Paper: arXiv 2501.08768

Authors: Elie Attal, Romain Allez

Abstract

We consider the singular vectors of any $m \times n$ submatrix of a rectangular $M \times N$ Gaussian matrix and study their asymptotic overlaps with those of the full matrix, in the macroscopic regime where $N ,/, M,$, $m ,/, M$ as well as $n ,/, N$ converge to fixed ratios. Our method makes use of the dynamics of the singular vectors and of specific resolvents when the matrix coefficients follow Brownian trajectories. We obtain explicit forms for the limiting rescaled mean squared overlaps for right and left singular vectors in the bulk of both spectra, for any initial matrix $A,$. When it is null, this corresponds to the Marchenko-Pastur setup for covariance matrices, and our formulas simplify into Cauchy-like functions.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical treatment of eigenvector overlaps for rectangular Gaussian matrices using random matrix theory and stochastic calculus. However, it is purely theoretical without empirical backtesting or real-world financial data validation, focusing instead on theoretical formulations under the Marchenko-Pastur setup.

Research Flowchart

  flowchart TD
    A[Research Goal: Compute Overlaps of Submatrix vs Full Matrix Singular Vectors] --> B[Inputs: M x N Gaussian Matrix & m x n Submatrix Dynamics]
    B --> C[Methodology: Evolve Matrix via Brownian Dynamics]
    C --> D[Analysis: Study Resolvents & Vector Dynamics in Macroscopic Limit]
    D --> E[Outcome 1: Limiting Rescaled Mean Squared Overlaps for General Initial A]
    D --> F[Outcome 2: Cauchy-like Formulas for Null Matrix / Marchenko-Pastur Setup]

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1 of 40 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.