  PLoM (Probabilistic Learning on Manifolds) is a method introduced in 2016 for
handling small training datasets by projecting an It\^o 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.
