Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Phase Transition of Spectral Fluctuations in Large Gram Matrices with a Variance Profile: A Unified Framework for Sparse CLTs

Rui Wang, Guangming Pan, Dandan Jiang

TL;DR

The paper investigates the spectral fluctuations of large sparse Gram matrices with a variance profile, identifying a phase transition between moderate sparsity and high sparsity. Using resolvent methods and martingale differences, it proves almost-sure LSD convergence and develops central limit theorems for linear spectral statistics in two regimes, including a necessary centering correction in the high-sparsity regime. The results hold for Gaussian and non-Gaussian entries and extend to a broad class of analytic test functions, providing a unified framework for sparse CLTs. The practical impact is demonstrated through applications to equality testing of large-scale fading matrices and outage probability analysis in sparse MIMO systems, offering statistically tractable tools for high-dimensional wireless inference.

Abstract

We study the asymptotic spectral behavior of high-dimensional random Gram matrices with sparsity and a given variance profile, motivated by applications in wireless communication. Specifically, we consider the Gram matrices $\mathbf S_n=\mathbf Y_n\mathbf Y_n^*$, where the entries of $\mathbf Y_n$ are independent, centered, heteroscedastic, and sparse through Bernoulli masking. The sparsity level is parameterized as $s=q^2/n$, with $q$ ranging from polynomial order to order $n^{1/2}$. We investigate two asymptotic regimes in a high-dimensional framework: a moderate-sparsity regime with fixed $s\in(0,1]$, and a high-sparsity regime where $s\to0$. In both regimes, we establish the convergence of the empirical spectral distribution of $\mathbf S_n$ to a deterministic limit, and further derive central limit theorems for linear spectral statistics using resolvent techniques and martingale difference arguments. Our analysis reveals a phase transition in the fluctuation behavior across the two regimes. In the high-sparsity regime, the asymptotic fluctuations are governed by fourth-moment effects, with sparsity-scaled contributions being suppressed. Moreover, a mismatch between the scaling of the mean and variance, of different orders in $q$, necessitates an explicit correction in the centering of the linear spectral statistic. The theory applies to both Gaussian and non-Gaussian entries, and its statistical utility is illustrated through applications to hypothesis testing and outage probability analysis in large-scale MIMO systems.

Phase Transition of Spectral Fluctuations in Large Gram Matrices with a Variance Profile: A Unified Framework for Sparse CLTs

TL;DR

The paper investigates the spectral fluctuations of large sparse Gram matrices with a variance profile, identifying a phase transition between moderate sparsity and high sparsity. Using resolvent methods and martingale differences, it proves almost-sure LSD convergence and develops central limit theorems for linear spectral statistics in two regimes, including a necessary centering correction in the high-sparsity regime. The results hold for Gaussian and non-Gaussian entries and extend to a broad class of analytic test functions, providing a unified framework for sparse CLTs. The practical impact is demonstrated through applications to equality testing of large-scale fading matrices and outage probability analysis in sparse MIMO systems, offering statistically tractable tools for high-dimensional wireless inference.

Abstract

We study the asymptotic spectral behavior of high-dimensional random Gram matrices with sparsity and a given variance profile, motivated by applications in wireless communication. Specifically, we consider the Gram matrices , where the entries of are independent, centered, heteroscedastic, and sparse through Bernoulli masking. The sparsity level is parameterized as , with ranging from polynomial order to order . We investigate two asymptotic regimes in a high-dimensional framework: a moderate-sparsity regime with fixed , and a high-sparsity regime where . In both regimes, we establish the convergence of the empirical spectral distribution of to a deterministic limit, and further derive central limit theorems for linear spectral statistics using resolvent techniques and martingale difference arguments. Our analysis reveals a phase transition in the fluctuation behavior across the two regimes. In the high-sparsity regime, the asymptotic fluctuations are governed by fourth-moment effects, with sparsity-scaled contributions being suppressed. Moreover, a mismatch between the scaling of the mean and variance, of different orders in , necessitates an explicit correction in the centering of the linear spectral statistic. The theory applies to both Gaussian and non-Gaussian entries, and its statistical utility is illustrated through applications to hypothesis testing and outage probability analysis in large-scale MIMO systems.
Paper Structure (18 sections, 11 theorems, 99 equations, 4 figures, 2 tables)

This paper contains 18 sections, 11 theorems, 99 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The model and preliminaries
  3. The model
  4. Notations
  5. Preliminary results
  6. Main results
  7. The limiting spectra distribution
  8. CLT in the moderate-sparsity regime
  9. CLT in the high-sparsity regime
  10. Applications
  11. Application to equality test of two large-scale fading matrices
  12. Application to outage probability analysis in sparse MIMO channels
  13. Numerical results
  14. Equality test of two large-scale fading matrices
  15. Outage probability analysis
  16. ...and 3 more sections

Key Result

Lemma 2.5

Assuming $\mathbf{x}_j=\mathbb B_j\mathbf w_j/\sqrt{s}$ follows our model, we consider any $p \times p$ nonrandom symmetric matrices $\mathbf{A}$ and $\mathbf{B}$. In this context, we establish the following result: where $\kappa=1$ for the real case and 0 for the complex case, and $\tilde{\nu}_4 = \mathbb{E}\!\left[\left|{w_{ij}}/{\sigma_{ij}}\right|^4\right]$ is the standardized fourth moment o

Figures (4)

  • Figure 1: Comparison of empirical and theoretical distributions of $\tilde{T}_{\log}$ when $n=10000$.
  • Figure 2: Comparison of empirical and theoretical powers of the test $T_x$ under the Gaussian distribution.
  • Figure 3: Comparison of empirical and theoretical distributions of $T_{\log}$.
  • Figure 4: Comparison of empirical and theoretical outage probability.

Theorems & Definitions (16)

  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Theorem 3.1
  • Remark 3.2
  • Theorem 3.3
  • Remark 3.4
  • Corollary 3.5
  • Remark 3.6
  • Theorem 4.1
  • ...and 6 more