Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Persistence diagrams of random matrices via Morse theory: universality and a new spectral diagnostic

Matthew Loftus

Abstract

We prove that the persistence diagram of the sublevel set filtration of the quadratic form f(x) = x^T M x restricted to the unit sphere S^{n-1} is analytically determined by the eigenvalues of the symmetric matrix M. By Morse theory, the diagram has exactly n-1 finite bars, with the k-th bar living in homological dimension k-1 and having length equal to the k-th eigenvalue spacing s_k = λ_{k+1} - λ_k. This identification transfers random matrix theory (RMT) universality to persistence diagram universality: for matrices drawn from the Gaussian Orthogonal Ensemble (GOE), we derive the closed-form persistence entropy PE = log(8n/π) - 1, and verify numerically that the coefficient of variation of persistence statistics decays as n^{-0.6}. Different random matrix ensembles (GOE, GUE, Wishart) produce distinct universal persistence diagrams, providing topological fingerprints of RMT universality classes. As a practical consequence, we show that persistence entropy outperforms the standard level spacing ratio \langle r \rangle for discriminating GOE from GUE matrices (AUC 0.978 vs. 0.952 at n = 100, non-overlapping bootstrap 95% CIs), and detects global spectral perturbations in the Rosenzweig-Porter model to which \langle r \rangle is blind. These results establish persistence entropy as a new spectral diagnostic that captures complementary information to existing RMT tools.

Persistence diagrams of random matrices via Morse theory: universality and a new spectral diagnostic

Abstract

We prove that the persistence diagram of the sublevel set filtration of the quadratic form f(x) = x^T M x restricted to the unit sphere S^{n-1} is analytically determined by the eigenvalues of the symmetric matrix M. By Morse theory, the diagram has exactly n-1 finite bars, with the k-th bar living in homological dimension k-1 and having length equal to the k-th eigenvalue spacing s_k = λ_{k+1} - λ_k. This identification transfers random matrix theory (RMT) universality to persistence diagram universality: for matrices drawn from the Gaussian Orthogonal Ensemble (GOE), we derive the closed-form persistence entropy PE = log(8n/π) - 1, and verify numerically that the coefficient of variation of persistence statistics decays as n^{-0.6}. Different random matrix ensembles (GOE, GUE, Wishart) produce distinct universal persistence diagrams, providing topological fingerprints of RMT universality classes. As a practical consequence, we show that persistence entropy outperforms the standard level spacing ratio \langle r \rangle for discriminating GOE from GUE matrices (AUC 0.978 vs. 0.952 at n = 100, non-overlapping bootstrap 95% CIs), and detects global spectral perturbations in the Rosenzweig-Porter model to which \langle r \rangle is blind. These results establish persistence entropy as a new spectral diagnostic that captures complementary information to existing RMT tools.

Paper Structure

This paper contains 22 sections, 3 theorems, 15 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Mathematical framework
  3. Morse theory of quadratic forms on spheres
  4. Sublevel set topology and persistence diagram
  5. Persistence statistics
  6. Analytical results
  7. Persistence statistics from eigenvalue density
  8. Closed form for GOE
  9. Closed form for Wishart
  10. Numerical verification
  11. Universality within GOE
  12. Analytical formula verification
  13. Ensemble comparison and fingerprinting
  14. Persistence entropy as a spectral diagnostic
  15. GOE vs. GUE discrimination
  16. ...and 7 more sections

Key Result

Theorem 1

For $\lambda_k < c < \lambda_{k+1}$ (with the convention $\lambda_0 = -\infty$), the sublevel set $f^{-1}(-\infty, c]$ is homotopy equivalent to $S^{k-1}$ for $k \ge 1$, and empty for $k = 0$.

Figures (5)

  • Figure 1: Eigenvalue empirical CDF for 200 independent GOE$(100)$ matrices (blue), overlaid with the Wigner semicircle CDF (black). The near-perfect collapse demonstrates eigenvalue universality, which by Theorem \ref{['thm:pd']} implies persistence diagram universality.
  • Figure 2: Unfolded bulk spacing distributions (central 80% of eigenvalues) for GOE, GUE, and Wishart ensembles ($n = 100$, 200 samples each). Histograms match the Wigner surmise: $p_1(s) = (\pi/2)s\exp(-\pi s^2/4)$ for $\beta = 1$ (GOE, Wishart) and $p_2(s) = (32/\pi^2)s^2\exp(-4s^2/\pi)$ for $\beta = 2$ (GUE). Since bar lengths equal eigenvalue spacings, these are also the universal bar length distributions of the persistence diagrams.
  • Figure 3: (a) Coefficient of variation of TP and PE vs. matrix size $n$, confirming universality (CV $\to 0$). (b) Persistence entropy: numerical values (200 GOE samples per $n$, error bars show $\pm 1$ std) vs. the closed-form prediction $\mathrm{PE} = \log(8n/\pi) - 1$ (solid line).
  • Figure 4: AUC for GOE vs. GUE discrimination as a function of matrix size $n$ (500 samples per class). PE (circles) consistently outperforms $\langle r \rangle$ (squares). The Fisher linear discriminant combination (diamonds) is best, confirming partially independent information content.
  • Figure 5: Signal-to-noise ratio (deviation from GOE reference divided by GOE standard deviation) for three spectral diagnostics across the Rosenzweig--Porter model ($n = 100$, 300 samples per $\lambda$). $\langle r \rangle$ (squares) remains at its GOE value for all $\lambda \le 5$, while PE (circles) and the spacing variance (triangles) detect the global density change at $3\sigma$ by $\lambda = 0.7$ and $0.5$, respectively.

Theorems & Definitions (7)

  • Theorem 1
  • proof
  • Theorem 2: Persistence diagram structure
  • proof
  • Remark 3
  • Proposition 4
  • proof