# 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 ϵ ℝn×n, target rank k, and error parameter ϵ > 0, one algorithm finds with constant probability a PSD matrix Y∼ of rank k such that ∥A - Y∼ ∥ 2 F ≤ (1 + ∈)∥2F 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/∈)+poly(k/∈), where nnz(A) denotes the number of nonzero entries of A, and poly(k=") denotes a polynomial in k=". (There are two difierent polynomials in the time bound.) Here the output matrix ~ Y has the form CUC>, 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/∈). 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.