# Sparse Bayesian Learning with Dynamic Filtering for Inference of Time-Varying Sparse Signals

## TL;DR

This paper introduces SBL-DF, a robust and efficient dynamic filtering algorithm for time-varying sparse signals that outperforms existing methods, especially in challenging dictionary conditions.

## Contribution

The paper proposes a novel SBL-DF algorithm that integrates dynamics models into the hyperpriors, improving robustness and efficiency in estimating time-varying sparse signals.

## Key findings

- SBL-DF converges faster and more accurately than standard SBL.
- The algorithm is robust to imperfect dynamics models and dictionary coherence.
- SBL-DF outperforms state-of-the-art methods in challenging scenarios.

## Abstract

Many signal processing applications require estimation of time-varying sparse signals, potentially with the knowledge of an imperfect dynamics model. In this paper, we propose an algorithm for dynamic filtering of time-varying sparse signals based on the sparse Bayesian learning (SBL) framework. The key idea underlying the algorithm, termed SBL-DF, is the incorporation of a signal prediction generated from a dynamics model and estimates of previous time steps into the hyperpriors of the SBL probability model. The proposed algorithm is robust to imperfect dynamics models (due to the propagation of dynamics information through higher-order statistics), robust to certain undesirable dictionary properties such as coherence (due to properties of the SBL framework), allows the use of arbitrary dynamics models, and requires the tuning of fewer parameters than many other dynamic filtering algorithms do. We also extend the fast marginal likelihood SBL inference procedure to the informative hyperprior setting to create a particularly efficient version of the SBL-DF algorithm. Numerical simulations show that SBL-DF converges much faster and to more accurate solutions than standard SBL and other dynamical filtering algorithms. In particular, we show that SBL-DF outperforms state of the art algorithms when the dictionary contains challenging structure such as coherence and column scaling, as is found in many practical applications.