^{\ast }{F}_Y))<\epsilon .$$

$$\mathrm{dis}(R):=\sup_{(x,y),(x^{\prime},y^{\prime})\in R}\big{|}d_{X}(x,x^{\prime})-d_{Y}(y,y^{\prime})\big{|}.$$

$$d_{\mathcal{GH}}(X,Y):=\frac{1}{2}\underset{R}{\inf }\mathrm{dis}(R),$$

$$F_X^{\mathrm{R}}(\sigma )={\mathbf{diam}}_X\left(\sigma \right):=\underset{x,x^{\prime }\in X}{\max }{d}_X(x,x^{\prime }).$$

$$d_{\mathcal{D}}\big{(}\mathrm{D}_{k}(F^{\mathrm{R}}_{X}),\mathrm{D}_{k}(F^{\mathrm{R}}_{Y})\big{)}\leq 2\,d_{\mathcal{G}\mathcal{H}}({X},{Y}).$$

$$d_{\mathcal{F}}\big{(}F^{\mathrm{R}}_{X},F^{\mathrm{R}}_{Y}\big{)}\leq 2\,d_{\mathcal{G}\mathcal{H}}({X},{Y}).$$

$$|d_X({\phi }_X(t),{\phi }_X(t^{\prime }))-d_Y({\phi }_Y(t),{\phi }_Y(t^{\prime }))|\le 2\eta$$

$${\phi }_X^{\ast }{F}_X^{\mathrm{R}}(\tau )=F_X^{\mathrm{R}}({\phi }_X(\tau ))=\underset{t,t^{\prime }\in \tau }{\max }{d}_X({\phi }_X(t),{\phi }_X(t^{\prime })).$$

$${\phi }_Y^{\ast }{F}_Y^{\mathrm{R}}(\tau )=\underset{t,t^{\prime }\in \tau }{\max }{d}_Y({\phi }_Y(t),{\phi }_Y(t^{\prime }))\le \underset{t,t^{\prime }\in \tau }{\max }{d}_X({\phi }_X(t),{\phi }_X(t^{\prime }))+2\eta ={\phi }_X^{\ast }{F}_X^{\mathrm{R}}(\tau )+2\eta ,$$

$$F_X^{\mathrm{C}}(\sigma ):={\mathbf{rad}}_X(\sigma )=\underset{p\in X}{\min }\underset{x\in \sigma }{\max }{d}_X(x,p).$$

$$d_{\mathcal{F}}\big{(}F^{W}_{X},F^{W}_{Y}\big{)}\leq 2\,d_{\mathcal{G}\mathcal{H}}({X},{Y}).$$

$$d_{\mathcal{D}}\big{(}\mathrm{D}_{k}(F^{\mathrm{C}}_{X}),\mathrm{D}_{k}(F^{\mathrm{C}}_{Y})\big{)}\leq 2\,d_{\mathcal{G}\mathcal{H}}({X},{Y}).$$

$${\phi }_X^{\ast }{F}_X^W(\tau )=F_X^W({\phi }_X(\tau ))=\underset{p\in X}{\min }\underset{t\in \tau }{\max }{d}_X(p,{\phi }_X(t))=\underset{t\in \tau }{\max }{d}_X(p_{\tau },{\phi }_X(t))$$

Full text

This work was supported by NSF grants IIS-1422400 and CCF-1526513.

1. 1 Introduction

2. 2 Simplicial Homology

3. 3 Filtrations and Persistent Homology

4. 4 Stability of filtrations

5. 4.1 Filtrations defined over different sets

6. 4.1.1 Common parametrizations of two spaces: tripods.

7. 5 Filtrations arising from metric spaces: Rips and Čech

8. 5.1 The Rips filtration

9. 5.2 The Čech filtration

10. 6 $d_{\mathcal{F}}$ is geodesic

11. 6.1 Geodesics

12. 6.2 A strengthening of Theorem 4.2

13. 7 Discussion

1. Introduction

The goal for the construction that I describe in this note was to lift the stability result of [CSEM06] to the setting when the simplicial filtrations are not necessarily defined on the same set. The ideas in this note were first presented at ATCMS in July 2012. A partial discussion appears in [Mḿissing].

