A Weighted Average of Sparse Representations is Better than the Sparsest One Alone

Structured Decompositions and Efficient Algorithms

计算机科学;物理学

Michael Elad ; Irad Yavneh

PID  :  82177

学科分类：计算机科学（综合）

来源: CEUR

【 摘 要 】

Cleaning of noise from signals is a classical and longstudied problem in signal processing. Algorithms for this task necessarily rely on an apriori knowledge about the signal characteristics, along with information about the noise properties. For signals that admit sparse representations over a known dictionary, a commonly used denoising technique is to seek the sparsest representation that synthesizes a signal close enough to the corrupted one. As this problem is too complex in general, approximation methods, such as greedy pursuit algorithms, are often employed. In this line of reasoning, we are led to believe that detection of the sparsest rep resentation is key in the success of the denoising goal. Does this mean that other competitive and slightly inferior sparse representations are meaningless? Suppose we are served with a group of competing sparse representations, each claiming to explain the signal differently. Can those be fused somehow to lead to a better result? Surpris ingly, the answer to this question is positive; merging these representations can form a more accurate, yet dense, estimate of the original signal even when the latter is known to be sparse. In this paper we demonstrate this behavior, propose a practical way to generate such a collection of representations by randomizing the Orthogonal Matching Pursuit (OMP) algorithm, and produce a clear analytical justification for the superiority of the associated Randomized OMP (RandOMP) algorithm. We show that while the Maximum aposterior Probability (MAP) estimator aims to find and use the sparsest representation, the Minimum MeanSquaredError (MMSE) estimator leads to a fusion of representations to form its result. Thus, working with an appropriate mixture of candidate representations, we are surpassing the MAP and tending towards the MMSE estimate, and thereby getting a far more accurate estimation, especially at medium and low SNR.