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 $(X,d_X,{\mu }_X)$ 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.