Bayesian Estimation for Continuous-Time Sparse Stochastic Processes

TL;DR

This paper develops Bayesian methods for denoising and interpolating continuous-time sparse stochastic processes using spline theory, providing a probabilistic framework that compares favorably with traditional regularization techniques.

Contribution

It introduces a Bayesian estimation framework for continuous-time sparse signals, deriving a tractable joint distribution and comparing statistical methods with existing regularization approaches.

Findings

MMSE estimator performs comparably to regularization methods under certain conditions.

The joint a priori distribution facilitates efficient Bayesian inference.

Simulation results validate the effectiveness of the proposed Bayesian methods.

Abstract

We consider continuous-time sparse stochastic processes from which we have only a finite number of noisy/noiseless samples. Our goal is to estimate the noiseless samples (denoising) and the signal in-between (interpolation problem). By relying on tools from the theory of splines, we derive the joint a priori distribution of the samples and show how this probability density function can be factorized. The factorization enables us to tractably implement the maximum a posteriori and minimum mean-square error (MMSE) criteria as two statistical approaches for estimating the unknowns. We compare the derived statistical methods with well-known techniques for the recovery of sparse signals, such as the $\ell_1$ norm and Log ($\ell_1$-$\ell_0$ relaxation) regularization methods. The simulation results show that, under certain conditions, the performance of the regularization techniques can be very close to that of the MMSE estimator.