Statistical Invariance of Betti Numbers in the Thermodynamic Regime
Year: 2020
Type: preprint
Abstract: Topological Data Analysis (TDA) has become a promising tool to uncover low-dimensional features from high-dimensional data. Notwithstanding the advantages afforded by TDA, its adoption in statistical methodology has been limited by several reasons. In this paper we study the framework of topological inference through the lens of classical parametric inference in statistics. Suppose $\mathcal{P} = \{\mathbb{P}_\theta : \theta \in \Theta\}$ is a parametric family of distributions indexed by a set $\Theta$, and $\mathbb{X}_n = \{\mathbf{X}_1, \mathbf{X}_2, \dots, \mathbf{X}_n\}$ is observed iid at random from a distribution $\mathbb{P}_\theta$. The asymptotic behaviour of the Betti numbers associated with the Cech complex of $\mathbb{X}_n$ contain both the topological and parametric information about the distribution of points. We investigate necessary and sufficient conditions under which topological inference is possible in this parametric setup.
Source: arXiv (Cornell University)
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2996843894