Low-Rank PSD Approximation in Input-Sparsity Time

Kenneth L. Clarkson, David P. Woodruff

Abstract: We give algorithms for approximation by low-rank positive semidefinite (PSD) matrices. For symmetric input matrix A ∊ ℝn×n, target rank k, and error parameter ∊ > 0, one algorithm finds with constant probability a PSD matrix Ỹ of rank k suchthat where Ak,+ denotes the best rank-k PSD approximation to A, and the norm is Frobenius. The algorithm takes time O(nnz(A) log n) + n poly((log n)k/∊) + poly(k/∊), where nnz(A) denotes the number of nonzero entries of A, and poly(k/∊) denotes a polynomial in k/∊. (There are two different polynomials in the time bound.) Here the output matrix Y has the form CUCT, where the O(k/∊) columns of c are columns of A. In contrast to prior work, we do not require the input matrix A to be PSD, our output is rank k (not larger), and our running time is O(nnz (A) log n) provided this is larger than npoly((log n)k/e). We give a similar algorithm that is faster and simpler, but whose rank- k PSD output does not involve columns of A, and does not require A to be symmetric. We give similar algorithms for best rank-k approximation subject to the constraint of symmetry. We also show that there are asymmetric input matrices that cannot have good symmetric column-selected approximations.