Section 6 proves that that this construction defines a geodesic metric on the collection of finite filtered spaces. There I give an improvement of the stability of persistence which uses these geodesics.

2. Simplicial Homology

Given a simplicial complex $L$ and simplices $\sigma,\tau\in L$, we write $\sigma\subseteq\tau$ whenever $\sigma$ is a face of $\tau$. For each integer $\ell\geq 0$ we denote by $L^{(\ell)}$ the $\ell$-skeleton of $L$.

Recall that given two finite simplicial complexes $L$ and $S$, a simplicial map between them arises from any map $f:L^{(0)}\rightarrow S^{(0)}$ with the property that whenever $p_{0},p_{1},\ldots,p_{k}$ span a simplex in $L$, then $f(p_{0}),f(p_{1}),\ldots,f(p_{k})$ span a simplex of $S$. One does not require that the vertices $f(p_{0}),f(p_{1}),\ldots,f(p_{k})$ be all distinct. Given a map $f:L^{(0)}\rightarrow S^{(0)}$ between the vertex sets of the finite simplicial complexes $L$ and $S$, we let $\overline{f}:L\rightarrow S$ denote the induced simplicial map.

We will make use of the following theorem in the sequel.

Theorem 2.1** (Quillen’s Theorem A in the simplicial category, [Qui73]).**

Let $\zeta:S\rightarrow L$ be a simplicial map between two finite complexes. Suppose that the preimage of each closed simplex of $L$ is contractible. Then $\zeta$ is a homotopy equivalence.

Corollary 2.1**.**

Let $L$ be a finite simplicial complex and $\varphi:Z\rightarrow L^{(0)}$ be any surjective map with finite domain $Z$. Let $S:=\{\tau\subseteq Z|\,\varphi(\tau)\in L\}$. Then $S$ is a simplicial complex and the induced simplicial map $\overline{\varphi}:S\rightarrow L$ is an homotopy equivalence.

Proof.

Note that $S=\bigcup_{\sigma\in L}\{\tau\subseteq Z|\,\varphi(\tau)=\sigma\}$ so it is clear that $S$ is a simplicial complex with vertex set $Z$. That the preimage of each $\sigma\in L$ is contractible is trivially true since those preimages are exactly the simplices in $S$. The conclusion follows directly from Quillen’s Theorem A. ∎

In this paper we consider homology with coefficients in a field $\mathbb{F}$ so that given a simplicial complex $L$, then for each $k\in\mathbb{N}$, $H_{k}(L,\mathbb{F})$ is a vector space. To simplify notation, we drop the argument $\mathbb{F}$ from the list and only write $H_{k}(L)$ for the homology of $L$ with coefficients in $\mathbb{F}$.

3. Filtrations and Persistent Homology

Let $\mathcal{F}$ denote the set of all finite filtered spaces: that is pairs $\mathbf{X}=(X,F_{X})$ where $X$ is a finite set and $F_{X}:\mathrm{pow}(X)\rightarrow\mathbb{R}$ is a monotone function. Any such function is called a filtration over $X$. Monotonicity in this context refers to the condition that $F_{X}(\sigma)\geq F_{X}(\tau)$ whenever $\sigma\supseteq\tau.$ Given a finite set $X$, by $\mathcal{F}(X)$ we denote the set of all possible filtrations $F_{X}:\mathrm{pow}({X})\rightarrow\mathbb{R}$ on $X$. Given a filtered space $\mathbf{X}=(X,F_{X})\in\mathcal{F}$, for each $\varepsilon\in\mathbb{R}$ define the simplicial complex

[TABLE]

One then considers the nested family of simplicial complexes

[TABLE]

where each $L_{\varepsilon}(\mathbf{X})$ is, by construction, finite. At the level of homology, for each $k\in\mathbb{N}$ the above inclusions give rise to a system of vector spaces and linear maps

[TABLE]

which is called a persistence vector space. Note that each $V_{\varepsilon}(\mathbf{X})$ is finite dimensional.

