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Mathematical Statistics and Learning

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Volume 1, Issue 3/4, 2018, pp. 257–315
DOI: 10.4171/MSL/7

Published online: 2019-05-18

Multiscale sparse microcanonical models

Joan Bruna[1] and Stéphane Mallat[2]

(1) New York University, USA
(2) Collège de France and École Normale Supérieure, Paris, France

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 $\mathbf{l}^1$ norms. Approximations of Gaussian, Ising and point processes are studied, as well as image and audio texture synthesis.

Keywords: Macrocanonical, microcanonical, wavelet, scattering, texture

Bruna Joan, Mallat Stéphane: Multiscale sparse microcanonical models. Math. Stat. Learn. 1 (2018), 257-315. doi: 10.4171/MSL/7