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Dominik Schröder

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

Afonso Bandeira, Giorgio Cipolloni, Ramon van Handel, Dominik Schröder



We determine the approximate location of the extreme eigenvalues for a large class of random matrix models. These two-sided bounds are fundamentally beyond the reach of classical matrix concentration inequalities.

Example: Sample covariance matrix

Consider a rank-one perturbation of the identity matrix IRp×pI\in \mathbb R^{p\times p} as a population covariance matrix

Σ=I+λvv,\Sigma = I + \lambda vv^\top,

where vRpv\in\mathbb{R}^p is a unit vector and λ>0\lambda>0 is a parameter. Then draw nn random vectors x1,,xnx_1, \ldots, x_n from the distribution N(0,Σ)\mathcal N(0, \Sigma) and consider the sample covariance matrix

Σ^=1ni=1nxixi.\hat \Sigma = \frac{1}{n}\sum_{i=1}^n x_i x_i^\top.

We are able to show that this model exhibits two phase transitions. Denoting the the ratio of dimensions p,np,n by δ:=p/n\delta:=p/n the largest eigenvalue λmax(Σ^Σ)\lambda_{\max}(\hat\Sigma-\Sigma) satisfies

λmax{(1+δ)21, if λ<1+δ,1+λ2λ(δ+δ+4λ)δ, if λ1+δ, \lambda_{\max} \approx \begin{cases} (1 + \sqrt\delta)^2 - 1, & \text{ if } \lambda < 1 + \sqrt\delta,\\ \frac{1 + \lambda}{2\lambda} (\sqrt\delta + \sqrt{\delta + 4 \lambda}) \sqrt\delta, & \text{ if } \lambda \geq 1 + \sqrt\delta, \end{cases}

while the smallest eigenvalue λmin(Σ^Σ)\lambda_{\min}(\hat\Sigma-\Sigma) satisfies

λmin{(1δ)21, if λ<1δ,1+λ2λ(δδ+4λ)δ, if λ1δ\lambda_{\min} \approx \begin{cases} (1 - \sqrt\delta)^2 - 1, & \text{ if } \lambda < 1 - \sqrt\delta,\\ \frac{1 + \lambda}{2\lambda} (\sqrt\delta - \sqrt{\delta + 4 \lambda}) \sqrt\delta, & \text{ if } \lambda \geq 1 - \sqrt\delta \end{cases}

exactly as the corresponding free model suggests.

Numerical illustration

MaxMin↑ ρ−2.5−2.0−1.5−1.0−
The grey histogram represents the empirical distribution of sample covariance eigenvalues, while the solid curve is the spectral density of the corresponding free model. The coloured histograms represent the empirical distribution of the largest and smallest eigenvalues of the sample covariance matrix. Here √δ ≈ 0.45 so that the two phase transitions occur at λ ≈ 0.55 and λ ≈ 1.45.


The first paper in this series introduced a new family of nonasymptotic matrix concentration inequalities that sharply capture the spectral properties of very general Gaussian (as well as non-Gaussian) 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 applications 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 undertake a detailed study of phase transition phenomena for spectral outliers of nonhomogeneous random matrices.