Statistical Invariance of Betti Numbers in the Thermodynamic Regime
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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 P = {P_θ : θ∈Θ} is a parametric family of distributions indexed by a set Θ, and X_n = {X_1, X_2, ..., X_n} is observed iid at random from a distribution P_θ. The asymptotic behaviour of the Betti numbers associated with the Čech complex of 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.
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