A Range-Deflated Stochastic Lanczos Quadrature Method for Large-Scale Log-Determinant Estimation

Abstract

Estimating the logarithm of the determinant of large sparse symmetric positive definite matrices is an important problem in numerical linear algebra, machine learning, Gaussian processes, and uncertainty quantification. We propose a range-deflated stochastic Lanczos quadrature method, termed DSLQ, for matrix-free log-determinant estimation inspired by Hutch ++. The method constructs a randomized approximation space directly from products with a scaled version of the original matrix, and uses the associated orthogonal projector to decompose the trace of matrix logarithm into projected and complementary contributions. Both contributions are evaluated through the Gauss-Lanczos quadrature, avoiding explicit formation of the matrix logarithm or a low-rank matrix-function surrogate. Since the approximation space is generated from the matrix rather than from the matrix logarithm, the standard Hutch++ approximation bound does not apply directly. We therefore derive an error analysis specific to the DSLQ construction, quantifying the residual induced by the matrix-generated subspace and propagating this effect through the stochastic residual estimator and the Lanczos quadrature approximations. Extensive numerical experiments on both large sparse synthetic matrices ( Gaussian Markov Random fields, Bayesian inverse problem.) and large-scale real-world matrices, demonstrate that our method substantially reduces computational time while retaining competitive accuracy, thereby confirming its effectiveness, scalability, and favorable accuracy-cost trade-off for large-scale log-determinant estimation.