Bayesian Inference of Log Determinants
This addresses a computational bottleneck in machine learning for problems like Gaussian processes, but it is incremental as it builds on existing probabilistic numerics and stochastic trace estimation methods.
The paper tackles the problem of computing log-determinants of large kernel matrices, which is intractable for sizes over a few thousand, by reinterpreting it as a Bayesian inference problem; the result is a method that provides competitive approximations with state-of-the-art approaches while quantifying uncertainty, though no specific numerical gains are mentioned.
The log-determinant of a kernel matrix appears in a variety of machine learning problems, ranging from determinantal point processes and generalized Markov random fields, through to the training of Gaussian processes. Exact calculation of this term is often intractable when the size of the kernel matrix exceeds a few thousand. In the spirit of probabilistic numerics, we reinterpret the problem of computing the log-determinant as a Bayesian inference problem. In particular, we combine prior knowledge in the form of bounds from matrix theory and evidence derived from stochastic trace estimation to obtain probabilistic estimates for the log-determinant and its associated uncertainty within a given computational budget. Beyond its novelty and theoretic appeal, the performance of our proposal is competitive with state-of-the-art approaches to approximating the log-determinant, while also quantifying the uncertainty due to budget-constrained evidence.