Persistence vector spaces admit a classification up to isomorphism in terms of collections of intervals so that to the persistence vector space $\mathbb{V}$ one assigns a multiset of intervals $I(\mathbb{V})$ [CDS10]. These collections of intervals are sometimes referred to as barcodes or also persistence diagrams, depending on the graphical representation that is adopted [EH10]. We denote by $\mathcal{D}$ the collection of all finite persistence diagrams. An element $D\in\mathcal{D}$ is a finite multiset of points

[TABLE]

for some (finite) index set $A$. Given $k\in\mathbb{N}$, to any filtered set $\mathbf{X}\in\mathcal{F}$ one can attach a persistence diagram via

[TABLE]

We denote by $\mathrm{D}_{k}:\mathcal{F}\rightarrow\mathcal{D}$ the resulting composite map. Given $\mathbf{X}=(X,F_{X})$, we will sometimes write $\mathrm{D}_{k}{(F_{X})}$ to denote $\mathrm{D}_{k}{(\mathbf{X})}$.

4. Stability of filtrations

The bottleneck distance is a useful notion of distance between persistence diagrams and we recall it’s definition next. We will follow the presentation on [Car14]. Let $\Delta\subset\mathbb{R}^{2}_{+}$ be comprised of those points which sit above the diagonal: $\Delta:=\{(x,y)|\,x\leq y\}.$

Define the persistence of a point $P=(x_{P},y_{P})\in\Delta$ by $\mathrm{pers}(P):=y_{P}-x_{P}$.

Let $D_{1}=\{P_{\alpha}\}_{\alpha\in A_{1}}$ and $D_{2}=\{Q_{\alpha}\}_{\alpha\in A_{2}}$ be two persistence diagrams indexed over the finite index sets $A_{1}$ and $A_{2}$, respectively. Consider subsets $B_{i}\subseteq A_{i}$ with $|B_{1}|=|B_{2}|$ and any bijection $\varphi:B_{1}\rightarrow B_{2}$, then define

[TABLE]

Finally, one defines the bottleneck distance between $D_{1}$ and $D_{2}$ by

[TABLE]

where $(B_{1},B_{2},\varphi)$ ranges over all $B_{1}\subset A_{1}$, $B_{2}\subset A_{2}$, and bijections $\varphi:B_{1}\rightarrow B_{2}$.

One of the standard results about the stability of persistent homology invariants, which is formulated in terms of the Bottleneck distance, is the proposition below which we state in a weaker form that will suffice for our presentation:111In [CSEM06] the authors do not assume that the underlying simplicial complex is the full powerset.

Theorem 4.1** ([CSEM06]).**

For all finite sets $X$ and filtrations $F,G:\mathrm{pow}(X)\rightarrow\mathbb{R}$,

[TABLE]

for all $k\in\mathbb{N}.$

The proof of this theorem offered in [CSEM06] is purely combinatorial and elementary. This result requires that the two filtrations be given on the same set. This restriction will be lifted using the ideas that follow.

4.1. Filtrations defined over different sets

A parametrization of a finite set $X$ is any finite set $Z$ and a surjective map $\varphi_{X}:Z\rightarrow X$. Consider a filtered space $\mathbf{X}=(X,F_{X})\in\mathcal{F}$ and a parametrization $\varphi_{X}:Z\rightarrow X$ of $X$. By $\varphi_{X}^{\ast}F_{X}$ we denote the pullback filtration induced by $F_{X}$ and the map $\varphi_{X}$ on $Z$. This filtration is given by $\tau\mapsto F_{X}(\varphi_{X}(\tau))$ for all $\tau\in\mathrm{pow}(Z).$

A useful corollary of the persistence homology isomorphism theorem [EH10, pp. 139] and Corollary 2.1 is that the persistence diagrams of the original filtration and the pullback filtration are identical.

