Matrix Concentration Inequalities and Free Probability II. Two-sided Bounds and Applications

TL;DR

The paper develops a two-sided theory of matrix concentration for spectral edges by comparing general Gaussian random matrices to a deterministic noncommutative model $X_{\rm free}$. It introduces sharp, nonasymptotic bounds for the whole spectrum via Hausdorff distance and extends them to non-Gaussian models through universality and quadratic perturbations. The authors prove isotropic and anisotropic phase-transition theorems that describe when spectral outliers detach from the bulk in broad classes of spiked models, with explicit thresholds and eigenvector overlap results. These general results yield concrete, nonasymptotic implications for decoding graphs, Tensor PCA, spike detection in block models, contextual SBMs, and sample covariance error, highlighting the practical impact on high-dimensional statistical inference and random matrix theory.

Abstract

The first paper in this series introduced a new family of nonasymptotic matrix concentration inequalities that sharply capture the spectral properties of very general random matrices in terms of an associated noncommutative model. These methods achieved matching upper and lower bounds for smooth spectral statistics, but only provided upper bounds for the spectral edges. Here we obtain matching lower bounds for the spectral edges, completing the theory initiated in the first paper. The resulting two-sided bounds enable the study of problems that require an exact determination of the spectral edges to leading order, which is fundamentally beyond the reach of classical matrix concentration inequalities. To illustrate their utility, we develop two general results that explain phase transitions of spectral outliers of a large class of nonhomogeneous random matrices. This enables us to elucidate phase transition phenomena that arise in diverse applications, including decoding node labels on graphs, tensor PCA, contextual stochastic block models, and centered sample covariance matrices.

Figures (5)

• Figure 1.1: Illustration of a hypothetical obstruction to the validity of Theorem \ref{['thm:normtwoside']}. The proof must show that this situation cannot occur.
• Figure 1.2: Spectral distribution of $X_{\rm free}$ for the spiked Wigner model.
• Figure 3.1: Outlier phase transition of a $2000\times 2000$ random band matrix with band width $101$. The markers represent the empricial result, the lines represent the theoretical prediction of Theorem \ref{['thm:simplebbp']}.
• Figure 3.2: Illustration of $200$ samples of $\hat{\Sigma}$ in Theorem \ref{['thm:scov']} with $p = 400$ and $n=2000$ (so that $\sqrt{\delta}\approx 0.45$). The light shaded area is the empirical histogram of all eigenvalues normalized to have total area $1$, while the colored shaded area is the empirical histogram of the largest eigenvalue normalized to have area $\frac{1}{p}$. The solid line is the spectrum of the free model, and the dashed vertical line marks $\mathrm{S}(\lambda,\delta)$. The vertical axis follows a square-root scale to visualize the density of the outlier.
• Figure 3.3: Illustration of $200$ samples of $\hat{\Sigma}-\Sigma$ in Theorem \ref{['thm:scov']} with the same parameters as in Figure \ref{['fig:scov1']}. Here the two colored shaded areas are the empirical histograms of the smallest and largest eigenvalues, and the dashed vertical lines mark the locations of $\mathrm{H}_\pm(\lambda,\delta)$.

Theorems & Definitions (115)

• Theorem 1.1
• Remark 2.1
• Theorem 2.2
• Corollary 2.3
• Theorem 2.4: TV23
• Theorem 2.5
• Remark 2.6
• Theorem 2.7
• Remark 2.8
• Theorem 2.9