Gordon's inequality and condition numbers in conic optimization

The probabilistic analysis of condition numbers has traditionally been approached from different angles; one is based on Smale's program in complexity theory and features integral geometry, while the other is motivated by geometric functional analysis and makes use of the theory of Gaussian processes. In this note we explore connections between the two approaches in the context of the biconic homogeneous feasiblity problem and the condition numbers motivated by conic optimization theory. Key tools in the analysis are Slepian's and Gordon's comparision inequalities for Gaussian processes, interpreted as monotonicity properties of moment functionals, and their interplay with ideas from conic integral geometry

Bayesian Inference of Log Determinants

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.Comment: 12 pages, 3 figure