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 Hk (X ; ℝ) of a compact metric-measure space (X, dX, μX) embedded in a metric-measure space (Y,dY,μY), given a finite set of points S sampled from a uniform distrbution μX on X, possibly corrupted with noise from Y. We present the results of several computational experiments in the case that X is embedded in ℝn, where two of the three algorithms performed well.