A recent proof of NP-hardness of Euclidean sum-of-squares clustering, due to Drineas et al.

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We give efficient algorithms for volume sampling, i.e., for picking k-subsets of the rows of any given matrix with probabilities proportional to the squared volumes of the simplices defined by them and the origin (or the squared volumes of the parallelepipeds defined by these subsets of rows). This solves an open problem from the monograph on spectral algorithms by Kannan and Vempala (see Section 7.4 of [15], also implicit in [1,5]).Our first algorithm for volume sampling k-subsets of rows from an m-by-n matrix runs in O(kmn ω log n) arithmetic operations and a second variant of it for (1 + ǫ)-approximate volume sampling runs in O(mn log m · k 2 /ǫ 2 + m log ω m · k 2ω+1 /ǫ 2ω · log(kǫ −1 log m)) arithmetic operations, which is almost linear in the size of the input (i.e., the number of entries) for small k.Our efficient volume sampling algorithms imply the following results for low-rank matrix approximation:1. Given A ∈ R m×n , in O(kmn ω log n) arithmetic operations we can find k of its rows such that projecting onto their span gives a √ k + 1-approximation to the matrix of rank k closest to A under the Frobenius norm. This improves the O(k √ log k)-approximation of Boutsidis, Drineas and Mahoney [1] and matches the lower bound shown in [5]. The method of conditional expectations gives a deterministic algorithm with the same complexity. The running time can be improved to O(mn log m·k 2 /ǫ 2 +m log ω m·k 2ω+1 /ǫ 2ω ·log(kǫ −1 log m)) at the cost of losing an extra (1 + ǫ) in the approximation factor.2. The same rows and projection as in the previous point give a (k + 1)(n − k)-approximation to the matrix of rank k closest to A under the spectral norm. In this paper, we show an almost matching lower bound of √ n, even for k = 1.

We prove that any real matrix A contains a subset of at most 4k/ + 2k log(k + 1) rows whose span "contains" a matrix of rank at most k with error only (1 + ) times the error of the best rank-k approximation of A. We complement it with an almost matching lower bound by constructing matrices where the span of any k/2 rows does not "contain" a relative (1 + )-approximation of rank k. Our existence result leads to an algorithm that finds such rank-k approximation in timei.e., essentially O(Mk/ ), where M is the number of nonzero entries of A. The algorithm maintains sparsity, and in the streaming model [12,14,15], it can be implemented using only 2(k + 1)(log(k + 1) + 1) passes over the input matrix and O min{m, n}( k + k 2 log k) additional space. Previous algorithms for low-rank approximation use only one or two passes but obtain an additive approximation.

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