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 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 $\mathbb{X}_n$?

Gardner et al. 'Toroidal topology of population activity in grid cells'. Nature. (2022)

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 $$d_{f,q}(x,y) = \inf_{\gamma} \int_{I}\frac{1}{f(\gamma_t)^{q}}||\dot{\gamma}_t|| dt$$ 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 $d_{f,q}$ if $q=(p-1)/d$.

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

• 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