Modeling and replicating statistical topology and evidence for CMB nonhomogeneity

Abstract

Under the banner of "big data," the detection and classification of structure in extremely large, high-dimensional, data sets are two of the central statistical challenges of our times. Among the most intriguing new approaches to this challenge is "TDA," or "topological data analysis," one of the primary aims of which is providing nonmetric, but topologically informative, preanalyses of data which make later, more quantitative, analyses feasible. While TDA rests on strong mathematical foundations from topology, in applications, it has faced challenges due to difficulties in handling issues of statistical reliability and robustness, often leading to an inability to make scientific claims with verifiable levels of statistical confidence. We propose a methodology for the parametric representation, estimation, and replication of persistence diagrams, the main diagnostic tool of TDA. The power of the methodology lies in the fact that even if only one persistence diagram is available for analysis-the typical case for big data applications-the replications permit conventional statistical hypothesis testing. The methodology is conceptually simple and computationally practical, and provides a broadly effective statistical framework for persistence diagram TDA analysis. We demonstrate the basic ideas on a toy example, and the power of the parametric approach to TDA modeling in an analysis of cosmic microwave background (CMB) nonhomogeneity.

Persistent homology and non-Gaussianity

“…It is worth noting the relation of this paper to [55], which aims to put TDA on a sta-tistically sound footing. Their method circumvents the need for simulations by using just one PD, then generating statistically similar PDs.…”

Homological percolation and the Euler characteristic

“…Persistent homology has been demonstrated to be a powerful tool in the analysis of various types of data (e.g. neuroscience [31], cosmology [1], and complex networks [21]).…”

Modeling of persistent homology

“…i . Then σ 2 H is the (un-normalized) variance of the horizontal points, and σ 2 V is the L 2 power of the vertical points (not centered because of the non-negativeness of x (2) ). As for the local interaction, for x ∈ X and for k ≥ 1 let x nn (k) ∈ X be the k-th nearest neighbor of x, and set…”

“…In a previous paper [2] we entered the diagram (1.2) at the intermediate step, by suggesting a new approach to providing multiple instances of a persistence diagram when, perhaps, only one such original diagram is available. This was done via probabilistic modeling of persistence diagrams.…”

“…This was done via probabilistic modeling of persistence diagrams. We shall briefly recall the basic ideas of [2] in Sections 2 and 3 below, and then develop them, in terms of more sophisticated examples than were treated there, in Section 4.…”