Normalizing constants of log-concave densities

2017/07/03 by Nicolas Brosse, Alain Durmus, Brosse, Nicolas +3 · 1 citation
Computer Science · Mathematics · #60F25 #60J05 #62L10 (Primary) 65C40 #65C05 #74G10 #74G15 (Secondary) #FOS: Computer and information sciences #Markov Chains and Monte Carlo Methods #Methodology (stat.ME) #Statistical Methods and Inference #Stochastic Gradient Optimization Techniques

paper · doi:10.48550/arxiv.1707.00460

openalex publication_date 2017/07/03 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

We derive explicit bounds for the computation of normalizing constants Z for log-concave densities π= exp(-U)/Z with respect to the Lebesgue measure on ℝ^d. Our approach relies on a Gaussian annealing combined with recent and precise bounds on the Unadjusted Langevin Algorithm (High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm, A. Durmus and E. Moulines). Polynomial bounds in the dimension d are obtained with an exponent that depends on the assumptions made on U. The algorithm also provides a theoretically grounded choice of the annealing sequence of variances. A numerical experiment supports our findings. Results of independent interest on the mean squared error of the empirical average of locally Lipschitz functions are established.

Cited by

• On the complexity of standard and waste-free SMC samplers

Related

• Recent progress in log-concave density estimation
• A new computational framework for log-concave density estimation
• A Polynomial Time Algorithm for Maximum Likelihood Estimation of Multivariate Log-concave Densities
• Log-concave Density Estimation with Independent Components
• Interpolating Thin-Shell and Sharp Large-Deviation Estimates For Isotropic Log-Concave Measures