2024-03-29

Section 4 of the paper Steinhaus Filtration and Stable Paths in the Mapper claims that (under certain conditions) the cover filtration is isomorphic to Cech and VR filtrations. It starts with the following conjecture:

Cech equivalence

Given a finite data set $X\subset {\mathbb{R}}^n$ and some radius $R>\text{diam}(X)$ the Cech filtration constructed from $X$ is isomorphic to the cover filtration on $X$ constructed from $C_R(X)$, given the Lebesgue measure.

In the conjecture above, $C_R(X)$ is the cover of $X$ by balls of radius $R$. Clearly, the nerve of $C_R(x)$ is the Cech complex. One gets the Cech filtration by varying the radius $R$.

I want to explore more what this conjecture is stating, i.e., what does it mean for two filtrations to be isomorphic?

The following is my educated guess:

Two filtrations

$$A_0\subset A_1\subset A_2\subset \cdots \subset A_m$$

and

$$B_0\subset B_1\subset B_2\subset \cdots \subset B_n$$

are isomorphic if $m=n$ and there exists bijections ${\phi }_i:A_i\to B_i$ for all $i=0,1,2,\dots ,m$.

Under this definition, the conjecture is claiming that there are bijections

$${\phi }_i:K^{t_i}\to \text{Cˇech}(r_i)$$

where $K^{t_i}:=\{\sigma \in {\text{Nrv}}_{St}(\mathcal{U}):w_{\sigma }\le t_i\}$.

The paper doesn’t provide a general proof, but it does provide a proof for the $n$-skeleton of a data set $X\subset \mathbb{R}$. They construct a bijection between birth times. Notice that for the Cech filtration, the birth radius of the simplex defined by the set of vertices $\{v_i\}$ is given by

$$C(\{v_i\})=\frac{{\max }_i(v_i)-{\min }_i(v_i)}{2},$$

i.e, when the balls around the two outer most points intersect.

If $\{V_i\}$ is the set of balls of radius $R$ centered on the set $\{v_i\}$, then the Steinhaus distance (recall we are using Lebesgue measure) is

$$d_{St}(\{V_i\})=1-\frac{\mu (\bigcap V_i)}{\mu (\bigcup V_i)}=1-\frac{{\min }_i(v_i+R)-{\max }_i(v_i-R)}{{\max }_i(v_i+R)-{\min }_i(v_i-R)}.$$

This can be seen with the visualization below:

The union is the long lower orange line, whereas the intersection is the blue line above it.

After some basic manipulation, we get that

$$C(\{v_i\})=\frac{R{d}_{St}(\{V_i\})}{2-d_{St}(\{V_i\})}.$$

establishing a bijection between birth times.

The paper then orders subsets of $X$ by increasing birth radius $(s_1,s_2,\dots ,s_n)$. This ensures that $(d_{St}(s_1),\dots ,d_{St}(s_n))$ are also in increasing order.

Question: How does the bijection between birth times result in a bijection between subcomplexes?