Bayesian Estimation for Continuous-Time Sparse Stochastic Processes

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.

To appear in IEEE TSP

References in corpus (3)

• Compressible Distributions for High-dimensional Statistics
• A unified formulation of Gaussian vs. sparse stochastic processes - Part I: Continuous-domain theory
• A unified formulation of Gaussian vs. sparse stochastic processes - Part II: Discrete-domain theory

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