Overview
Estimation of the logarithm of the determinant of large sparse symmetric positive definite matrices is a significant problem in numerical linear algebra, machine learning, Gaussian processes, and uncertainty quantification. A proposed range-deflated stochastic Lanczos quadrature (DSLQ) method addresses this challenge. Inspired by Hutch++, DSLQ constructs a randomized approximation space directly from products involving a scaled version of the original matrix. This method then utilizes an associated orthogonal projector to decompose the trace of the matrix logarithm into distinct projected and complementary contributions.
Approach
The DSLQ method operates by generating an approximation space from products with a scaled version of the target matrix. The orthogonal projector derived from this space facilitates the decomposition of the trace of the matrix logarithm into two components: a projected contribution and a complementary contribution. Both of these contributions are subsequently evaluated through the Gauss-Lanczos quadrature. This approach avoids the explicit formation of the matrix logarithm itself, as well as the use of a low-rank matrix-function surrogate.
Given that the approximation space is generated from the matrix directly, rather than from the matrix logarithm, standard Hutch++ approximation bounds are not directly applicable. Consequently, a specific error analysis for the DSLQ construction was derived. This analysis quantifies the residual introduced by the matrix-generated subspace and tracks its effects through both the stochastic residual estimator and the Lanczos quadrature approximations.
Findings
Extensive numerical experiments were conducted to evaluate the DSLQ method. These experiments involved both large sparse synthetic matrices, specifically Gaussian Markov Random fields and Bayesian inverse problems, and large-scale real-world matrices. The results indicated that the DSLQ method:
- Substantially reduced computational time.
- Retained competitive accuracy.
- Demonstrated effectiveness for large-scale log-determinant estimation.
- Exhibited scalability.
- Presented a favorable accuracy-cost trade-off.
Why This Matters
The problem of estimating the logarithm of the determinant of large sparse symmetric positive definite matrices holds importance across several computational fields. These include numerical linear algebra, machine learning, Gaussian processes, and uncertainty quantification. The development of methods like DSLQ that can efficiently and accurately address this estimation problem can contribute to advancements in these areas by providing more effective computational tools.