The topology of probability distributions on manifolds

By Omer Bobrowski, Sayan Mukherjee. 2013

60 citations recorded by the source corpus.

Let $$\mathcal {P}$$P be a set of $$n$$n random points in $$\mathbb {R}^d$$Rd, generated from a probability measure on a $$m$$m-dimensional manifold $$\mathcal {M}\subset \mathbb {R}^d$$M⊂Rd. In this paper we study the homology of $$\mathcal {U}(\mathcal {P},r)$$U(P,r)—the union of $$d$$d-dimensional balls of radius $$r$$r around $$\mathcal {P}$$P, as $$n\rightarrow \infty $$n→∞, and $$r\rightarrow 0$$r→0. In addition we study the critical points of $$d_{\mathcal {P}}$$dP—the distance function from the set $$\mathcal {P}$$P. These two objects are known to be related via Morse theory. We present limit theorems for the Betti numbers of $$\mathcal {U}(\mathcal {P},r)$$U(P,r), as well as for number of critical points of index $$k$$k for $$d_{\mathcal {P}}$$dP. Depending on how fast $$r$$r decays to zero as $$n$$n grows, these two objects exhibit different types of limiting behavior. In one particular case ($$n r^m \ge C \log n$$nrm≥Clogn), we show that the Betti numbers of $$\mathcal {U}(\mathcal {P},r)$$U(P,r) perfectly recover the Betti numbers of the original manifold $$\mathcal {M}$$M, a result which is of significant interest in topological manifold learning.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/328d3ba46b7f14439f65bbc55c72daaf79a89ac3

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Citations in the corpus

On the topology of random complexes built over stationary point processes. (2012, background): “This has lead, for example, to the Čech and Vietoris–Rips complexes on random Euclidean point sets, studied in papers such as [22, 24], with an extension to the manifold setting in [7].”
Maximally Persistent Cycles in Random Geometric Complexes (2015, background): “Several recent papers have studied various aspects of the topology of these complexes (see [6, 9, 11, 38, 40, 50, 51] and the survey [8]).”
Random geometric complexes in the thermodynamic regime (2014, background): “Recently [4] has established important – from the point of view of applications – extensions to the results of [3, 18, 20] in which the underlying point process lies on a manifold of lower dimension than an ambient Euclidean space in which the balls of (1.”
Topological Consistency via Kernel Estimation (2014, methodology): “Theorem 3.6 (Theorem 4.9 in [9]) .”