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 the homology 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 $\mathbb{X}_n$?

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 $\mathbb{X}_n = \{x_1,...,x_n\}\subseteq \mathbb{R}^D$ 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 $f\colon \mathcal M\to \mathbb R$.

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 $$d_{f,q}(x,y) = \inf_{\gamma} \int_{I}\frac{1}{f(\gamma_t)^{q}}||\dot{\gamma}_t|| dt$$ 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 $\{x_1, x_2, \dots, x_{r}\}\subseteq \mathbb{X}_n$.

• 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 the delay embedding $\mathcal{M}_{T,D} (\varphi_{x_0})$.

• 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).


• X. Fernandez, D. Mateos Topological biomarkers for real-time detection of epileptic seizures. arXiv:2211.02523 (2022).


• Github Repositories: ximenafernandez/intrinsicPH $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$ximenafernandez/epilepsy


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


• Python Library: fermat