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

We propose three completely data-driven methods for estimating the real cohomology groups $H^k (X ; \mathbb{R})$ of a compact metric-measure space $(X, d_X, μ_X)$ embedded in a metric-measure space $(Y,d_Y,μ_Y)$, given a finite set of points $S$ sampled from a uniform distrbution $μ_X$ 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.