Corollary 4.1**.**

Let $\mathbf{X}=(X,F_{X})\in\mathcal{F}$ and $\varphi:Z\twoheadrightarrow X$ a parametrization of $X$. Then, for all $k\in\mathbb{N}$, $\mathrm{D}_{k}(\varphi^{\ast}F_{X})=\mathrm{D}_{k}(F_{X}).$

4.1.1. Common parametrizations of two spaces: tripods.

Now, given $\mathbf{X}=(X,F_{X})$ and $\mathbf{Y}=(Y,F_{Y})$ in $\mathcal{F}$, the main idea in comparing filtrations defined on different spaces is to consider parametrizations $\varphi_{X}:Z\twoheadrightarrow X$ and $\varphi_{Y}:Z\twoheadrightarrow Y$ of $X$ and $Y$ from a common parameter space $Z$, i.e. tripods:

[TABLE]

and compare the pullback filtrations $\varphi_{X}^{*}F_{X}$ and $\varphi_{Y}^{*}F_{Y}$ on $Z$. Formally, define

[TABLE]

Remark 4.1**.**

Notice that in case $X=\{\ast\}$ and $F_{\{\ast\}}(\ast)=c\in\mathbb{R}$, then d_{\mathcal{F}}(X,Y)=\max_{\sigma\subset Y}\big{|}F_{Y}(\sigma)-c\big{|}, for any filtered space $Y$. If $c=0$, $Y=\{y_{1},y_{2}\}$ with $F_{Y}(y_{1})=F_{Y}(y_{2})=0$ and $F_{Y}(\{y_{1},y_{2}\})=1$. Then, $d_{\mathcal{F}}(X,Y)=1.$

However, still with $c=0$ and $Y=\{y_{1},y_{2}\}$, but $F_{Y}(y_{1})=F_{Y}(y_{2})=F_{Y}(\{y_{1},y_{2}\})=0$, one has $d_{\mathcal{F}}(X,Y)=0$. This means that $d_{\mathcal{F}}$ is at best a pseudometric on filtered spaces.

Proposition 4.1**.**

$d_{\mathcal{F}}$* is a pseudometric on $\mathcal{F}$.*

Proof.

Symmetry and non-negativity are clear. We need to prove the triangle inequality. Let $\mathbf{X}=(X,F_{X})$, $\mathbf{Y}=(Y,F_{Y})$, and $\mathbf{W}=(W,F_{W})$ in $\mathcal{F}$ be non-empty and $\eta_{1},\eta_{2}>0$ be s.t.

[TABLE]

Choose, $\psi_{X}:Z_{1}\twoheadrightarrow X$, $\psi_{Y}:Z_{1}\twoheadrightarrow Y$, $\zeta_{Y}:Z_{2}\twoheadrightarrow Y$, and $\zeta_{W}:Z_{2}\twoheadrightarrow W$ surjective such that

[TABLE]

and

[TABLE]

Let $Z\subseteq Z_{1}\times Z_{2}$ be defined by $Z:=\{(z_{1},z_{2})\in Z_{1}\times Z_{2}|\psi_{Y}(z_{1})=\zeta_{Y}(z_{2})\}$ and consider the following (pullback) diagram:

[TABLE]

Clearly, since $\psi_{Y}$ and $\zeta_{Y}$ are surjective, $Z$ is non-empty. Now, consider the following three maps with domain $Z$: $\phi_{X}:=\psi_{X}\circ\pi_{1}$, $\phi_{Y}:=\psi_{Y}\circ\pi_{1}=\zeta_{Y}\circ\pi_{2}$, and $\phi_{W}:=\zeta_{W}\circ\pi_{2}$. These three maps are surjective and therefore constitute parametrizations of $X$, $Y$, and $W$, respectively. Then, since $\pi_{i}:Z\rightarrow Z_{i}$, $i=1,2$, are surjective and $\psi_{Y}\circ{\pi_{1}}=\zeta_{Y}\circ\pi_{2}$, we have

