On the smoothed analysis of the smallest singular value with discrete noise
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Let A be an n× n real matrix, and let M be an n× n random matrix whose entries are i.i.d sub-Gaussian random variables with mean 0 and variance 1. We make two contributions to the study of s_n(A+M), the smallest singular value of A+M. (1) We show that for all ϵ≥ 0, ℙ[s_n(A + M) ≤ϵ] = O(ϵ√(n)) + 2e^-Ω(n), provided only that A has Ω (n) singular values which are O(√(n)). This extends a well-known result of Rudelson and Vershynin, which requires all singular values of A to be O(√(n)). (2) We show that any bound of the form sup_A≤ n^C_1ℙ[s_n(A+M)≤ n^-C_3] ≤ n^-C_2 must have C_3 = Ω (C_1 √(C_2)). This complements a result of Tao and Vu, who proved such a bound with C_3 = O(C_1C_2 + C_1 + 1), and counters their speculation of possibly taking C_3 = O(C_1 + C_2).
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