Transient anisotropic kernel for probabilistic learning on manifolds

PLoM (Probabilistic Learning on Manifolds) is a method introduced in 2016 for handling small training datasets by projecting an Itô equation from a stochastic dissipative Hamiltonian dynamical system, acting as the MCMC generator, for which the KDE-estimated probability measure with the training dataset is the invariant measure. PLoM performs a projection on a reduced-order vector basis related to the training dataset, using the diffusion maps (DMAPS) basis constructed with a time-independent isotropic kernel. In this paper, we propose a new ISDE projection vector basis built from a transient anisotropic kernel, providing an alternative to the DMAPS basis to improve statistical surrogates for stochastic manifolds with heterogeneous data. The construction ensures that for times near the initial time, the DMAPS basis coincides with the transient basis. For larger times, the differences between the two bases are characterized by the angle of their spanned vector subspaces. The optimal instant yielding the optimal transient basis is determined using an estimation of mutual information from Information Theory, which is normalized by the entropy estimation to account for the effects of the number of realizations used in the estimations. Consequently, this new vector basis better represents statistical dependencies in the learned probability measure for any dimension. Three applications with varying levels of statistical complexity and data heterogeneity validate the proposed theory, showing that the transient anisotropic kernel improves the learned probability measure.

PLOM（对流形的概率学习）是一种在2016年介绍的方法，用于通过从随机耗散的汉密尔顿动力系统中投射一个ITô方程，作为MCMC生成器，KDE估算的概率衡量的概率衡量数据集是使用PLOM进行了训练的范围，该数据与训练数据相关。地图）构建的基础在本文中，与时间无关的核心构建的新的ISD投影基础是瞬时的，为DMAPS基础提供了一个替代，我们在本文中构建了一个新的ISD投影矢量基础，这为DMAPS基础提供了替代方案，以改善与较大时间的构建的统计替代。他们的角度跨度的矢量亚空间。最佳的瞬时瞬时基础是通过信息理论估算的，这是通过熵估计来归一化的，以说明估计中使用的实现数量。异质性验证了所提出的理论，表明瞬时各向异性内核改善了学习概率的测量。