2025/10/13 by François Clément, Stefan Steinerberger, Clément, François +1 · 1 voice
Computer Science · Mathematics · #math.OC #cs.DS #math.ST
paper · pdf · doi:10.48550/arxiv.2510.11571
We suppose we are given a list of points x1, …, xn ∈ ℝ, a target probability measure μ and are asked to add additional points xn+1, …, xn+m so that x1, …, xn+m is as close as possible to the distribution of μ; additionally, we want this to be true uniformly for all m. We propose a simple method that achieves this goal. It selects new points in regions where the existing set is lacking points and avoids regions that are already overly crowded. If we replace μ by another measure μ2 in the middle of the computation, the method dynamically adjusts and allows us to keep the original sampling points. xn+1 can be computed in O(n) steps and we obtain state-of-the-art results. It appears to be an interesting dynamical system in its own right; we analyze a continuous mean-field version that reflects much of the same behavior.