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Comparison theorems for the minimum eigenvalue of a random positive-semidefinite matrix

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arxiv 2501.16578 v1 pith:GLITWF74 submitted 2025-01-27 math.PR cs.NAmath.NAmath.STstat.TH

Joel A. Tropp This is my paper
classification math.PRcs.NAmath.NAmath.STstat.TH
keywords randomeigenvalueminimummatricesmatrixcomparisongaussianpositive-semidefinite
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This paper establishes a new comparison principle for the minimum eigenvalue of a sum of independent random positive-semidefinite matrices. The principle states that the minimum eigenvalue of the matrix sum is controlled by the minimum eigenvalue of a Gaussian random matrix that inherits its statistics from the summands. This methodology is powerful because of the vast arsenal of tools for treating Gaussian random matrices. As applications, the paper presents short, conceptual proofs of some old and new results in high-dimensional statistics. It also settles a long-standing open question in computational linear algebra about the injectivity properties of very sparse random matrices.

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