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

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

$\bullet ~ ~\mathrm{Rips}_\epsilon(\mathcal{M}, d_E)\simeq \mathcal{M}$ 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)

Theorem (Groisman, Jonckheere, Sapienza, 2022)

Given $p>1$ and $q=(p-1)/d$, there exists a constant $\mu = \mu(p,d)$ such that, for any $x,y\in\mathcal{M}$, \[ \lim_{n\to +\infty} n^q d_{\mathbb{X}_n, p}(x,y) = \mu d_{f,q}(x,y )~ \text{almost surely.} \]

Theorem (Hwang, Damelin, Hero, 2016)

Given $\varepsilon > 0$ and $b>0$, there exists constants $\mu = \mu(d,p)>0$ and $\theta = \theta(\varepsilon)>0$ such that, for all sufficiently large $n$, \[ \mathbb {P} \left(\sup_{\substack{x,y: d_{\mathcal M}(x,y)\geq b}}\left|\frac{n^{q}L_{\mathbb X_n, p}(x,y)}{ \mu d_{f,q}(x,y)}-1\right|>\varepsilon\right)\leq \exp(-\theta n^{1/(d+2p)}) \] In particular, for every $x,y\in \mathcal M$, $ \lim_{n\to +\infty} n^{(p-1)/d} L_{\mathbb X_n, p}(x,y) = \mu d_{f,p}(x,y )~ \text{almost surely.} $

Steps of the proof:

$\bullet$ Starting points

Lemma

Given $\delta >0$, there exists $\epsilon>0$ such that for every $x,y\in \mathcal M$ with $|x-y|<\epsilon$, \[ d_{\mathcal M}(x,y)\leq (1+\delta)|x-y|. \]

Theorem (Hwang, Damelin, Hero, 2016)

Given $\varepsilon > 0$ and $b>0$, there exists constants $\mu = \mu(d,p)>0$ and $\theta = \theta(\varepsilon)>0$ such that, for all sufficiently large $n$, \[ \mathbb {P} \left(\sup_{\substack{x,y: d_{\mathcal M}(x,y)\geq b}}\left|\frac{n^{(q}L_{\mathbb X_n, p}(x,y)}{ \mu d_{f,q}(x,y)}-1\right|>\varepsilon\right)\leq \exp(-\theta n^{1/(d+2p)}) \]

Steps of the proof:

$\bullet$ Lemma 1 (Segment length)

Let $\epsilon>0$. Let $x,y\in \mathbb X_n$ such that they belong to some minimal path between points in $\mathcal M$ with respect to $d_{\mathbb X_n, p}$. Then, for some constant $\theta_1 > 0$ there exist $n$ large enough such that \[\mathbb P(|x-y|>\epsilon)\leq \exp(-\theta_1 n^{\alpha})\] for $\alpha = 1/(d+2p)$.

Steps of the proof:

$\bullet$ Lemma 1 (Segment length)

Let $\epsilon>0$. Let $x,y\in \mathbb X_n$ such that they belong to some minimal path between points in $\mathcal M$ with respect to $d_{\mathbb X_n, p}$. Then, for some constant $\theta_1 > 0$ there exist $n$ large enough such that \[\mathbb P(|x-y|>\epsilon)\leq \exp(-\theta_1 n^{\alpha})\] for $\alpha = 1/(d+2p)$.

$\bullet$ Proposition 2

Fix $\varepsilon > 0$ and a sequence of positive real numbers $(b_n)_{n \geq 1}$ satisfying that $\frac{\log(n)}{n b_n^{d}} \to 0$ when $n \to \infty$. Then, for every $p>1$, there exists $\theta>0$ such that \[ \mathbb{P}\left(\sup_{x,y}\left| \frac{n^{(p-1)/d} d_{\mathbb{X}_n, p}(x,y)}{d_{f,q}(x,y )} - \mu \right|>\varepsilon\right) \leq \exp\left(-\theta (n b_n^{d})^\alpha\right) \] for $n$ large enough, where the supremum is taken over $x,y\in \mathcal M$ with $d_{\mathcal M}(x,y)\geq b_n$. Here, $\alpha = 1/(d+2p)$.

Steps of the proof:

$\bullet$ Proposition 2

Fix $\varepsilon > 0$ and a sequence of positive real numbers $(b_n)_{n \geq 1}$ satisfying that $\frac{\log(n)}{n b_n^{d}} \to 0$ when $n \to \infty$. Then, for every $p>1$, there exists $\theta>0$ such that \[ \mathbb{P}\left(\sup_{x,y}\left| \frac{n^{(p-1)/d} d_{\mathbb{X}_n, p}(x,y)}{d_{f,q}(x,y )} - \mu \right|>\varepsilon\right) \leq \exp\left(-\theta (n b_n^{d})^\alpha\right) \] for $n$ large enough, where the supremum is taken over $x,y\in \mathcal M$ with $d_{\mathcal M}(x,y)\geq b_n$. Here, $\alpha = 1/(d+2p)$.

$\bullet$ Proposition 3

For every $p>1$ and $\lambda \in \big((p-1)/pd, 1/d\big)$, given $\varepsilon>0$ there exist $\mu, \theta>0$ such that \[ \mathbb P\left(\sup_{x,y}\left| n^{(p-1)/d} d_{\mathbb X_n, p}(x,y) - \mu d_{f,q}(x,y )\right|>\varepsilon\right) \le \exp\left(-\theta n^{(1 - \lambda d)\alpha}\right) \] for $n$ large enough, where the supremum is taken over $x,y\in \mathcal M$. Here $\alpha = 1/(d+2p)$.

Steps of the proof:

$\bullet~$ Recall \[d_{GH}\big( (\mathbb X_n, C(n,p,d)d_{\mathbb X_n, p}), (\mathcal M, d_{f,q})\big) = \frac{1}{2}\inf_R \sup_{(x,y), (x', y')\in R} | C(n, p, d) d_{\mathbb X_n, p}(x, x')- d_{f,q}(y, y')|,\] where the infimum is taken over subsets $R\subseteq \mathbb X_n\times \mathcal M$ such that the projections $\pi_{\mathcal M}(R) = \mathcal M$, $\pi_{\mathbb X_n}(R) = \mathbb X_n$.

$\bullet~$ Consider $ R=\{(x_y, y)\colon y \in \mathcal M, d_{f,q}(x_y, y) = d_{f,q}(\mathbb X_n, y)\}. $

$\bullet~$ By triangular inequality \[d_{GH}\big( (\mathbb X_n, C(n, p, d) d_{\mathbb X_n,p}), (\mathcal M, d_{f,q})\big) \le \frac{1}{2}\left(\sup_{x,x'\in \mathbb X_n} | C(n, p, d) d_{\mathbb X_n,p}(x,x')-d_{f,q}(x, x')| + 2 \sup_{y\in \mathcal M}d_{f,q}(\mathbb X_n, y) \right).\]

Prop (F., Borghini, Mindlin, Groisman, 2020/2023)

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 $k$.

• 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:

  In progress

• Computational experiments:

  ximenafernandez/intrinsicPH

• 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. F., E. Borghini, G. Mindlin, P. Groisman, Intrinsic persistent homology via density-based metric learning. Journal of Machine Learning Research (to appear), 2023. ArXiv:2012.07621 (2020)


• Github Repository: ximenafernandez/intrinsicPH


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