[TABLE]

The conclusion follows by letting $\eta_{1}\searrow d_{\mathcal{F}}(X,Y)$ and $\eta_{2}\searrow d_{\mathcal{F}}(Y,W)$. ∎

We now obtain a lifted version of Theorem 4.1

Theorem 4.2**.**

For all finite filtered spaces $\mathbf{X}=(X,F_{X})$ and $\mathbf{Y}=(Y,F_{Y})$, and all $k\in\mathbb{N}$ one has:

[TABLE]

Proof of Theorem 4.2.

Assume $\varepsilon>0$ is such that $d_{\mathcal{F}}(F_{X},F_{Y})<\varepsilon.$ Then, let $\varphi_{X}:Z\rightarrow X$ and $\varphi_{Y}:Z\rightarrow Y$ be surjective maps from the finite set $Z$ into $X$ and $Y$, respectively, such that $|\varphi_{X}^{\ast}F_{X}(\tau)-\varphi_{Y}^{\ast}F_{Y}(\tau)|<\varepsilon$ for all $\tau\in\mathrm{pow}(Z)$. Then, by Theorem 4.1,

[TABLE]

for all $k\in\mathbb{N}.$ Now apply Corollary 4.1 and conclude by letting $\varepsilon$ approach $d_{\mathcal{F}}(\mathbf{X},\mathbf{Y})$. ∎

Remark 4.2**.**

Consider the case of $\mathbf{Y}$ being the one point filtered space $\{\ast\}$ such that $F_{\{\ast\}}(\{\ast\})=0$, and $\mathbf{X}$ such that $X=\{x_{2},x_{2}\}$, and $F_{X}(\{x_{1}\})=F_{X}(\{x_{2}\})=0$, $F_{X}(\{x_{1},x_{2}\})=1$. In this case $d_{\mathcal{F}}(\mathbf{X},\mathbf{Y})=1$. However, notice that for $k=0$ $\mathrm{D}_{0}(\mathbf{X})=\{[0,\infty),[0,1)\}$ and $\mathrm{D}_{0}(\mathbf{Y})=\{[0,\infty)\}$. Additionaly, for all $k\geq 1$ one has $\mathrm{D}_{k}(\mathbf{X})=\mathrm{D}_{k}(\mathbf{Y})=\emptyset$. This means that the lower bound provided by Theorem 4.2 is equal to $\frac{1}{2}<1=d_{\mathcal{F}}(\mathbf{X},\mathbf{Y})$.

5. Filtrations arising from metric spaces: Rips and Čech

Recall [BBI01] that for two compact metric spaces $(X,d_{X})$ and $(Y,d_{Y})$, a correspondence between them is amy subset $R$ of $X\times Y$ such that the natural projections $\pi_{X}:X\times Y\rightarrow X$ and $\pi_{Y}:X\times Y\rightarrow Y$ are such that $\pi_{X}(R)=X$ and $\pi_{Y}(R)=Y$. The distortion of any such correspondence is given by

[TABLE]

Then, Gromov-Hausdorff distance between $(X,d_{X})$ and $(Y,d_{Y})$ is defined as

[TABLE]

where the infimum is taken over all correspondences $R$ between $X$ and $Y$.

5.1. The Rips filtration

Recall the definition of the Rips filtration of a finite metric space $(X,d_{X})$: for $\sigma\in\mathrm{pow}({X})$,

[TABLE]

The following theorem was first proved in [CCSG*+*09]. A different proof (also applicable to compact metric spaces) relying on the interleaving distance and multivalued maps was given in [CDSO14]. Yet another different proof avoiding multivalued maps is given in [CM16b].

