Close to optimal column approximations with a single SVD

08/17/2023

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by   Alexander Osinsky, et al.

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The best column approximation in the Frobenius norm with r columns has an error at most √(r+1) times larger than the truncated singular value decomposition. Reaching this bound in practice involves either expensive random volume sampling or at least r executions of singular value decomposition. In this paper it will be shown that the same column approximation bound can be reached with only a single SVD (which can also be replaced with approximate SVD). As a corollary, it will be shown how to find a highly nondegenerate submatrix in r rows of size N in just O(Nr^2) operations, which mostly has the same properties as the maximum volume submatrix.