Noncommutative Model Selection and the Data-Driven Estimation of Real Cohomology Groups
This work addresses a domain-specific problem in computational topology for researchers dealing with noisy data in metric spaces, but it appears incremental as it builds on existing methods without broad SOTA claims.
The authors tackled the problem of estimating real cohomology groups from noisy, finite point samples of a compact metric-measure space, proposing three data-driven methods and finding that two performed well in computational experiments.
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.