The Pseudospectrum of Random Compressions of Matrices
Rikhav Shah
TL;DR
The paper analyzes the stability of eigenvalues for random compressions $Q^*AQ$, where $Q$ is Haar-distributed on the complex Grassmannian. It develops a reduction to the numerical-measure problem of a random Schur complement $M=(A/Q')$, establishes anti-concentration-based lower-tail bounds for $\sigma_{\min}(Q^*AQ)$, and then derives non-asymptotic bounds on the expected area of the $\varepsilon$-pseudospectrum ${\bf E}\,\mathrm{Area}\Lambda_\varepsilon(Q^*AQ)$. Depending on mild assumptions on $A$ relating to its numerical range and shifted singular-values, the authors obtain bounds of the form $\text{poly}(n)\log^2(1/\varepsilon)\varepsilon^\beta$ with $\beta\in\{6/5,4/3,2\}$, along with tail bounds for the least singular value and small-ball estimates for non-Hermitian quadratic forms. This advances understanding of pseudospectral stability under random compressions and provides tools for anti-concentration in non-Hermitian matrix settings, with implications for randomized Rayleigh-Ritz-type methods.
Abstract
The compression of a matrix $A\in\mathbb C^{n\times n}$ onto a subspace $V\subset\mathbb C^n$ is the matrix $Q^*AQ$ where the columns of $Q$ form an orthonormal basis for $V$. This is an important object in both operator theory and numerical linear algebra. Of particular interest are the eigenvalues of the compression and their stability under perturbations. This paper considers compressions onto subspaces sampled from the Haar measure on the complex Grassmannian. We show the expected area of the $\varepsilon$-pseudospectrum of such compressions is bounded by $\text{poly}(n)\log^2(1/\varepsilon)\cdot\varepsilon^β$, where $β=6/5,4/3$, or $2$ depending on some mild assumptions on $A$. Along the way, we obtain (a) tail bounds for the least singular value of compressions and (b) non-asymptotic small-ball estimates for random non-Hermitian quadratic forms surpassing bounds achieved by existing methods.