Entrywise Approximation for Matrix Inversion and Linear Systems

We study matrix inversion and solving linear systems on diagonally dominant matrices. These are associated with random walk quantities such as hitting times and escape probabilities in graphs. Such quantities can be exponentially small, even on undirected unit-weighted graphs. However, their nonnegativity suggests that they can be approximated entrywise, leading to a stronger notion of approximation than vector norm-based error.

Under this notion of error, existing Laplacian solvers and fast matrix multiplication approaches require Ω(mn 2 ) and Ω(n ω+1 ) bit operations, respectively, where m is the number of nonzero entries in the matrix, n is its size, and ω is the matrix multiplication exponent.

We present algorithms that compute entrywise exp(ϵ)-approximate inverses of row diagonally dominant L-matrices (RDDL) in two settings: (1) when the matrix entries are given in floatingpoint representation; (2) when they are given in fixed-point representation.

For floating-point inputs, we present a cubic-time algorithm and show that it has an optimal running time under the all-pairs shortest paths (APSP) conjecture.

For fixed-point inputs, we present several algorithms for solving linear systems and inverting RDDL and SDDM matrices (the latter being symmetric RDDL matrices), all with high probability.