Accuracy vs. Cost in Parallel Fixed-Precision Low-Rank Approximations of Sparse Matrices

Abstract

We study a randomized and a deterministic algorithm for the fixed-precision low-rank approximation problem of large sparse matrices. The Randomized QB Factorization (RandQB_EI) constructs a reduced and dense representation of the originally sparse matrix based on randomization. The representation resulting from the deterministic Truncated LU Factorization with Column and Row Tournament Pivoting (LU_CRTP) is sparse, but fill-in introduced in the factorization process can affect sparsity and performance. We therefore attempt to mitigate fill-in with an incomplete LU_CRTP variant with thresholding (ILUT_CRTP). We analyze this approach and identify potential problems that may arise in practice. We design parallel implementations of RandQB_EI, LU_CRTP and ILUT_CRTP. We experimentally evaluate strong scaling properties for different problems and the runtime required for achieving a given approximation quality. Our results show that LU_CRTP tends to be particularly competitive for low approximation quality. However, when a lot of fill-in occurs, LU_CRTP is outperformed by RandQB_EI especially for higher approximation quality. ILUT_CRTP outperforms both LU_CRTP and RandQB_EI and can achieve speedups up to 40 over LU_CRTP, depending on the amount of fill-in.