Wavelet Conditional Renormalization Group

Abstract

We develop a multiscale approach to estimate high-dimensional probability distributions from a dataset of physical fields or configurations observed in experiments or simulations. In this way we can estimate energy functions (or Hamiltonians) and efficiently generate new samples of many-body systems in various domains, from statistical physics to cosmology. Our method -- the Wavelet Conditional Renormalization Group (WC-RG) -- 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. WC-RG 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, it 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 ${\phi }^4$ field theories. We show that multiscale WC-RG 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. WC-RG has a large number of potential applications in non-equilibrium systems, where the underlying distribution is not known {\it a priori}. Finally, we discuss the connection between WC-RG and deep network architectures.

arXiv comment36 pages, 21 figures

References in corpus (35)

• Weak Gravitational Lensing
• Simulated Tempering: A New Monte Carlo Scheme
• Weak pairwise correlations imply strongly correlated network states in a neural population
• Non-Perturbative Renormalization Flow in Quantum Field Theory and Statistical Physics
• Machine learning phases of matter
• Understanding Deep Convolutional Networks
• Macroscopic fluctuation theory
• Cosmology with cosmic shear observations: a review
• A multifractal random walk
• Why does deep and cheap learning work so well?
• Rigorous Inequalities between Length and Time Scales in Glassy Systems
• The Principles of Deep Learning Theory
• Large Deviation of the Density Profile in the Steady State of the Open Symmetric Simple Exclusion Process
• CFHTLenS: Cosmological constraints from a combination of cosmic shear two-point and three-point correlations
• Cosmology constraints from shear peak statistics in Dark Energy Survey Science Verification data
• Neural Network Renormalization Group
• KiDS-450: Cosmological Constraints from Weak Lensing Peak Statistics - II: Inference from Shear Peaks using N-body Simulations
• Non-Gaussian information from weak lensing data via deep learning
• KiDS-450: Cosmological Constraints from Weak Lensing Peak Statistics-I: Inference from Analytical Prediction of High Signal-to-Noise Ratio Convergence Peaks
• Cosmological Constraints From Weak Lensing Peak Statistics With CFHT Stripe 82 Survey
• New Interpretable Statistics for Large Scale Structure Analysis and Generation
• A Monte Carlo study of leading order scaling corrections of phi^4 theory on a three dimensional lattice
• Entanglement renormalization and wavelets
• Scale-invariant Feature Extraction of Neural Network and Renormalization Group Flow
• Non-perturbative Approach to Critical Dynamics
• Weak lensing scattering transform: dark energy and neutrino mass sensitivity
• Field theories and exact stochastic equations for interacting particle systems
• Do dark matter halos explain lensing peaks?
• Maximum Entropy Models from Phase Harmonic Covariances
• Hidden scale invariance of intermittent turbulence in a shell model
• Inverse Renormalization Group in Quantum Field Theory
• Inverse Renormalization Group based on Image Super-Resolution using Deep Convolutional Networks
• Unifying renormalization group and the continuous wavelet transform
• Critical Dynamical Exponent of the Two-Dimensional Scalar ${\varphi }^4$ Model with Local Moves
• Inverse Renormalization Group Transformation in Bayesian Image Segmentations

Cited by in corpus (7)

• Scattering Spectra Models for Physics
• Policy-guided Monte Carlo on general state spaces: Application to glass-forming mixtures
• Performance of machine-learning-assisted Monte Carlo in sampling from simple statistical physics models
• Inverse renormalization group of spin glasses
• Dreaming up scale invariance via inverse renormalization group
• Computational Methods toward Ultrastable Glasses
• Journey from the Wilson exact RG towards the Wegner-Morris Fokker-Planck RG and the Carosso field-coarsening via Langevin stochastic processes