2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS) (2016)
New Brunswick, New Jersey, USA
Oct. 9, 2016 to Oct. 11, 2016
DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/FOCS.2016.68
We show how to perform sparse approximate Gaussian elimination for Laplacian matrices. We present a simple, nearly linear time algorithm that approximates a Laplacian by the product of a sparse lower triangular matrix with its transpose. This gives the first nearly linear time solver for Laplacian systems that is based purely on random sampling, and does not use any graph theoretic constructions such as low-stretch trees, sparsifiers, or expanders. Our algorithm performs a subsampled Cholesky factorization, which we analyze using matrix martingales. As part of the analysis, we give a proof of a concentration inequality for matrix martingales where the differences are sums of conditionally independent variables.
Laplace equations, Sparse matrices, Symmetric matrices, Matrix decomposition, Approximation algorithms, Linear matrix inequalities, Algorithm design and analysis
R. Kyng and S. Sachdeva, "Approximate Gaussian Elimination for Laplacians - Fast, Sparse, and Simple," 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS), New Brunswick, New Jersey, USA, 2016, pp. 573-582.