2024-09-27

We seek stability results on the Steinhaus filtration. Below is the current strategy:

1. Construct a correspondence $C$ between two Steinhaus filtrations $\mathbb{S}$, $\mathbb{T}$
2. Show that $C$ and $C^{\top }$ are both $ϵ$-simplicial
3. Conclude that $H(C)$ and $H(C^{\top })$ induce an $ϵ$-interleaving on $H(\mathbb{S})$ and $H(\mathbb{T})$ (see 2024-09-06 proposition 2).
4. Show that $H(\mathbb{S})$ and $H(\mathbb{T})$ are $q$-tame.
5. Conclude that the bottleneck distance is bounded above by $ϵ$ (see 2024-09-06 theorem 1).

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Consider two Steinhaus filtrations ${\text{Nrv}}_{St}(\mathcal{U})$ and ${\text{Nrv}}_{St}(\mathcal{V})$ given by

$${\text{Nrv}}_{St}(\mathcal{U},t_1)\hookrightarrow {\text{Nrv}}_{St}(\mathcal{U},t_2)\hookrightarrow \cdots \hookrightarrow {\text{Nrv}}_{St}(\mathcal{U},t_n)$$

where ${\text{Nrv}}_{St}(\mathcal{U},t_i)$ is the subcomplex of ${\text{Nrv}}_{St}(\mathcal{U})$ formed by including all simplices $\sigma$ whose Steinhaus distance

$$d_{St}(\{U_i\mid i\in \sigma \})=1-\frac{\mu \left(\bigcap U_i\right)}{\mu \left(\bigcup U_i\right)}$$

is bounded above by $t_i$.

Based on the current $ϵ$-interleaving result, we likely want $ϵ\ge 1/m$ where

$$m=\min \left\{\underset{U\in \mathcal{U}}{\min }\mu (U),\underset{V\in \mathcal{V}}{\min }\mu (V)\right\},$$

i.e., we want to find a correspondence such that $C$ and $C^{\top }$ are both $ϵ$-simplicial whenever $ϵ\ge 1/m$. So for any simplex $\sigma \in {\text{Nrv}}_{St}(\mathcal{U},\alpha )$, every finite subset $\tau \in C(\sigma )$ should be a simplex in ${\text{Nrv}}_{St}(\mathcal{V},\alpha +ϵ)$ and for any simplex $\sigma \in {\text{Nrv}}_{St}(\mathcal{V},\alpha )$, every finite subset $\tau \in C^{\top }(\sigma )$ should be a simplex in ${\text{Nrv}}_{St}(\mathcal{U},\alpha +ϵ)$.

Borrowing from Persistence stability for geometric complexes, define the distortion of a correspondence $C:\mathcal{U}\rightrightarrows \mathcal{V}$ to be

$$\text{dis}(C)=\sup \{|d_{St}(\{U_i\mid i\in \sigma \})-d_{St}(\{V_i\mid i\in \tau \})|:(\sigma ,\tau )\in C\},$$

i.e., the largest difference in Steinhaus distance for each simplex. We then define the pseudometric (does the triangle inequality hold?)

$$d(\mathcal{U},\mathcal{V})=\frac{1}{2}\inf \{\text{dis}(C)\mid C:\mathcal{U}\rightrightarrows \mathcal{V}\text{ is a correpondence}\}.$$

TODO: Figure out what exactly $d(\mathcal{U},\mathcal{V})$ measures here.

Assume that $2d(\mathcal{U},\mathcal{V})<ϵ$. Then there is a correspondence $C$ such $\text{dis}(C)\le ϵ$. So for any simplex $\sigma \in {\text{Nrv}}_{St}(\mathcal{U},\alpha )$ and simplex $\tau \subset C(\sigma )$,

$$|d_{St}(\{U_i\mid i\in \sigma \})-d_{St}(\{V_i\mid i\in \tau \})|\le ϵ.$$

By the reverse triangle-inequality:

$$d_{St}(\{V_i\mid i\in \tau \})\le d_{St}(\{U_i\mid i\in \sigma \})+ϵ\le \alpha +ϵ,$$

i.e., $\tau \in {\text{Nrv}}_{St}(\mathcal{V},\alpha +ϵ)$. So $C$ is $ϵ$-simplicial.

We claim that if $\text{dis}(C)=\text{dis}(C^{\top })$. Notice that

$$\begin{array}{rl}\text{dis}(C^{\top })& =\sup \{|d_{St}(\{U_i\mid i\in \sigma \})-d_{St}(\{V_i\mid i\in \tau \})|:(\tau ,\sigma )\in C^{\top }\}\\ & =\sup \{|d_{St}(\{U_i\mid i\in \sigma \})-d_{St}(\{V_i\mid i\in \tau \})|:(\sigma ,\tau )\in C\}\\ & =\text{dis}(C).\end{array}$$

So for any simplex $\tau \in {\text{Nrv}}_{St}(\mathcal{V},\alpha )$ and $\sigma \subset C(\sigma )$,

$$|d_{St}(\{U_i\mid i\in \sigma \})-d_{St}(\{V_i\mid i\in \tau \})|\le ϵ.$$

By an identical argument we get

$$d_{St}(\{U_i\mid i\in \sigma \})\le d_{St}(\{V_i\mid i\in \tau \})+ϵ\le \alpha +ϵ,$$

i.e., $\sigma \in {\text{Nrv}}_{St}(\mathcal{U},\alpha +ϵ)$. So $C^{\top }$ is $ϵ$-simplicial.

Therefore, $C$ and $C^{\top }$ are both $ϵ$-simplicial. Hence, we get that for some correspondence $C$ the induced maps $H(C)$ and $H(C^{\top })$ form $ϵ$-interleaving between $H({\text{Nrv}}_{St}(\mathcal{U}))$ and $H({\text{Nrv}}_{St}(\mathcal{V}))$.

Prp

Let ${\text{Nrv}}_{St}(\mathcal{U})$ and ${\text{Nrv}}_{St}(\mathcal{V})$ be Steinhaus filtrations with $ϵ>2d(\mathcal{U},\mathcal{V})$, then the persistence modules $H({\text{Nrv}}_{St}(\mathcal{U}))$ and $H({\text{Nrv}}_{St}(\mathcal{V}))$ are $ϵ$-interleaved.

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Claim: If $\mathcal{U}$ and $\mathcal{V}$ are both finite with the same cardinality such that $d_B(\mathcal{U},\mathcal{V})\le \alpha$, then

$$d(\mathcal{U},\mathcal{V})\le \alpha /m.$$

We want to show that there is a correspondence $C:\mathcal{U}\rightrightarrows \mathcal{V}$ such that $\text{dis}(C)\le 2\alpha /m$. It is shown in the paper that

$$d_{St}(V_I)-\frac{\alpha }{m}\le d_{St}(U_I)\le d_{St}(V_I)+\frac{\alpha }{m}$$

for all index sets $I$. If we can construct a correspondence $C$ so that $U_i\mapsto V_i$ for all $i\in I$, then

$$\text{dis}(C)=\underset{I}{\sup }|d_{St}(U_I)-d_{St}(V_I)|\le \underset{I}{\sup }2\frac{\alpha }{m}=2\frac{\alpha }{m}.$$

Therefore,

$$d(\mathcal{U},\mathcal{V})\le \alpha /m.$$

TODO: Figure out what exactly the relationship between $U_I$ and $V_I$ is in the paper so we can verify that $C$ is a correspondence.

Question: Is $d(\mathcal{U},\mathcal{V})=\alpha /m$?