Low-Rank PSD Approximation in Input-Sparsity Time

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 ε > 0, one algorithm finds with constant probability a PSD matrix Ỹ of rank k such that ‖A− Ỹ ‖2F ≤ (1+ε)‖A−Ak,+‖ 2 F , 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) + npoly((log n)k/ε) + poly(k/ε), where nnz(A) denotes the number of nonzero entries of A, and poly(k/ε) denotes a polynomial in k/ε. (There are two different polynomials in the time bound.) Here the output matrix Ỹ has the form CUC>, where the O(k/ε) 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/ ). 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.