Transient anisotropic kernel for probabilistic learning on manifolds
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Abstract (arXiv)
PLoM (Probabilistic Learning on Manifolds) is a method introduced in 2016 for handling small training datasets by projecting an Ito 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.
Key facts. Cited by 3 papers in the map (data through 14 Oct 2026). Builds on 9 of its 88 references that are in the map. First posted on arXiv 31 Jul 2024.
Builds on
9 of 88 references.
References · 9
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Citing papers per year: 2025: 3.
3 papers.
- 2025IIKL: Isometric Immersion Kernel Learning with Riemannian Manifold for Geometric Preservationcited by 0
- 2025Enabling Probabilistic Learning on Manifolds through Double Diffusion Mapscited by 0
- 2025Supervised Learning of Random Neural Architectures Structured by Latent Random Fields on Compact Boundaryless Multiply-Connected Manifoldscited by 0
- 2025Supervised Learning of Random Neural Architectures Structured by Latent Random Fields on Compact Boundaryless Multiply-Connected Manifolds11 Dec 2025
- 2025Enabling Probabilistic Learning on Manifolds through Double Diffusion Maps2 Jun 2025
- 2025IIKL: Isometric Immersion Kernel Learning with Riemannian Manifold for Geometric Preservation7 May 2025