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Fast and accurate log-determinant approximations

Owen Deen, Colton River Waller, John Paul Ward

TL;DR

This paper addresses the challenge of estimating the log-determinant $\log|A|$ for large, sparse, positive definite matrices by introducing a multi-pattern sparse inverse approach coupled with graph-based spline interpolation. The method builds a sequence of sparse approximations $D^1,\dots,D^m$ from nested sparsity patterns and refines them with a spline $S^m$ defined on a graph-embedded density axis, achieving improved accuracy while reducing memory and computational costs. A monotonicity theorem guarantees that larger pattern inclusions yield non-increasing approximation error, and experiments on Laplacian matrices demonstrate substantial time and memory savings over Sparse-LU, with splines providing notable accuracy gains. The technique has practical impact for applications in image processing, pattern recognition, and related areas where scalable log-determinant computations for SPD matrices are essential.

Abstract

We consider the problem of estimating log-determinants of large, sparse, positive definite matrices. A key focus of our algorithm is to reduce computational cost, and it is based on sparse approximate inverses. The algorithm can be implemented to be adaptive, and it uses graph spline approximation to improve accuracy. We illustrate our approach on classes of large sparse matrices.

Fast and accurate log-determinant approximations

TL;DR

This paper addresses the challenge of estimating the log-determinant for large, sparse, positive definite matrices by introducing a multi-pattern sparse inverse approach coupled with graph-based spline interpolation. The method builds a sequence of sparse approximations from nested sparsity patterns and refines them with a spline defined on a graph-embedded density axis, achieving improved accuracy while reducing memory and computational costs. A monotonicity theorem guarantees that larger pattern inclusions yield non-increasing approximation error, and experiments on Laplacian matrices demonstrate substantial time and memory savings over Sparse-LU, with splines providing notable accuracy gains. The technique has practical impact for applications in image processing, pattern recognition, and related areas where scalable log-determinant computations for SPD matrices are essential.

Abstract

We consider the problem of estimating log-determinants of large, sparse, positive definite matrices. A key focus of our algorithm is to reduce computational cost, and it is based on sparse approximate inverses. The algorithm can be implemented to be adaptive, and it uses graph spline approximation to improve accuracy. We illustrate our approach on classes of large sparse matrices.
Paper Structure (14 sections, 5 theorems, 13 equations, 1 figure, 3 tables, 2 algorithms)

This paper contains 14 sections, 5 theorems, 13 equations, 1 figure, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background: positive definite matrices
  3. Background: graphs and splines
  4. Sparse approximate inverses
  5. Sparsity model
  6. Properties of sparse approximate inverses
  7. Algorithm
  8. Sparse inverses
  9. Spline Construction
  10. Python code notes
  11. Experimental results
  12. Motivating examples
  13. Spline improvement experiments
  14. Summary and discussion

Key Result

Proposition 1

For a positive definite matrix $A$, there are unique matrices $L$, which is lower triangular, and $U$, which is unit upper triangular, such that $A=LU$.

Figures (1)

  • Figure 1: Convergence of the approximations $D^j$ and spline approximations $S^j$ to the exact log-determinant of $\mathcal{L}(35,3)$. The $x$-axis represents the density of the approximation being considered.

Theorems & Definitions (11)

  • Proposition 1
  • Proposition 2
  • Proposition 3: Hadamard-Fischer Inequalities johnson85
  • Definition 1
  • Remark 1
  • Proposition 4
  • proof
  • Theorem 1
  • proof
  • Remark 2
  • ...and 1 more