Eigenvector Overlaps of Random Covariance Matrices and their Submatrices (2501.08768v1)

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.

• The paper analyzes the asymptotic behavior of eigenvector overlaps in random covariance matrices and their submatrices using eigenvector dynamics and resolvent techniques.
• It derives explicit Cauchy-like formulas for mean squared overlaps, providing insights into Principal Component Analysis (PCA) of noisy data, particularly in the Marchenko-Pastur case.
• The findings have implications for large-dimensional data applications in signal processing, image analysis, and finance, informing methodologies like PCA in noisy environments.

Summary of "Eigenvector Overlaps of Random Covariance Matrices and their Submatrices"

This paper by Attal Elie and Allez Romain examines the asymptotic behavior of eigenvector overlaps in random covariance matrices and their submatrices. The paper is conducted within the framework of the macroscopic regime, where matrix dimensions increase while maintaining constant ratios.

Main Contributions

1. Analysis Framework: The authors analyze the singular vectors of submatrices derived from an $M \times N$ Gaussian matrix perturbed by Brownian motion. They introduce a matrix $X_t$ as a noisy observation of a deterministic matrix $A$, and its submatrix $\tilde{X}_t$. They derive explicit forms for the asymptotic rescaled mean squared overlaps for both left and right singular vectors.
2. Asymptotic Formulas: Using a combination of eigenvector dynamics and resolvent techniques, the paper derives explicit Cauchy-like formulas for the mean squared overlaps. These results are significant when the initial matrix $A$ is null, leading to simplified expressions under the Marchenko-Pastur distribution.
3. Implications on PCA: The findings provide insights into the information preserved in a subimage of a noisy rectangular image via singular vectors. This is particularly applicable to comparing the Principal Component Analysis (PCA) of a random matrix and its submatrices.
4. Derivation of the Burgers Equation: The paper solves the dynamic behavior of the Stieltjes transform associated with the spectra of these matrices. The resulting Burgers equation characterizes the deterministic limits of the spectral densities.
5. Computation of Overlaps: The authors develop a system of coupled differential equations that describe the evolution of overlaps. This system is subsequently solved to yield explicit asymptotic forms for various overlap measures, such as $$N \, \mathbb{E}[\langle \tilde{v}_i^t, v_j^t \rangle^2]$$, capturing the expected squared overlaps of singular vectors from the full and submatrices.
6. The Marchenko-Pastur Case: Special attention is given to the simplification of results in the null matrix setup, corresponding to the standard Marchenko-Pastur setting. This yields Cauchy distribution-like results for overlaps and clear expressions regarding PCA dimensionality reduction under matrix perturbation.

Implications and Future Directions

The results have significant implications for applications involving large dimensional data, including signal processing, image analysis, and financial data correlations. The understanding of eigenvector stability and overlaps underpin crucial methodologies like PCA in noisy environments, which can be extended to incremental PCA and PCA with missing data. Moreover, these findings contribute to enhancing algorithmic performance in areas like sectoral portfolio optimization and adaptive process monitoring.

Additionally, this work provides a theoretical foundation for future investigations of eigenvector overlaps in diverse matrix forms, potentially impacting areas like wireless communications and statistical learning where covariance structure plays a pivotal role. More broadly, the methodologies outlined could initiate further explorations into AI model robustness and optimization techniques involving large data matrices under perturbation.