Sampling with Riemannian Hamiltonian Monte Carlo in a Constrained Space
This enables efficient sampling for constrained high-dimensional problems in fields like systems biology and linear programming, representing a significant practical advance rather than an incremental improvement.
The authors tackled the problem of sampling from ill-conditioned, non-smooth, constrained distributions in very high dimensions (up to 100,000) by incorporating constraints into Riemannian Hamiltonian Monte Carlo while maintaining sparsity, achieving a mixing rate independent of smoothness and condition numbers. They demonstrated a 1,000-fold speed-up for sampling from the largest published human metabolic network (RECON3D) and outperformed existing packages by orders of magnitude on benchmark datasets.
We demonstrate for the first time that ill-conditioned, non-smooth, constrained distributions in very high dimension, upwards of 100,000, can be sampled efficiently $\textit{in practice}$. Our algorithm incorporates constraints into the Riemannian version of Hamiltonian Monte Carlo and maintains sparsity. This allows us to achieve a mixing rate independent of smoothness and condition numbers. On benchmark data sets in systems biology and linear programming, our algorithm outperforms existing packages by orders of magnitude. In particular, we achieve a 1,000-fold speed-up for sampling from the largest published human metabolic network (RECON3D). Our package has been incorporated into the COBRA toolbox.