# Multiscale Sparse Microcanonical Models

arXiv: https://arxiv.org/abs/1801.02013 · HTML: https://mlsanity.com/paper/1801.02013/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Jan 2018.

## Built on by (most cited first, 14 of 14)

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- [Wavelet Conditional Renormalization Group](https://mlsanity.com/paper/2207.04941/) (2022)
- [Scale Dependencies and Self-Similar Models with Wavelet Scattering Spectra](https://mlsanity.com/paper/2204.10177/) (2022)
- [Equivariant Wavelets: Fast Rotation and Translation Invariant Wavelet Scattering Transforms](https://mlsanity.com/paper/2104.11244/) (2021)
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- [BLIS-Net: Classifying and Analyzing Signals on Graphs](https://mlsanity.com/paper/2310.17579/) (2023)
- [WhaleNet: a Novel Deep Learning Architecture for Marine Mammals Vocalizations on Watkins Marine Mammal Sound Database](https://mlsanity.com/paper/2402.17775/) (2024)
- [Texture synthesis via projection onto multiscale, multilayer statistics](https://mlsanity.com/paper/2105.10825/) (2021)
- [MGD: Moment Guided Diffusion for Maximum Entropy Generation](https://mlsanity.com/paper/2602.17211/) (2026)
- [Statistical Component Separation for Targeted Signal Recovery in Noisy Mixtures](https://mlsanity.com/paper/2306.15012/) (2023)
- [Mean-Field Microcanonical Gradient Descent](https://mlsanity.com/paper/2403.08362/) (2024)
- [Beyond sparse denoising in frames: minimax estimation with a scattering transform](https://mlsanity.com/paper/2510.19612/) (2025)
- [Generative Modeling via Kernelized Stochastic Interpolants](https://mlsanity.com/paper/2602.20070/) (2026)
- [Specific Algorithmic Interpretability of Neural Networks: A Case Study on Textures](https://mlsanity.com/paper/2610.04413/) (2026)

## Abstract (arXiv)

We study approximations of non-Gaussian stationary processes having long range correlations with microcanonical models. These models are conditioned by the empirical value of an energy vector, evaluated on a single realization. Asymptotic properties of maximum entropy microcanonical and macrocanonical processes and their convergence to Gibbs measures are reviewed. We show that the Jacobian of the energy vector controls the entropy rate of microcanonical processes. Sampling maximum entropy processes through MCMC algorithms require too many operations when the number of constraints is large. We define microcanonical gradient descent processes by transporting a maximum entropy measure with a gradient descent algorithm which enforces the energy conditions. Convergence and symmetries are analyzed. Approximations of non-Gaussian processes with long range interactions are defined with multiscale energy vectors computed with wavelet and scattering transforms. Sparsity properties are captured with bf l1 norms. Approximations of Gaussian, Ising and point processes are studied, as well as image and audio texture synthesis.