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Can Adaptive Gradient Methods Converge under Heavy-Tailed Noise? 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Comparison theorems for the extreme eigenvalues of a random symmetric matrix
[Submitted on 4 Mar 2026 (v1), last revised 5 Jul 2026 (this ver · 2026-03-05 · via stat updates on arXiv.org

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Abstract:This paper establishes a comparison theorem for the maximum eigenvalue of a sum of independent random symmetric matrices. The theorem states that the maximum eigenvalue of the matrix sum is dominated by the maximum eigenvalue of a Gaussian random matrix whose statistics match the sum, and it strengthens previous results of this type. Corollaries address the minimum eigenvalue and the spectral norm; the proof strategy also extends to matrix martingale sequences.
The comparison methodology is powerful because of the vast arsenal of tools for treating Gaussian random matrices. As applications, the paper improves on existing eigenvalue bounds for random matrices arising in spectral graph theory, quantum information theory, high-dimensional statistics, and numerical linear algebra. In particular, these techniques deliver the first complete proof that a sparse random dimension reduction map has the injectivity properties conjectured by Nelson & Nguyen in 2013.

Submission history

From: Joel Tropp [view email]
[v1] Wed, 4 Mar 2026 18:30:04 UTC (121 KB)
[v2] Sat, 14 Mar 2026 18:12:03 UTC (117 KB)
[v3] Sun, 5 Jul 2026 00:09:16 UTC (164 KB)