Multiscale Data-Driven Energy Estimation and Generation

Abstract

We develop a multiscale approach to estimate high-dimensional probability distributions. Our approach applies to cases in which the energy function (or Hamiltonian) is not known from the start. Using data acquired from experiments or simulations we can estimate the energy function. We obtain a representation of the approximate probability distribution based on a multiscale cascade of conditional probabilities. This representation allows for fast sampling of many-body systems in various domains, from statistical physics to cosmology. Our method—the wavelet-conditional renormalization group (WCRG)—proceeds scale by scale, estimating models for the conditional probabilities of “fast degrees of freedom” conditioned by coarse-grained fields. These probability distributions are modeled by energy functions associated with scale interactions, and are represented in an orthogonal wavelet basis. The WCRG decomposes the microscopic energy function as a sum of interaction energies at all scales and can efficiently generate new samples by going from coarse to fine scales. Near phase transitions, this representation of the approximate probability distribution completely avoids the “critical slowing-down” of direct estimation and sampling algorithms. This is explained theoretically by combining results from RG and wavelet theories, and verified numerically for the Gaussian and φ^{4}-field theories. We show that multiscale WCRG energy-based models are more general than local potential models and can capture the physics of complex many-body interacting systems at all length scales. This is demonstrated for weak-gravitational-lensing fields reflecting dark-matter distributions in cosmology, which include long-range interactions with long-tail probability distributions. The WCRG has a large number of potential applications in nonequilibrium systems, where the underlying distribution is not known a priori. Finally, we discuss the connection between the WCRG and deep network architectures.