Hierarchic Flows to Estimate and Sample High-dimensional Probabilities
This work addresses the long-standing problem of modeling complex physical fields for researchers in physics and machine learning, though it appears incremental by building on existing wavelet and renormalization group concepts.
The paper tackles the challenge of estimating and sampling high-dimensional probability distributions for complex physical fields like turbulence by introducing hierarchic probability flows and wavelet scattering models, achieving efficient sampling that avoids critical slowing down at phase transitions and generating realistic 2D vorticity fields and dark matter images.
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 $\varphi^4$ scalar potential, sampling these hierarchic models avoids the critical slowing down at the phase transition. An outstanding issue is to also approximate non-Gaussian fields having long-range interactions in space and across scales. We introduce low-dimensional models with robust multiscale approximations of high order polynomial energies. They are 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.