Information Limits of Low-Rank Approximation Certification

Abstract:Low-rank approximation can require additional matrix--vector products to verify that its error meets a prescribed tolerance. We characterize this certification cost for both relative matrix error and mean-square output error. For a single approximation matrix candidate, we determine the exact dimension-uniform minimax query constant as the allowed failure probability vanishes. Our main result concerns reusing validation responses as the approximation space expands. For a candidate family constructed independently of validation, one batch supports an entire nested path without increasing the query budget with the number of checks. Across \(W\) paths, a concentration bound exploiting shared residual energy yields a \(\sqrt{\log(W+1)}\) dependence. A matching lower bound establishes its optimality for fixed interior error targets and sufficiently small separation gaps. Finally, we compare two uniformly valid certificates on the same dispersed-spectrum family. Optimizing the validation budget within each rule family yields costs of orders \(N^{1/3}\) and \(N^{2/3}\) for validation and construction beyond the true target. Code is available at this https URL

Optimization and Control (math.OC); Artificial Intelligence (cs.AI); Information Theory (cs.IT)

arXiv:2610.03321 [math.OC]

(or arXiv:2610.03321v1 [math.OC] for this version)

https://doi.org/10.48550/arXiv.2610.03321 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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