# Hierarchic Flows to Estimate and Sample High-dimensional Probabilities

arXiv: https://arxiv.org/abs/2405.03468 · HTML: https://mlsanity.com/paper/2405.03468/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 88 references that are in the map.
- First posted on arXiv 6 May 2024.

## Builds on (most cited first, 18 of 18)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices](https://mlsanity.com/paper/1903.08568/) (2019)
- [Statistical Efficiency of Score Matching: The View from Isoperimetry](https://mlsanity.com/paper/2210.00726/) (2022)
- [Wavelet Score-Based Generative Modeling](https://mlsanity.com/paper/2208.05003/) (2022)
- [Minimax Mixing Time of the Metropolis-Adjusted Langevin Algorithm for Log-Concave Sampling](https://mlsanity.com/paper/2109.13055/) (2021)
- [Sqrt(d) Dimension Dependence of Langevin Monte Carlo](https://mlsanity.com/paper/2109.03839/) (2021)
- [Learning multi-scale local conditional probability models of images](https://mlsanity.com/paper/2303.02984/) (2023)
- [Renormalizing Diffusion Models](https://mlsanity.com/paper/2308.12355/) (2023)
- [Non-Uniform Diffusion Models](https://mlsanity.com/paper/2207.09786/) (2022)
- [Maximum Entropy Models from Phase Harmonic Covariances](https://mlsanity.com/paper/1911.10017/) (2019)
- [Wavelet Conditional Renormalization Group](https://mlsanity.com/paper/2207.04941/) (2022)
- [Scattering Spectra Models for Physics](https://mlsanity.com/paper/2306.17210/) (2023)
- [Scale Dependencies and Self-Similar Models with Wavelet Scattering Spectra](https://mlsanity.com/paper/2204.10177/) (2022)
- [Log-Sobolev inequality for near critical Ising models](https://mlsanity.com/paper/2202.02301/) (2022)
- [Conditionally Strongly Log-Concave Generative Models](https://mlsanity.com/paper/2306.00181/) (2023)

## Built on by (most cited first, 3 of 3)

- [Generalization Dynamics of Linear Diffusion Models](https://mlsanity.com/paper/2505.24769/) (2025)
- [Scale-Adaptive Generative Flows for Multiscale Scientific Data](https://mlsanity.com/paper/2509.02971/) (2025)
- [Renormalization Group Flow Matching for Scalable Local Generative Modeling](https://mlsanity.com/paper/2608.23696/) (2026)

## Abstract (arXiv)

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 vvarphi4 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.