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Publication# Gaussian and sparse processes are limits of generalized Poisson processes

Julien René Fageot, Virginie Sophie Uhlmann, Michaël Unser

*ACADEMIC PRESS INC ELSEVIER SCIENCE, *2020

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Résumé

The theory of sparse stochastic processes offers a broad class of statistical models to study signals, far beyond the more classical class of Gaussian processes. In this framework, signals are represented as realizations of random processes that are solution of linear stochastic differential equations driven by Levy white noises. Among these processes, generalized Poisson processes based on compound-Poisson noises admit an interpretation as random L-splines with random knots and weights. We demonstrate that every generalized Levy process-from Gaussian to sparse-can be understood as the limit in law of a sequence of generalized Poisson processes. This enables a new conceptual understanding of sparse processes and suggests simple algorithms for the numerical generation of such objects. (C) 2018 Elsevier Inc. All rights reserved.

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Although our work lies in the field of random processes, this thesis was originally motivated by signal processing applications, mainly the stochastic modeling of sparse signals. We develop a mathematical study of the innovation model, under which a signal is described as a random process s that can be linearly and deterministically transformed into a white noise. The noise represents the unpredictable part of the signal, called its innovation, and is part of the family of Lévy white noises, which includes both Gaussian and Poisson noises. In mathematical terms, s satisfies the equation Ls=w where L is a differential operator and w a Lévy noise. The problem is therefore to study the solution of a stochastic differential equation driven by a Lévy noise. Gaussian models usually fail to reproduce the empirical sparsity observed in real-world signals. By contrast, Lévy models offer a wide range of random processes going from typically non-sparse (Gaussian) to very sparse ones (Poisson), and with many sparse signals standing between these two extremes. Our contributions can be divided in four parts. First, the cornerstone of our work is the theory of generalized random processes. Within this framework, all the considered random processes are seen as random tempered generalized functions and can be observed through smooth and rapidly decaying windows. This allows us to define the solutions of Ls=w, called generalized Lévy processes, in the most general setting. Then, we identify two limit phenomenons: the approximation of generalized Lévy processes by their Poisson counterparts, and the asymptotic behavior of generalized Lévy processes at coarse and fine scales. In the third part, we study the localization of Lévy noise in notorious function spaces (Hölder, Sobolev, Besov). As an application, characterize the local smoothness and the asymptotic growth rate of the Lévy noise. Finally, we quantify the local compressibility of the generalized Lévy processes, understood as a measure of the decreasing rate of their approximation error in an appropriate basis. From this last result, we provide a theoretical justification of the ability of the innovation model to represent sparse signals. The guiding principle of our research is the duality between the local and asymptotic properties of generalized Lévy processes. In particular, we highlight the relevant quantities, called the local and asymptotic indices, that allow quantifying the local regularity, the asymptotic growth rate, the limit behavior at coarse and fine scales, and the level of compressibility of the solutions of generalized Lévy processes.

Arash Amini, Ulugbek Kamilov, Pedram Pad, Michaël Unser

We investigate a stochastic signal-processing framework for signals with sparse derivatives, where the samples of a Levy process are corrupted by noise. The proposed signal model covers the well-known Brownian motion and piecewise-constant Poisson process; moreover, the Levy family also contains other interesting members exhibiting heavy-tail statistics that fulfill the requirements of compressibility. We characterize the maximum-a-posteriori probability (MAP) and minimum mean-square error (MMSE) estimators for such signals. Interestingly, some of the MAP estimators for the Levy model coincide with popular signal-denoising algorithms (e.g., total-variation (TV) regularization). We propose a novel non-iterative implementation of the MMSE estimator based on the belief-propagation (BP) algorithm performed in the Fourier domain. Our algorithm takes advantage of the fact that the joint statistics of general Levy processes are much easier to describe by their characteristic function, as the probability densities do not always admit closed-form expressions. We then use our new estimator as a benchmark to compare the performance of existing algorithms for the optimal recovery of gradient-sparse signals.

Thomas Marie Jean-Baptiste Humeau

We study various aspects of stochastic partial differential equations driven by Lévy white noise. This driving noise, which is a generalization of Gaussian white noise, can be viewed either as a generalized random process or as an independently scattered random measure. After unifying these approaches and establishing appropriate stochastic integral representations, we show that a necessary and sufficient condition for a Lévy white noise to have values in the space of tempered Schwartz distributions, is that the underlying Lévy measure have a positive absolute moment.
In the case of a linear stochastic partial differential equation with a general differential operator and driven by a symmetric pure jump Lévy white noise, we show that when the mild solution is locally Lebesgue integrable, then it is equal to the generalized solution, and that a random field representation exists for the generalized solution if and only if the fundamental solution of the operator has certain integrability properties. In that case, we show that the random field representation is equal to the mild solution. For this purpose, a new stochastic Fubini theorem is proved. These results are applied to the linear stochastic heat and wave equations driven by a symmetric alpha-stable noise.
We then study the non-linear stochastic heat equation driven by a general type of Lévy white noise, possibly with heavy tails and non-summable small jumps. Our framework includes in particular the alpha-stable noise. In the case of the equation on the whole space, we show that the law of the solution that we construct does not depend on the space variable. Then we show in various domains D that the solution u to the stochastic heat equation is such that t -> u(t,·) has a càdlàg version in a fractional Sobolev space of order r < -d /2. Finally, we show that the partial functions have a continuous version under some optimal moment conditions. In the alpha-stable case, we show that for the choices of alpha for which this moment condition is not satisfied, the sample paths of the partial functions are unbounded on any non-empty open subset.