Theorem 5.1**.**

For all finite metric spaces $X$ and $Y$, and all $k\in\mathbb{N}$,

[TABLE]

A different proof of Theorem 5.1 can be obtained by combining Theorem 4.2 and Proposition 5.1 below.

Proposition 5.1**.**

For all finite metric spaces $X$ and $Y$,

[TABLE]

Proof of Proposition 5.1.

Let $X$ and $Y$ be s.t. $d_{\mathcal{G}\mathcal{H}}({X},{Y})<\eta$, and let $R\subset X\times Y$ be a surjective relation with $|d_{X}(x,x^{\prime})-d_{Y}(y,y^{\prime})|\leq 2\eta$ for all $(x,y),(x^{\prime},y^{\prime})\in R$. Consider the parametrization $Z=R$, and $\varphi_{X}=\pi_{1}:Z\rightarrow X$ and $\varphi_{Y}=\pi_{2}:Z\rightarrow Y$, then

[TABLE]

for all $t,t^{\prime}\in Z$. Pick any $\tau\in Z$ and notice that

[TABLE]

Now, similarly, write

[TABLE]

where the last inequality follows from (2). The proof follows by interchanging the roles of $X$ and $Y$. ∎

5.2. The Čech filtration

Another interesting and frequently used filtration is the Čech filtration: for each $\sigma\in\mathrm{pow}({X})$,

[TABLE]

That is, the filtration value of each simplex corresponds to its circumradius.

Proposition 5.2**.**

For all finite metric spaces $X$ and $Y$,

[TABLE]

Again, as a corollary of Theorem 4.2 and Proposition 5.2 we have the following

Theorem 5.2**.**

For all finite metric spaces $X$ and $Y$, and all $k\in\mathbb{N}$,

[TABLE]

A proof of this theorem via the interleaving distance and multi-valued maps has appeared in [CDSO14].222The version in [CDSO14] applies to compact metric spaces. Another proof avoiding multivalued maps is given in [CM16b].

Proof of Proposition 5.2.

The proof is similar to that of Proposition 5.1. Pick any $\tau\in Z$, then,

[TABLE]

for some $p_{\tau}\in X$. Let $t_{\tau}\in Z$ be s.t. $\varphi_{X}(t_{\tau})=p_{\tau}$, and from the above obtain

[TABLE]

Now, similarly, write

[TABLE]

where the last inequality follows from (2). The proof follows by interchanging the roles of $X$ and $Y$. ∎

6. $d_{\mathcal{F}}$ is geodesic

In this section we construct geodesics between any pair $\mathbf{X}$ and $\mathbf{Y}$ of filtered spaces and obtain a strengthening of Theorem 4.2.

6.1. Geodesics

Given $\mathbf{X}$ and $\mathbf{Y}$ in $\mathcal{F}$ consider $\mathcal{T}^{\mathrm{opt}}(\mathbf{X}.\mathbf{Y})$ the set of all minimizing tripods: That is, for any $(Z,\varphi_{X},\varphi_{Y})\in\mathcal{T}(\mathbf{X},\mathbf{Y})$ we have $\|\varphi_{X}^{\ast}F_{X}-\varphi_{Y}^{*}F_{Y}\|_{\ell^{\infty}(\mathrm{pow}({Z})}=d_{\mathcal{F}}(\mathbf{X},\mathbf{Y}).$

For each minimizing tripod $T=(Z,\varphi_{X},\varphi_{Y})\in\mathcal{T}^{\mathrm{opt}}(\mathbf{X},\mathbf{Y})$ consider the curve

[TABLE]

where

[TABLE]

Theorem 6.1**.**

For each $T\in\mathcal{T}^{\mathrm{opt}}(\mathbf{X},\mathbf{Y})$ the curve $\gamma_{T}$ is a geodesic between $\mathbf{X}$ and $\mathbf{Y}$. Namely, for all $s,t\in[0,1]$ one has:

[TABLE]

Proof.

