Linear-Complexity Black-Box Randomized Compression of Hierarchically Block Separable Matrices
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A randomized algorithm for computing a compressed representation of a given rank structured matrix A ∈ℝ^N× N is presented. The algorithm interacts with A only through its action on vectors. Specifically, it draws two tall thin matrices Ω, Ψ∈ℝ^N× s from a suitable distribution, and then reconstructs A by analyzing the matrices AΩ and A^*Ψ. For the specific case of a "Hierarchically Block Separable (HBS)" matrix (a.k.a. Hierarchically Semi-Separable matrix) of block rank k, the number of samples s required satisfies s = O(k), with s ≈ 3k being a typical scaling. While a number of randomized algorithms for compressing rank structured matrices have previously been published, the current algorithm appears to be the first that is both of truly linear complexity (no Nlog(N) factors) and fully black-box in nature (in the sense that no matrix entry evaluation is required).
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