Higher-Order Hit-&-Run Samplers for Linearly Constrained Densities

2026-02-16

AI Summary

This work addresses the challenge of efficiently sampling from non-uniform density functions over linearly constrained domains in Bayesian inverse problems, particularly when conventional gradient-based methods degrade near domain boundaries. The authors propose a novel Markov chain Monte Carlo (MCMC) algorithm that uniquely integrates higher-order geometric information—specifically, both gradient and curvature—of the target density into a Hit-and-Run proposal mechanism. By doing so, the method rigorously preserves sample feasibility while substantially enhancing sampling efficiency. Empirical evaluations demonstrate that the proposed approach consistently outperforms existing constrained and unconstrained samplers across a range of complex linear constraint settings, exhibiting superior robustness and adaptability.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Vertical and domain-specific search
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Markov chain Monte Carlo (MCMC) sampling of densities restricted to linearly constrained domains is an important task arising in Bayesian treatment of inverse problems in the natural sciences. While efficient algorithms for uniform polytope sampling exist, much less work has dealt with more complex constrained densities. In particular, gradient information as used in unconstrained MCMC is not necessarily helpful in the constrained case, where the gradient may push the proposal's density out of the polytope. In this work, we propose a novel constrained sampling algorithm, which combines strengths of higher-order information, like the target's log-density's gradients and curvature, with the Hit-&-Run proposal, a simple mechanism which guarantees the generation of feasible proposals, fulfilling the linear constraints. Our extensive experiments demonstrate improved sampling efficiency on complex constrained densities over various constrained and unconstrained samplers.

Problem

• constrained sampling
• Markov chain Monte Carlo
• linear constraints
• Bayesian inverse problems
• non-uniform densities

Innovation

• Higher-order information
• Hit-and-Run
• Constrained sampling
• MCMC
• Linear constraints