Hierarchic Flows to Estimate and Sample High-dimensional Probabilities

TL;DR

This paper introduces a hierarchical wavelet-based probabilistic modeling approach to efficiently estimate and sample high-dimensional physical fields like turbulence, overcoming traditional dimensionality and critical slowing down issues.

Contribution

It proposes a novel hierarchic probability flow framework using wavelet bases and scattering covariance to model complex fields with long-range interactions.

Findings

Successfully estimated and sampled turbulence vorticity fields.

Generated dark matter density images with the proposed models.

Avoided critical slowing down in phase transition sampling.

Abstract

Finding low-dimensional interpretable models of complex physical fields such as turbulence remains an open question, 80 years after the pioneer work of Kolmogorov. Estimating high-dimensional probability distributions from data samples suffers from an optimization and an approximation curse of dimensionality. It may be avoided by following a hierarchic probability flow from coarse to fine scales. This inverse renormalization group is defined by conditional probabilities across scales, renormalized in a wavelet basis. For a $\vvarphi^4$ scalar potential, sampling these hierarchic models avoids the critical slowing down at the phase transition. In a well chosen wavelet basis, conditional probabilities can be captured with low dimensional parametric models, because interactions between wavelet coefficients are local in space and scales. An outstanding issue is also to approximate non-Gaussian fields having long-range interactions in space and across scales. We introduce low-dimensional models of wavelet conditional probabilities with the scattering covariance. It is calculated with a second wavelet transform, which defines interactions over two hierarchies of scales. We estimate and sample these wavelet scattering models to generate 2D vorticity fields of turbulence, and images of dark matter densities.