Incremental Reformulated Automatic Relevance Determination

In this work, the relationship between the incremental version of sparse Bayesian learning (SBL) with automatic relevance determination (ARD)-a fast marginal likelihood maximization (FMLM) algorithm-and a recently proposed reformulated ARD scheme is established. The FMLM algorithm is an incremental approach to SBL with ARD, where the corresponding objective function-the marginal likelihood-is optimized with respect to the parameters of a single component provided that the other parameters are fixed; the corresponding maximizer is computed in closed form, which enables a very efficient SBL realization. Wipf and Nagarajan have recently proposed a reformulated ARD (R-ARD) approach, which optimizes the marginal likelihood using auxiliary upper bounding functions. The resulting algorithm is then shown to correspond to a series of reweighted <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -constrained convex optimization problems. This correspondence establishes and analyzes the relationship between the FMLM and R-ARD schemes. Specifically, it is demonstrated that the FMLM algorithm realizes an incremental approach to the optimization of the R-ARD objective function. This relationship allows deriving the R-ARD pruning conditions similar to those used in the FMLM scheme to analytically detect components that are to be removed from the model, thus regulating the estimated signal sparsity and accelerating the algorithm convergence.