Truncated Log-concave Sampling with Reflective Hamiltonian Monte Carlo
This addresses the problem of efficient truncated sampling for high-dimensional data in statistics and machine learning, representing a strong specific gain rather than a broad paradigm shift.
The paper tackles sampling from log-concave distributions restricted to convex bodies by introducing Reflective Hamiltonian Monte Carlo (ReHMC), achieving mixing within accuracy ε in Õ(κd²ℓ² log(1/ε)) steps from a warm start, with experiments showing it outperforms Hit-and-Run methods and enables practical sampling in thousands of dimensions.
We introduce Reflective Hamiltonian Monte Carlo (ReHMC), an HMC-based algorithm, to sample from a log-concave distribution restricted to a convex body. We prove that, starting from a warm start, the walk mixes to a log-concave target distribution $π(x) \propto e^{-f(x)}$, where $f$ is $L$-smooth and $m$-strongly-convex, within accuracy $\varepsilon$ after $\widetilde O(κd^2 \ell^2 \log (1 / \varepsilon))$ steps for a well-rounded convex body where $κ= L / m$ is the condition number of the negative log-density, $d$ is the dimension, $\ell$ is an upper bound on the number of reflections, and $\varepsilon$ is the accuracy parameter. We also developed an efficient open source implementation of ReHMC and we performed an experimental study on various high-dimensional data-sets. The experiments suggest that ReHMC outperfroms Hit-and-Run and Coordinate-Hit-and-Run regarding the time it needs to produce an independent sample and introduces practical truncated sampling in thousands of dimensions.