MMSE Estimation of Sparse Lévy Processes

Abstract

Abstract-We investigate a stochastic signal-processing framework for signals with sparse derivatives, where the samples of a Lévy process are corrupted by noise. The proposed signal model covers the well-known Brownian motion and piecewise-constant Poisson process; moreover, the Lévy 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 Lévy 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 Lévy 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.

MMSE denoising of sparse and non-Gaussian AR(1) processes

“…Since for any value of κ there is a Markov dependency between entries of s, we notice that the structure of the factor-graph is exactly the same as that of Lévy processes (κ = 0) described in [4]. This implies that the network through which the messages are passed remains the same.…”

“…In the case of the Lévy processed considered in [4] where κ = 0 and ρ = 1, the authors could evaluate (10) and (11) simply by saving all messages and probability density functions over a fine enough grid and using a Riemann sum approximation of the integrals (notice that when ρ = 1 the integrals are simple convolutions). In the more general AR(1) scenario for which κ > 0 and ρ < 1, if we start with a certain grid on x for storing the values of the μ functions, the grid will expand in the case of μ − and shrink in the case of μ + by a factor of ρ after each iteration of the algorithm.…”

“…In [4], the authors considered the problem of denoising Lévy processes [5], which may be interpreted as the (unstable) limit of a first order auto-regressive (AR(1)) process (i.e., with pole at the origin) [6]. Based on the property that Lévy processes have independent increments, the authors proposed a message-passing estimation method to denoise such processes.…”

“…This implies that the decomposition in a standard wavelet basis is no longer appropriate. Also, the lack of independence of their increments makes the message-passing scheme proposed in [4] inadequate.…”

Autocalibrated signal reconstruction from linear measurements using adaptive GAMP

“…Although estimators based on 1 -regularization in some suitable transform domain offer a good reconstruction performance, there is often a large performance gap between 1 -based approaches and MMSE estimation [14].…”