[Paper Review] Tracy-Widom distribution for the edge eigenvalues of Gram type random matrices.

Xiucai Ding, Fan Yang|arXiv (Cornell University)|Aug 10, 2020

Random Matrices and Applications97 references7 citations

TL;DR

This paper establishes the Tracy-Widom distribution for edge eigenvalues of high-dimensional Gram-type random matrices under nearly sharp moment conditions and regularity assumptions. It proves asymptotic edge universality via rectangular Dyson Brownian motion, enabling adaptive inference for signal detection, sphericity testing, and block model structure analysis in high-dimensional statistics and machine learning.

ABSTRACT

Large dimensional Gram type matrices are common objects in high-dimensional statistics and machine learning. In this paper, we study the limiting distribution of the edge eigenvalues for a general class of high-dimensional Gram type random matrices, including separable sample covariance matrices, sparse sample covariance matrices, bipartite stochastic block model and random Gram matrices with general variance profiles. Specifically, we prove that under (almost) sharp moment conditions and certain tractable regularity assumptions, the edge eigenvalues, i.e., the largest few eigenvalues of non-spiked Gram type random matrices or the extremal bulk eigenvalues of spiked Gram type random matrices, satisfy the Tracy-Widom distribution asymptotically. Our results can be used to construct adaptive, accurate and powerful statistics for high-dimensional statistical inference. In particular, we propose data-dependent statistics to infer the number of signals under general noise structure, test the one-sided sphericity of separable matrix, and test the structure of bipartite stochastic block model. Numerical simulations show strong support of our proposed statistics. The core of our proof is to establish the edge universality and Tracy-Widom distribution for a rectangular Dyson Brownian motion with regular initial data. This is a general strategy to study the edge statistics for high-dimensional Gram type random matrices without exploring the specific independence structure of the target matrices. It has potential to be applied to more general random matrices that are beyond the ones considered in this paper.

Motivation & Objective

• To derive the limiting distribution of edge eigenvalues for a broad class of high-dimensional Gram-type random matrices.
• To establish edge universality and Tracy-Widom asymptotic behavior without relying on specific independence structures of the matrices.
• To develop data-dependent statistics for high-dimensional inference under general noise and network structures.
• To enable accurate detection of signals, sphericity testing, and block model structure testing in high-dimensional settings.

Proposed method

• Analyzes rectangular Dyson Brownian motion with regular initial data to derive edge universality.
• Applies a general strategy that avoids dependence on specific matrix independence structures.
• Uses nearly sharp moment conditions and tractable regularity assumptions to ensure asymptotic validity.
• Derives the Tracy-Widom distribution for extremal eigenvalues in non-spiked and spiked matrix models.
• Proposes data-driven test statistics for signal count estimation, one-sided sphericity, and bipartite stochastic block model testing.
• Validates the method through numerical simulations demonstrating strong empirical support.

Experimental results

Research questions

• RQ1 Do edge eigenvalues of high-dimensional Gram-type random matrices converge to the Tracy-Widom distribution under general moment and regularity conditions?
• RQ2 Can edge universality be established without exploiting specific independence structures in the matrix entries?
• RQ3 How can the Tracy-Widom limit be leveraged to construct adaptive and powerful inference statistics in high-dimensional settings?
• RQ4 Can the proposed framework detect the number of signals under general noise structures?
• RQ5 Can the method test the one-sided sphericity of separable covariance matrices and the structure of bipartite stochastic block models?

Key findings

• The edge eigenvalues of non-spiked Gram-type matrices and extremal bulk eigenvalues of spiked ones converge to the Tracy-Widom distribution under nearly sharp moment conditions.
• Edge universality is established via a general framework based on rectangular Dyson Brownian motion with regular initial data.
• The method applies broadly to separable sample covariance matrices, sparse covariance matrices, bipartite stochastic block models, and random Gram matrices with general variance profiles.
• Data-dependent statistics are proposed for signal count detection, one-sided sphericity testing, and block model structure inference.
• Numerical simulations confirm the strong empirical performance and robustness of the proposed inference methods.
• The approach is transferable to other random matrix classes beyond those explicitly studied, due to its structural generality.