Low-rank PSD approximation in input-sparsity time

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 e > 0, one algorithm finds with constant probability a PSD matrix Ỹ of rank k such that ||A − Ỹ ||F2 ≤ (1 + e) || A − Ak, +||F2, 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((logn)k/e) + poly(k/e), where nnz(A) denotes the number of nonzero entries of A, and poly(k/e) denotes a polynomial in k/e. (There are two different polynomials in the time bound.) Here the output matrix Ỹ has the form CUC⊤, where the O(k/e) 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.