XIMENA FERNANDEZ
Durham University
CENTRE FOR TOPOLOGICAL DATA ANALYSIS
Group Meeting - 19th November 2021
∙ Fernandez X., Borghini E., Mindlin G., Groisman P. Intrinsic persistent homology via density-based metric learning, 2021. arXiv:2012.07621
Let Xn={x1,...,xn}⊆RD be a finite sample.
Let Xn={x1,...,xn}⊆RD be a finite sample.
Assume that:
Goal: Infer H∙(M)
Let Xn={x1,...,xn}⊆RD be a finite sample.
For p>1, the Fermat distance between x,y∈RD is defined by dXn,p(x,y)=inf 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.
\big(\mathbb{X}_n, C(n,p,d) d_{\mathbb{X}_n,p})\big)\xrightarrow[n\to \infty]{GH}\big(\mathcal{M}, d_{f,q}\big) ~~~ \text{ for } q = (p-1)/d
Theorem (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}\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.
\mathrm{dgm}(\mathrm{Filt}(\mathbb{X}_n, {C(n,p,d)} d_{\mathbb{X}_n,p}))\xrightarrow[n\to \infty]{B}\mathrm{dgm}(\mathrm{Filt}(\mathcal{M}, d_{f,q})) ~~~ \text{ for } q = (p-1)/d
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.
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).
fermat
Coming soon :)
ximenafernandez/intrinsicPH
\bullet Fernandez X., Borghini E., Mindlin G., Groisman P. Intrinsic persistent homology via density-based metric learning, 2021. arXiv:2012.07621
\bullet Fernandez X., Mateos D. Topological prediction of epileptic seizures. Work in progress, 2021.
Source data: PhysioNet Database https://physionet.org/about/database/
Source data: Private experiments. Laboratory of Dynamical Systems, University of Buenos Aires.
Source data: Private experiments. Institute of Applied Math of Litoral (IMAL-CONICET-UNL).