Multiscale Sparse Microcanonical Models

01/06/2018
by   Joan Bruna, et al.
0

We study density estimation of stationary processes defined over an infinite grid from a single, finite realization. Gaussian Processes and Markov Random Fields avoid the curse of dimensionality by focusing on low-order and localized potentials respectively, but its application to complex datasets is limited by their inability to capture singularities and long-range interactions, and their expensive inference and learning respectively. These are instances of Gibbs models, defined as maximum entropy distributions under moment constraints determined by an energy vector. The Boltzmann equivalence principle states that under appropriate ergodicity, such macrocanonical models are approximated by their microcanonical counterparts, which replace the expectation by the sample average. Microcanonical models are appealing since they avoid computing expensive Lagrange multipliers to meet the constraints. This paper introduces microcanonical measures whose energy vector is given by a wavelet scattering transform, built by cascading wavelet decompositions and point-wise nonlinearities. We study asymptotic properties of generic microcanonical measures, which reveal the fundamental role of the differential structure of the energy vector in controlling e.g. the entropy rate. Gradient information is also used to define a microcanonical sampling algorithm, for which we provide convergence analysis to the microcanonical measure. Whereas wavelet transforms capture local regularity at different scales, scattering transforms provide scale interaction information, critical to restore the geometry of many physical phenomena. We demonstrate the efficiency of sparse multiscale microcanonical measures on several processes and real data exhibiting long-range interactions, such as Ising, Cox Processes and image and audio textures.

READ FULL TEXT

page 24

page 27

page 29

page 30

research
02/10/2021

Differential Entropy Rate Characterisations of Long Range Dependent Processes

A quantity of interest to characterise continuous-valued stochastic proc...
research
02/06/2015

Quantum Energy Regression using Scattering Transforms

We present a novel approach to the regression of quantum mechanical ener...
research
07/11/2022

Wavelet Conditional Renormalization Group

We develop a multiscale approach to estimate high-dimensional probabilit...
research
04/19/2022

Scale Dependencies and Self-Similarity Through Wavelet Scattering Covariance

We introduce a scattering covariance matrix which provides non-Gaussian ...
research
05/30/2023

Characterization of p-exponents by continuous wavelet transforms, applications to the multifractal analysis of sum of random pulses

The theory of orthonormal wavelet bases is a useful tool in multifractal...
research
05/16/2016

Wavelet Scattering Regression of Quantum Chemical Energies

We introduce multiscale invariant dictionaries to estimate quantum chemi...
research
10/27/2020

Particle gradient descent model for point process generation

This paper introduces a generative model for planar point processes in a...

Please sign up or login with your details

Forgot password? Click here to reset