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.

翻译：受从静态快照数据学习动力学结构的启发，本文提出了一种标量分布回归方法，用于从含噪声的时序点云中估计随机过程的密度演化。我们提出了一种熵正则化非参数最大似然估计器（E-NPMLE），该方法利用熵最优传输作为密度流的平滑正则化器。我们证明E-NPMLE对真实分布具有几乎与维度无关的统计收敛速率，该速率在快照数量和每快照样本量方面表现出显著的相变现象。为高效计算E-NPMLE，我们设计了一种新颖的基于粒子且无需网格的坐标KL散度梯度下降（CKLGD）算法，并证明了其多项式迭代复杂度。此外，我们在合成数据上提供了数值证据以支持理论结果。这项工作为从任意维度的含噪声观测中估计密度演化提供了理论理解和实用计算方法。