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arXiv Machine Learning By Shi Fu, Huibo Xu, Qixin Zhang, Dacheng Tao

Restricted Eigenvalues Beyond Gaussian Width: Threshold Occupancy under Heavy Tails

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The paper investigates restricted eigenvalue (RE) bounds for norm‑regularized estimators under heavy‑tailed designs. It shows that the previously conjectured sample‑size law based on Gaussian width fails for heavy‑tailed measurements, due to a phenomenon called simultaneous threshold occupancy. The authors provide explicit counterexamples, derive worst‑case sample‑complexity bounds, and compare the behavior of heavy‑tailed versus Gaussian designs on constant‑width polyhedral descent cones.

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