Noncommutative Model Selection and the Data-Driven Estimation of Real Cohomology Groups

TL;DR

This paper introduces three data-driven methods for estimating real cohomology groups of a space from sampled data, demonstrating their effectiveness through computational experiments, especially in Euclidean embeddings.

Contribution

The paper presents novel algorithms for cohomology estimation that are fully data-driven and applicable to noisy, finite samples from metric-measure spaces.

Findings

Two of the three proposed algorithms performed well in experiments.

Methods are effective for spaces embedded in Euclidean space.

Algorithms handle noise in sampled data.

Abstract

We propose three completely data-driven methods for estimating the real cohomology groups $H^k (X ; \mathbb{R})$ of a compact metric-measure space $(X, d_X, \mu_X)$ embedded in a metric-measure space $(Y,d_Y,\mu_Y)$, given a finite set of points $S$ sampled from a uniform distrbution $\mu_X$ on $X$, possibly corrupted with noise from $Y$. We present the results of several computational experiments in the case that $X$ is embedded in $\mathbb{R}^n$, where two of the three algorithms performed well.