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

Araceli Guzmán-Tristán, Antonio Rieser, Eduardo Velázquez-Richards

TL;DR

This work tackles the challenge of estimating the real cohomology groups $H^q(X;\mathbb{R})$ of a compact metric-measure space from finite, uniformly sampled points by introducing three data-driven methods that translate topology estimation into the analytic problem of fitting heat semigroups generated by combinatorial Hodge-Laplacians on weighted Vietoris-Rips models. The authors leverage model-selection-inspired criteria, notably relative von Neumann entropy, plus two alternative distance measures, to identify a scale $\hat{r}$ at which the $q$-dimensional spectral information best reveals the invariant, then recover $H^q(X;\mathbb{R})$ from the kernel of $\Delta_{q,\hat{r}}$. Empirical tests on datasets including points on a circle, two circles, and nonuniform two-circle configurations show strong performance for the entropy- and trace-based criteria, providing the first fully data-driven Betti-number estimations in a uniform-sampling setting and signaling directions for extending to broader distributions and convergence analysis. The approach offers a novel, nonpersistent, operator-theoretic pathway for topology-aware data analysis with potential computational advantages and opens avenues in noncommutative statistics for topological inference from finite samples.

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 $(X, d_X, μ_X)$ 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.

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

TL;DR

This work tackles the challenge of estimating the real cohomology groups of a compact metric-measure space from finite, uniformly sampled points by introducing three data-driven methods that translate topology estimation into the analytic problem of fitting heat semigroups generated by combinatorial Hodge-Laplacians on weighted Vietoris-Rips models. The authors leverage model-selection-inspired criteria, notably relative von Neumann entropy, plus two alternative distance measures, to identify a scale at which the -dimensional spectral information best reveals the invariant, then recover from the kernel of . Empirical tests on datasets including points on a circle, two circles, and nonuniform two-circle configurations show strong performance for the entropy- and trace-based criteria, providing the first fully data-driven Betti-number estimations in a uniform-sampling setting and signaling directions for extending to broader distributions and convergence analysis. The approach offers a novel, nonpersistent, operator-theoretic pathway for topology-aware data analysis with potential computational advantages and opens avenues in noncommutative statistics for topological inference from finite samples.

Abstract

We propose three completely data-driven methods for estimating the real cohomology groups of a compact metric-measure space embedded in a metric-measure space , given a finite set of points sampled from a uniform distrbution on , possibly corrupted with noise from . We present the results of several computational experiments in the case that is embedded in , where two of the three algorithms performed well.

Paper Structure

This paper contains 10 sections, 4 theorems, 10 equations, 3 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Contributions
  4. Organization
  5. The Weighted Vietoris-Rips Complex and Combinatorial Hodge Theory for Metric Spaces
  6. The Combinatorial Hodge Theorem and Weighted Simplicial Complexes
  7. Estimation of the $q$-Dimensional Heat Semigroups
  8. Relative Entropy in Operator Spaces
  9. Algorithms and Results
  10. Discussion and Future Work

Key Result

Theorem 2.1

For any abstract simplicial complex $K$ with Hodge-Laplacians $\Delta_n(K)$,

Figures (3)

  • Figure 4.1: An example showing the typical values of the quantities used for the combinatorial Hodge-Laplacian selection in the experiment with points uniformly sampled from a single circle.
  • Figure 4.2: An example showing the typical values of the quantities used for the combinatorial Hodge-Laplacian selection in the experiment with points uniformly sampled from a two circles embedded in $\mathbb{R}^3$. (The circles appear distorted due to the viewing angle of the image.)
  • Figure 4.3: An example showing the typical values of the quantities used for the combinatorial Hodge-Laplacian selection in the experiment with points non-uniformly sampled from a two circles embedded in $\mathbb{R}^3$. (The circles appear distorted due to the viewing angle of the image.)

Theorems & Definitions (10)

  • Theorem 2.1: Eckmann 1945
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6: Latschev Latschev_2001
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof