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 Riemannian manifold $\mathcal M$ of dimension $d$.
• The points are sampled according to a density $f\colon \mathcal M\to \mathbb R$.

Goal: Infer $H_\bullet(\mathcal M)$

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 $\{x_1, x_2, \dots, x_{r}\}\subseteq \mathbb{X}_n$.

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

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

Corollary (Borghini, F., Groisman, Mindlin)

Let $\mathcal{M}$ be a closed smooth $d$-dimensional Riemannian manifold embedded in $\mathbb{R}^D$. Let $\mathbb X_n\subseteq \mathcal{M}$ be a set of $n$ independent sample points with common smooth density $f:\mathcal{M}\to \mathbb{R}_{>0}$.

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 \[ \mathbb{P}\Big( d_B\big(\mathrm{dgm}(\mathrm{Filt}(\mathcal{M}, d_{f,q})),\mathrm{dgm}(\mathrm{Filt}(\mathbb{X}_n, {\scriptstyle \frac{n^{q}}{\mu}} d_{\mathbb{X}_n,p}))\big)>\varepsilon\Big)\\\leq \exp{\big(-\theta n^{(1 - \lambda d)/(d+2p)}\big)}\] for $n$ large enough.

Prop (Borghini, F., Groisman, Mindlin)

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. There exists $\delta >0$ such that 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 $k$.

• 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

• Tool in Giotto-TDA:

  Coming soon :)

• Computational experiments:

  ximenafernandez/intrinsicPH

• Signal: $X:[t_0,t_1] \to \mathbb{R}$
  
• Delay embedding: Given $T$ the time delay and $D$ the embedding dimension. \[\mathcal{M}_{[t_0, t_1]} = \{\big(X(t), X(t+T), X(t+2 T) \dots, X(t+(D-1)T)\big): t\in [t_0, t_1-(D-1)T]\}\subseteq \mathbb{R}^D\]