Matrix anti-concentration inequalities with applications
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We provide a polynomial lower bound on the minimum singular value of an m× m random matrix M with jointly Gaussian entries, under a polynomial bound on the matrix norm and a global small-ball probability bound inf_x,y∈ S^m-1ℙ(|x^* M y|>m^-O(1))≥1/2. With the additional assumption that M is self-adjoint, the global small-ball probability bound can be replaced by a weaker version. We establish two matrix anti-concentration inequalities, which lower bound the minimum singular values of the sum of independent positive semidefinite self-adjoint matrices and the linear combination of independent random matrices with independent Gaussian coefficients. Both are under a global small-ball probability assumption. As a major application, we prove a better singular value bound for the Krylov space matrix, which leads to a faster and simpler algorithm for solving sparse linear systems. Our algorithm runs in Õ(n^3ω-4/ω-1)=O(n^2.2716) time where ω<2.37286 is the matrix multiplication exponent, improving on the previous fastest one in Õ(n^5ω-4/ω+1)=O(n^2.33165) time by Peng and Vempala.
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