Let $X$ be a space and let $\mathbb{X}_n = \{x_1,...,x_n\}$ be a finite sample of $X$.

Q: How to infer topological properties of $X$ from $\mathbb{X}_n$?

Let $X$ be a space and let $\mathbb{X}_n = \{x_1,...,x_n\}$ be a finite sample of $X$.

Q: How to infer topological properties of $X$ from $\mathbb{X}_n$?

Point cloud
$\mathbb{X}_n \subset \mathbb{R}^D$

For $\epsilon>0$, the $\epsilon$-thickening of $\mathbb{X}_n$: \[\displaystyle U_\epsilon = \bigcup_{x\in \mathbb{X}_n}B_{\epsilon}(x)\]

Let $X$ be a topological space and let $\mathbb{X}_n = \{x_1,...,x_n\}$ be a finite sample of $X$.

Q: How to infer topological properties of $X$ from $\mathbb{X}_n$?

Let $X$ be a space and let $\mathbb{X}_n = \{x_1,...,x_n\}$ be a finite sample of $X$.

Q: How to infer topological properties of $X$ from $\mathbb{X}_n$?

Point cloud
$\mathbb{X}_n \subset \mathbb{R}^D$

Evolving thickenings

Let $X$ be a topological space and let $\mathbb{X}_n = \{x_1,...,x_n\}$ be a finite sample of $X$.

Q: How to infer topological properties of $X$ from $\mathbb{X}_n$?

Let $X$ be a topological space and let $\mathbb{X}_n = \{x_1,...,x_n\}$ be a finite sample of $X$.

Q: How to infer topological properties of $X$ from $\mathbb{X}_n$?

$\bullet ~ ~\mathrm{Rips}_\epsilon(\mathcal{M}, d_E)\simeq \mathcal{M}$ for $\epsilon < 2 \sqrt{\frac{D+1}{2D}}\mathrm{rch}(\mathcal{M})~~$ (Kim, Shin, Chazal, Rinaldo & Wasserman, 2020)

$\bullet ~ ~\mathrm{Rips}_\epsilon(\mathcal{M}, d_\mathcal{M})\simeq \mathcal{M}$ for $\epsilon < \mathrm{conv}(\mathcal{M}, d_{\mathcal{M}})~~$ (Hausmann, 1995; Latschev, 2001)

Let $\mathbb{X}_n = \{x_1,...,x_n\}\subseteq \mathbb{R}^D$ be a finite sample.

Let $\mathbb{X}_n = \{x_1,...,x_n\}\subseteq \mathbb{R}^D$ be a finite sample.

Assume that:

• $\mathbb{X}_n$ is a sample of a compact manifold $\mathcal M$ of dimension $d$.
• The points are sampled according to a density $f\colon \mathcal M\to \mathbb R$.

The path taken by a ray between two given points is the path that can be traversed in the least time.

That is, it is the extreme of the functional \[ \gamma\mapsto \int_{0}^1\eta(\gamma_t)||\dot{\gamma}_t|| dt \] with $\eta$ is the refraction index.

Let $\mathcal M \subseteq \mathbb{R}^D$ be a manifold and let $f\colon\mathcal{M}\to \mathbb{R}_{>0}$ be a smooth density.

For $q>0$, the deformed Riemannian distance* in $\mathcal{M}$ is \[d_{\mathcal M, f,q}(x,y) = \inf_{\gamma} \int_{I}\frac{1}{f(\gamma_t)^{q}}||\dot{\gamma}_t|| dt \] over all $\gamma:I\to \mathcal{M}$ with $\gamma(0) = x$ and $\gamma(1)=y$.

* Here, if $g$ is the inherited Riemannian tensor, then $d_{f,q}$ is the Riemannian distance induced by $g_q= f^{-2q} g$.

Let $\mathbb{X}_n = \{x_1,...,x_n\}\subseteq \mathbb{R}^D$ be a finite sample.

For $p> 1$, the Fermat distance between $x,y\in \mathbb{R}^D$ is defined by \[ d_{\mathbb{X}_n, p}(x,y) = \inf_{\gamma} \sum_{i=0}^{r}|x_{i+1}-x_i|^{p} \] over all paths $\gamma=(x_0, \dots, x_{r+1})$ of finite length with $x_0=x$, $x_{r+1} = y$ and $\{x_1, x_2, \dots, x_{r}\}\subseteq \mathbb{X}_n$.

$d_{\mathbb X_n, p}$ is an estimator of $d_{\mathcal M, f,q}$ if $q=(p-1)/d$.

\[d_{\mathbb{X}_n, p}(x,y) = \inf_{\gamma} \sum_{i=0}^{r}|x_{i+1}-x_i|^{p}\]

\[d_{\mathcal M, f,q}(x,y) = \inf_{\gamma} \int_{I}\frac{1}{f(\gamma_t)^{q}}||\dot{\gamma}_t|| dt \]

Theorem (F., Borghini, Mindlin, Groisman, 2023)

Let $\mathbb{X}_n$ be a sample of a closed manifold $\mathcal M$ of dimension $d$, drawn according to a density $f\colon \mathcal M\to \mathbb R$.
Given $p>1$ and $q=(p-1)/d$, there exists a constant $\mu = \mu(p,d)$ such that for every $\lambda \in \big((p-1)/pd, 1/d\big)$ and $\varepsilon>0$ there exist $\theta>0$ satisfying \[ \mathbb{P}\left( d_{GH}\left(\big(\mathcal{M}, d_{f,q}\big), \big(\mathbb{X}_n, {\scriptstyle \frac{n^{q}}{\mu}} d_{\mathbb{X}_n, p}\big)\right) > \varepsilon \right) \leq \exp{\left(-\theta n^{(1 - \lambda d) /(d+2p)}\right)} \] for $n$ large enough.

