Wavelet Score-Based Generative Modeling

Score-based generative models (SGMs) synthesize new data samples from Gaussian white noise by running a time-reversed Stochastic Differential Equation (SDE) whose drift coefficient depends on some probabilistic score. The discretization of such SDEs typically requires a large number of time steps and hence a high computational cost. This is because of ill-conditioning properties of the score that we analyze mathematically. We show that SGMs can be considerably accelerated, by factorizing the data distribution into a product of conditional probabilities of wavelet coefficients across scales. The resulting Wavelet Score-based Generative Model (WSGM) synthesizes wavelet coefficients with the same number of time steps at all scales, and its time complexity therefore grows linearly with the image size. This is proved mathematically over Gaussian distributions, and shown numerically over physical processes at phase transition and natural image datasets.

Multiscale Sparse Microcanonical Models

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 \emph{macrocanonical} models are approximated by their \emph{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.

Deep Haar Scattering Networks

An orthogonal Haar scattering transform is a deep network, computed with a hierarchy of additions, subtractions and absolute values, over pairs of coefficients. It provides a simple mathematical model for unsupervised deep network learning. It implements non-linear contractions, which are optimized for classification, with an unsupervised pair matching algorithm, of polynomial complexity. A structured Haar scattering over graph data computes permutation invariant representations of groups of connected points in the graph. If the graph connectivity is unknown, unsupervised Haar pair learning can provide a consistent estimation of connected dyadic groups of points. Classification results are given on image data bases, defined on regular grids or graphs, with a connectivity which may be known or unknown.