Learning Density Evolution from Snapshot Data

TL;DR

This paper introduces a new method for estimating how probability densities evolve over time from static snapshot data, using an entropy-regularized approach and a novel computational algorithm, with theoretical guarantees and empirical validation.

Contribution

It proposes the E-NPMLE estimator with entropic optimal transport regularization and a particle-based gradient descent algorithm, advancing the estimation of density evolution from noisy data.

Findings

E-NPMLE achieves near dimension-free convergence rates.

The phase transition phenomenon depends on snapshot number and sample size.

Numerical experiments support theoretical convergence results.

Abstract

Motivated by learning dynamical structures from static snapshot data, this paper presents a distribution-on-scalar regression approach for estimating the density evolution of a stochastic process from its noisy temporal point clouds. We propose an entropy-regularized nonparametric maximum likelihood estimator (E-NPMLE), which leverages the entropic optimal transport as a smoothing regularizer for the density flow. We show that the E-NPMLE has almost dimension-free statistical rates of convergence to the ground truth distributions, which exhibit a striking phase transition phenomenon in terms of the number of snapshots and per-snapshot sample size. To efficiently compute the E-NPMLE, we design a novel particle-based and grid-free coordinate KL divergence gradient descent (CKLGD) algorithm and prove its polynomial iteration complexity. Moreover, we provide numerical evidence on synthetic data to support our theoretical findings. This work contributes to the theoretical understanding and practical computation of estimating density evolution from noisy observations in arbitrary dimensions.