• Complexity:

  $O(n^3)$

  reducible to $O(n^2*k*\log(n))$ using the $k$-NN-graph (for $k = O(\log n)$ the geodesics belong to the $k$-NN graph with high probability).

• Python library:

  fermat

• Computational experiments:

  ximenafernandez/intrinsicPH

Prop (F., Borghini, Mindlin, Groisman, 2023)

Let $\mathbb{X}_n$ be a sample of $\mathcal{M}$ and let $Y\subseteq \mathbb{R}^D\smallsetminus \mathcal{M}$ be a finite set of outliers. Let $\delta = \displaystyle \min\Big\{\min_{y\in Y} d_E(y, Y\smallsetminus \{y\}), ~d_E(\mathbb X_n, Y)\Big\}$.
Then, for all $k>0$ and $p>1$, \[ \mathrm{dgm}_k(\mathrm{Rips}_{<\delta^p}(\mathbb{X}_n \cup Y, d_{\mathbb{X}_n\cup Y, p})) = \mathrm{dgm}_k(\mathrm{Rips}_{<\delta^p}(\mathbb{X}_n, d_{\mathbb{X}_n, p})) \] where $\mathrm{Rips}_{<\delta^p}$ stands for the Rips filtration up to parameter $\delta^{p}$ and $\mathrm{dgm}_k$ for the persistent homology of deg $k$.

• Signal: $\varphi:\mathbb R \to \mathbb{R}$
  

We assume that $\varphi$ is an observation of an underlying dynamical system $(\mathcal M, \phi)$, with $\mathcal M$ a topological space and $\phi\colon \mathbb R \times \mathcal M\to \mathcal M$ the evolution function.

That is, there exist an observation function $F:\mathcal M\to \mathbb R$ and an initial state $x_0\in \mathcal M$ such that \begin{align}\varphi = \varphi_{x_0}:\mathbb{R}&\to \mathbb{R}\\ t&\mapsto F(\phi_t(x_0)) \end{align}

• Signal: $\varphi:\mathbb R \to \mathbb{R}$
  
• Delay embedding: Given $T$ the time delay and $D$ the embedding dimension. \[\mathcal{M}_{T,D} = \{\big(\varphi(t), \varphi(t+T), \varphi(t+2 T) \dots, \varphi(t+(D-1)T)\big): t\in \mathbb R\}\subseteq \mathbb{R}^D\]

• Signal: $\varphi:\mathbb R \to \mathbb{R}$
  
• Delay embedding: Given $T$ the time delay and $D$ the embedding dimension. \[\mathcal{M}_{T,D} = \{\big(\varphi(t), \varphi(t+T), \varphi(t+2 T) \dots, \varphi(t+(D-1)T)\big): t\in \mathbb R\}\subseteq \mathbb{R}^D\]
• Limit set: Given $(\mathcal M, \phi)$ a dynamical system and $x_0\in \mathcal M$, \[\mathcal A_{x_0} = \{x\in \mathcal M: \exists t_i\to \infty \text { with } \phi_{t_i}(x_0)\to x\}.\]

• Signal: $\varphi:\mathbb R \to \mathbb{R}$
  
• Delay embedding: Given $T$ the time delay and $D$ the embedding dimension. \[\mathcal{M}_{T,D} (\phi) = \{\big(\varphi(t), \varphi(t+T), \varphi(t+2 T) \dots, \varphi(t+(D-1)T)\big): t\in \mathbb R\}\subseteq \mathbb{R}^D\]
• Limit set: Given $(\mathcal M, \phi)$ a dynamical system and $x_0\in \mathcal M$, \[\mathcal A_{x_0} = \{x\in \mathcal M: \exists t_i\to \infty \text { with } \phi_{t_i}(x_0)\to x\}.\]
• Theorem (Takens).* Let $\mathcal{M}$ be a smooth, compact, Riemannian manifold. Let $T> 0$ be a real number and let $D > 2 \mathrm{dim}(\mathcal{M})$ be an integer. Then, for generic $\phi \in C^2(\mathbb{R} \times \mathcal{M}, \mathcal{M})$, $F\in C^2(\mathcal{M}, \mathbb{R})$ and $x_0\in \mathcal M$, if $\varphi_{x_0} = F(\phi_\bullet(x_0))$ is an observation of $(\mathcal M, \phi)$, then the limit set $\mathcal A_{x_0}$ is 'diffeomorphic'$^{**}$ to $\mathcal{M}_{T,D} (\varphi_{x_0})$.

• Source: X. Fernandez, E. Borghini, G. Mindlin, P. Groisman. Intrinsic persistent homology via density-based metric learning. Journal of Machine Learning Research 24(75):1−42 (2023).


• Github Repository: ximenafernandez/intrinsicPH


• Tutorial: Intrinsic persistent homology. AATRN Youtube Channel (2021)


• Python Library: fermat