2017/02/14 by Fan Yang, Yang, Fan
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.1702.04050
openalex publication_date 2017/02/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove an estimate on the smallest singular value of a multiplicatively and additively deformed random rectangular matrix. Suppose n≤ N ≤ M ≤ ΛN for some constant Λ≥ 1. Let X be an M× n random matrix with independent and identically distributed entries, which have zero mean, unit variance and arbitrarily high moments. Let T be an N× M deterministic matrix with comparable singular values c≤ sN(T) ≤ s1(T) ≤ c-1 for some constant c>0. Let A be an N× n deterministic matrix with ‖A‖=O(√(N)). Then we prove that for any ε>0, the smallest singular value of TX-A is larger than N-ε(√(N)-√(n-1)) with high probability. If we assume further the entries of X have subgaussian decay, then the smallest singular value of TX-A is at least of the order √(N)-√(n-1) with high probability, which is an essentially optimal estimate.