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Wavelet Conditional Renormalization Group

Tanguy Marchand, Misaki Ozawa, Giulio Biroli, Stéphane Mallat

TL;DR

The paper introduces Wavelet Conditional Renormalization Group (WC-RG), a multiscale energy-based framework that learns the unknown microscopic energy $E_0(\varphi_0)$ from data by modeling a product of conditional wavelet-scale distributions. By representing scale interactions with Gibbs energies in an orthogonal wavelet basis, WC-RG enables fast coarse-to-fine sampling and avoids critical slowing down near phase transitions; this is supported by a Gaussian (Ornstein–Uhlenbeck) analysis and non-perturbative tests on the $\varphi^4$ model and cosmological weak-lensing data. It provides a practical procedure to recover $E_0$ through scale-interaction free energies and thermodynamic integration, yielding energy representations that capture long-range interactions and rare-event statistics. The work also links WC-RG to deep networks, highlighting potential architectural parallels with convolutional U-Nets and conditioning schemes, while offering a robust framework for modeling complex many-body systems across physics and cosmology.

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 $\varphi^4$ 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.

Wavelet Conditional Renormalization Group

TL;DR

The paper introduces Wavelet Conditional Renormalization Group (WC-RG), a multiscale energy-based framework that learns the unknown microscopic energy from data by modeling a product of conditional wavelet-scale distributions. By representing scale interactions with Gibbs energies in an orthogonal wavelet basis, WC-RG enables fast coarse-to-fine sampling and avoids critical slowing down near phase transitions; this is supported by a Gaussian (Ornstein–Uhlenbeck) analysis and non-perturbative tests on the model and cosmological weak-lensing data. It provides a practical procedure to recover through scale-interaction free energies and thermodynamic integration, yielding energy representations that capture long-range interactions and rare-event statistics. The work also links WC-RG to deep networks, highlighting potential architectural parallels with convolutional U-Nets and conditioning schemes, while offering a robust framework for modeling complex many-body systems across physics and cosmology.

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 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.
Paper Structure (49 sections, 1 theorem, 155 equations, 21 figures, 1 table)

This paper contains 49 sections, 1 theorem, 155 equations, 21 figures, 1 table.

Table of Contents

  1. Introduction
  2. A Brief Introduction to WC-RG
  3. Renormalization Group and Wavelets
  4. RG in a Wavelet Orthogonal Basis
  5. Coarse graining and renormalization
  6. Fast wavelet transform
  7. Wavelet renormalization
  8. The ${\varphi}^4$ Model
  9. Energy Ansatz
  10. Local Potentials
  11. RG Flow Equation
  12. Wavelet Conditional RG
  13. Conditional Probability Representation
  14. Conditional Couplings Parameters
  15. Scale Interactions
  16. ...and 34 more sections

Key Result

Theorem 4.1

Let ${\varphi}(x)$ be stationary field over $x \in {\mathbb R}^d$, whose covariance has eigenvalues $\lambda_{\varphi}(k) = c\,|k|^{-\zeta}$ for $k \in {\mathbb R}^d$. If $\widehat{G}(k) = \sqrt{2} + \mathcal{O}(|k|^q)$ for $q \geq \zeta / 2$ then there exists $A >0$ and $B$ such that for all $j \in

Figures (21)

  • Figure 1: A major challenge in physics and machine learning is to estimate the probability distribution $p_0$ of a field $\varphi_0$ and its microscopic energy function $E_0$, from examples of fields or configurations. New fields can then be generated by sampling this probability distribution. The Wavelet-Conditional Renormalization Group (WC-RG) is a fast multiscale approach which eliminates "critical slowing down" phenomena near phase transitions.
  • Figure 2: Coarse-grained fields ${\varphi}_j$ and wavelet fields ${\overline \varphi}_j$ at length scales $2^j$ are iteratively computed from a field ${\varphi}_{j-1}$ at a finer scale $2^{j-1}$. It is implemented with orthogonal convolutional and subsampling operators $G$ and $\overline G$. A wavelet field ${\overline \varphi}_j$ represents the "fast degrees of freedom" of ${\varphi}_{j-1}$ which have disappeared in ${\varphi}_j$. It contains three sub-fields in dimension $d=2$, corresponding to spatial fluctuations along different orientations. The inverse wavelet transform reconstructs ${\varphi}_{j-1}$ from ${\varphi}_j$ and ${\overline \varphi}_j$ with the adjoint operators ${G}^T$ and ${\overline G}^T$. Near the phase transition, a coarse-grained field ${\varphi}_j$ has long-range spatial correlations, whereas ${\overline \varphi}_j$ has short range correlations.
  • Figure 3: The left graphs show one-dimensional wavelets $\psi(x)$ in real space, and the right graphs give their Fourier transform amplitude $|\widehat{\psi}(k)|$. (a, b): Haar wavelet. (c, d): Shannon wavelet. (e, f): Daubechies wavelet with $q=4$ vanishing moments daubechies1992ten.
  • Figure 4: A quadratic function (original curve, dashed) is represented by a piecewise linear approximation (red) given by a linear combination of hat functions (black).
  • Figure 5: (a, c): Training samples of the Ornstein-Uhenbeck process for $\xi=4$ (a) and $\xi=32$ (c). (b, d): Synthetized fields generated by WC-RG sampling for $\xi=4$ (b) and $\xi=32$ (d).
  • ...and 16 more figures

Theorems & Definitions (1)

  • Theorem 4.1