• Clustering.
  $K$-medoids method aims to partition $\mathbb X_n$ in $S_1, S_2, \dots, S_k$ with associated $x_{S_i}\in S_i$ to minimize \[\sum_{i=1}^k\sum_{x\in S_i}d(x, x_{S_i})\]

• Clustering.
  $K$-medoids method aims to partition $\mathbb X_n$ in $S_1, S_2, \dots, S_k$ with associated $x_{S_i}\in S_i$ to minimize \[\sum_{i=1}^k\sum_{x\in S_i}d(x, x_{S_i})\]
• Dimensionality reduction.
  Multi-dimensional scaling (MDS) aims to find a projection $p:(\mathbb X_n, d)\to (\mathbb R^k, d_E)$ that minimizes \[\sum_{i\neq j=1,\dots,n} d(x_i,x_j)-d_E\big(p(x_i), p(x_j)\big)\]

• Clustering.
  $K$-medoids method aims to partition $\mathbb X_n$ in $S_1, S_2, \dots, S_k$ with associated $x_{S_i}\in S_i$ to minimize \[\sum_{i=1}^k\sum_{x\in S_i}d(x, x_{S_i})\]
• Dimensionality reduction.
  Multi-dimensional scaling (MDS) aims to find a projection $p:(\mathbb X_n, d)\to (\mathbb R^k, d_E)$ that minimizes \[\sum_{i\neq j=1,\dots,n} d(x_i,x_j)-d_E\big(p(x_i), p(x_j)\big)\]

• Clustering.
  $K$-medoids method aims to partition $\mathbb X_n$ in $S_1, S_2, \dots, S_k$ with associated $x_{S_i}\in S_i$ to minimize \[\sum_{i=1}^k\sum_{x\in S_i}d(x, x_{S_i})\]
• Dimensionality reduction.
  Multi-dimensional scaling (MDS) aims to find a projection $p:(\mathbb X_n, d)\to (\mathbb R^k, d_E)$ that minimizes \[\sum_{i\neq j=1,\dots,n} d(x_i,x_j)-d_E\big(p(x_i), p(x_j)\big)\]
• Persistent homology
  Filtration from the metric structure. The $k$-simplices of the Vietoris-Rips complex $VR_\epsilon(\mathbb X_n)$ are \[\{x_0, \dots, x_k\}\subseteq \mathbb X_n ~\text{ s.t. }~d(x_i, x_j)\leq \epsilon ~~~\forall i\neq j.\]

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 $f\colon \mathcal M\to \mathbb R$.

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 $q=(p-1)/d$.

Set $q = (p-1)/d.$

• (Hwang, Damelin & Hero, 2016) If $L_{\mathbb X_n, p}(x,y) := \inf_{\gamma}\sum_{i=0}^{k}d_{\mathcal M}(x_{i+1},x_i)^{p}$ over all paths $\gamma=(x_0, \dots, x_{r+1})$ of finite length with $x_0=x$, $x_{r+1} = y$ and $\{x_1, \dots, x_r\}\subseteq \mathbb{X}_n$, then \[C(n,p,d) L_{\mathbb{X}_n,p}\underset{n \to \infty}{\overset{a.s}{\rightrightarrows}}d_{f,q}~~~ \text{ in } \{(x,y)\in \mathcal{M}: d_{\mathcal M}(x,y)\geq b\}.\]
• (Groisman, Jonckheere & Sapienza, 2022) If there exists $S\subseteq \mathbb{R}^d$ an open connected set and $\phi:S \to \mathbb{R}^D$ an isometric transformation such that $\phi(\bar S)=\mathcal{M}.$ \[ \lim_{n\to +\infty} C(n,p,d) d_{\mathbb{X}_n, p}(x,y) = d_{f,q}(x,y )~ \text{almost surely.}\]

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 $k$.

Results in the noisy case
If $\widetilde{\mathbb{X}}_n = \{\widetilde {x}_1, \widetilde {x}_2, \dots, \widetilde {x}_n\} $ such that $\widetilde{x}_i = x_i + \xi_i $ with $x_i \in \mathcal M$ and $\xi_i\in \mathbb R^D$ 'noise', \[\big(\mathbb{\widetilde X}_n, C(n,p,d) d_{\mathbb{\widetilde X}_n,p}\big)\xrightarrow[n\to \infty]{GH}\big(\mathcal{M}, d_{f,q}\big) ?\]



Density-based homology theory
Given $X$ a topological space and $\mu:X\to \mathbb R$ a density, 'understand' the homology of $(X,\mu)$.

• 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