Let $\eta=d_{\mathcal{F}}(\mathbf{X},\mathbf{Y})$. We check that

[TABLE]

and notice that this is enough. Otherwise, let $s<t$ in $[0,1]$ be such that $d_{\mathcal{F}}(\gamma_{T}(s),\gamma_{T}(t))\leq(t-s)\cdot\eta$. Then, by the triangle inequality for $d_{\mathcal{F}}$ we would have

[TABLE]

which by $(*)$ and the definition of $s$ and $t$ would be strictly smaller than $\eta\cdot s+(t-s)\cdot\eta+(1-t)\cdot\eta=\eta$, a contradiction.

Now, in order to verify $(\ast)$, we need to construct a tripod between $\gamma_{T}(s)=(Z,F_{s})$ and $\gamma_{T}(t)=(Z,F_{t})$. We consider the tripod $(Z,\mathrm{id}_{Z},\mathrm{id}_{Z})$ and notice that this tripod gives

[TABLE]

∎

6.2. A strengthening of Theorem 4.2

Recall the definition of length in a metric space $(M,d_{M})$. For a curve $\alpha:[0,1]\rightarrow M$, its length is

[TABLE]

We now use the construction of geodesics above to strengthen Theorem 4.2.

Theorem 6.2**.**

For all $\mathbf{X},\mathbf{Y}\in\mathcal{F}$ and $k\in\mathbb{N}$ one has

[TABLE]

Remark 6.1** (Strengthening).**

That this theorem is a strengthening can be argued as follows. Firstly, by definition of length $\mathrm{L}_{\mathcal{D}}(\mathrm{D}_{k}(\gamma_{T}))\geq d_{\mathcal{D}}(\mathrm{D}_{k}(\gamma_{T}(0)),\mathrm{D}_{k}(\gamma_{T}(1)))=d_{\mathcal{D}}(\mathrm{D}_{k}(\mathbf{X}),\mathrm{D}_{k}(\mathbf{Y})).$ Therefore the lower bound provided by Theorem 6.2 cannot be smaller than the one provided by Theorem 4.2.

Now, to argue about the improvement offered by the new lower bound consider the two spaces from Remark 4.2. For those spaces, a minimizing tripod is $T=(X,\mathrm{id}_{X},\varphi)$ where $\varphi:X\rightarrow\{\ast\}$ is the unique map. In that case, one sees that $F_{t}=(1-t)\cdot F_{X}$ and therefore $\alpha_{T}(t):=\mathrm{D}_{0}(\gamma_{T}(t))=\{[0,\infty),[0,(1-t))\}.$ Notice that for ant two $t,s\in[0,1]$ such that $|s-t|$ is sufficiently small, the bottleneck distance $d_{\mathcal{D}}(\alpha_{T}(t),\alpha_{T}(s))$ equals $|t-s|.$ Then, $\mathrm{L}_{\mathcal{D}}(\alpha)=1=d_{\mathcal{F}}(\gamma_{T}(0),\gamma_{T}(1))>\frac{1}{2}=d_{\mathcal{D}}(\alpha_{T}(0),\alpha_{T}(1)),$ which is the bound provided by Theorem 4.2.

Proof of Theorem 6.2.

Let $T\in\mathcal{T}^{\mathrm{opt}}(\mathbf{X},\mathbf{Y})$ and let $0=t_{0}<\cdots<t_{n}=1$ be any partition of $[0,1].$ Then, write

[TABLE]

where the first inequality follows from Theorem 4.2, and the equality immediately after it follows from the fact that $\gamma_{T}$ is a geodesic.

Thus, we have for any partition $0=t_{0}<\cdot<t_{1}=1$ that

[TABLE]

Taking supremum over all possible partitions yields the claim. ∎

Remark 6.2**.**

The techniques of the above theorem and the results in [CM16a] imply similar strengthenings of Propositions 5.1 and 5.2.

7. Discussion

It seems possible to extend some of these ideas to the case of non necessarily finite filtered spaces.