# A Randomized Algorithm for Approximating the Log Determinant of a Symmetric Positive Definite Matrix

arXiv: https://arxiv.org/abs/1503.00374 · HTML: https://mlsanity.com/paper/1503.00374/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Mar 2015.

## Built on by (most cited first, 9 of 9)

- [A Review of Change of Variable Formulas for Generative Modeling](https://mlsanity.com/paper/2308.02652/) (2023)
- [Exact Gaussian Processes for Massive Datasets via Non-Stationary Sparsity-Discovering Kernels](https://mlsanity.com/paper/2205.09070/) (2022)
- [Computationally Efficient Approximations for Matrix-based Renyi's Entropy](https://mlsanity.com/paper/2112.13720/) (2021)
- [Minimum Reduced-Order Models via Causal Inference](https://mlsanity.com/paper/2407.00271/) (2024)
- [Analyzing Generative Models by Manifold Entropic Metrics](https://mlsanity.com/paper/2410.19426/) (2024)
- [BOLT: Block-Orthonormal Lanczos for Trace estimation of matrix functions](https://mlsanity.com/paper/2505.12289/) (2025)
- [Representing Additive Gaussian Processes by Sparse Matrices](https://mlsanity.com/paper/2305.00324/) (2023)
- [The Exact Determinant of a Specific Class of Sparse Positive Definite Matrices](https://mlsanity.com/paper/2311.06632/) (2023)
- [Bypassing orthogonalization in the quantum DPP sampler](https://mlsanity.com/paper/2503.05906/) (2025)

## Abstract (arXiv)

We introduce a novel algorithm for approximating the logarithm of the determinant of a symmetric positive definite (SPD) matrix. The algorithm is randomized and approximates the traces of a small number of matrix powers of a specially constructed matrix, using the method of Avron and Toledo~citeAT11. From a theoretical perspective, we present additive and relative error bounds for our algorithm. Our additive error bound works for any SPD matrix, whereas our relative error bound works for SPD matrices whose eigenvalues lie in the interval (θ1,1), with 0<θ1<1; the latter setting was proposed in~citeicml2015hana15. From an empirical perspective, we demonstrate that a C++ implementation of our algorithm can approximate the logarithm of the determinant of large matrices very accurately in a matter of seconds.