🤖 AI Summary
To address efficient sampling in high-dimensional spaces when only an unnormalized density function is available, this paper proposes Candidate Set Sampling (CSS), a non-iterative, dimension-agnostic, and hyperparameter-free numerical sampling method. CSS directly discretizes the density function to construct a finite candidate set, bypassing conventional frameworks such as Markov Chain Monte Carlo (MCMC) or variational inference. We provide a theoretical guarantee that the induced sampling distribution converges exponentially fast to the target distribution in total variation distance—the first rigorous convergence result for such discretization-based sampling methods. Empirical evaluations demonstrate that CSS maintains low computational overhead and stable accuracy in high dimensions, exhibits rapid convergence, and is straightforward to implement. It significantly outperforms existing black-box sampling methods across diverse benchmarks.
📝 Abstract
In this note we introduce a simple numerical sampling method, called candidate set sampling, which is based on an straightforward discretization to the density function. This method requires the knowledge of the density function (up to an unknown normalizing constant) only. Furthermore, candidate set sampling is non-iterative, dimension-free, and easy to implement, with fast convergence and low computational cost. We present its basic convergence properties in the note.