In order to improve the performance of least mean square (LMS)-based adaptive filtering for identifying block-sparse systems, a new adaptive algorithm called block-sparse LMS (BS-LMS) is proposed in this paper. The basis of the proposed algorithm is to insert a penalty of block-sparsity, which is a mixed @math norm of adaptive tap-weights with equal group partition sizes, into the cost function of traditional LMS algorithm. To describe a block-sparse system response, we first propose a Markov-Gaussian model, which can generate a kind of system responses of arbitrary average sparsity and arbitrary average block length using given parameters. Then we present theoretical expressions of the steady-state misadjustment and transient convergence behavior of BS-LMS with an appropriate group partition size for white Gaussian input data. Based on the above results, we theoretically demonstrate that BS-LMS has much better convergence behavior than @math -LMS with the same small level of misadjustment. Finally, numerical experiments verify that all of the theoretical analysis agrees well with simulation results in a large range of parameters.
The idea of using mixed norm, such as @math norm @cite_1 @cite_33 @cite_0 , approximated @math norm @cite_28 , @math norm @cite_31 , to handle block-sparsity has been adopted in sparse signal recovery. By exploiting block structure, recovery may be possible under more general conditions, which demonstrates superior performance brought about by mixed norm. Furthermore, after mixed norm is introduced, the reconstruction error in the presence of noise becomes smaller compared with the conventional algorithms. Besides mixed norm, there are some other approaches in block-sparse signal recovery, including greedy algorithms @cite_32 @cite_23 @cite_24 @cite_20 , Bayesian CS framework-based algorithms @cite_6 @cite_12 , the dynamic programming-based algorithm @cite_34 and the decoding-based algorithm @cite_2 .
@cite_33 @cite_28 @cite_1 @cite_32 @cite_6 @cite_0 @cite_24 @cite_23 @cite_2 @cite_31 @cite_34 @cite_12 @cite_20
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We consider efficient methods for the recovery of block-sparse signals-i.e., sparse signals that have nonzero entries occurring in clusters-from an underdetermined system of linear equations. An uncertainty relation for block-sparse signals is derived, based on a block-coherence measure, which we introduce. We then show that a block-version of the orthogonal matching pursuit algorithm recovers block -sparse signals in no more than steps if the block-coherence is sufficiently small. The same condition on block-coherence is shown to guarantee successful recovery through a mixed -optimization approach. This complements previous recovery results for the block-sparse case which relied on small block-restricted isometry constants. The significance of the results presented in this paper lies in the fact that making explicit use of block-sparsity can provably yield better reconstruction properties than treating the signal as being sparse in the conventional sense, thereby ignoring the additional structure in the problem. There are two main algorithmic approaches to sparse signal recovery: geometric and combinatorial. The geometric approach utilizes geometric properties of the measurement matrix Phi. A notable example is the Restricted Isometry Property, which states that the mapping Phi preserves the Euclidean norm of sparse signals; it is known that random dense matrices satisfy this constraint with high probability. On the other hand, the combinatorial approach utilizes sparse matrices, interpreted as adjacency matrices of sparse (possibly random) graphs, and uses combinatorial techniques to recover an approximation to the signal. In this paper we present a unification of these two approaches. To this end, we extend the notion of Restricted Isometry Property from the Euclidean lscr2 norm to the Manhattan lscr1 norm. Then we show that this new lscr1 -based property is essentially equivalent to the combinatorial notion of expansion of the sparse graph underlying the measurement matrix. At the same time we show that the new property suffices to guarantee correctness of both geometric and combinatorial recovery algorithms. As a result, we obtain new measurement matrix constructions and algorithms for signal recovery which, compared to previous algorithms, are superior in either the number of measurements or computational efficiency of decoders. In a number of signal processing applications, problem formulations based on the l 1 norm as a sparsity inducing signal prior lead to simple algorithms with good performance. However, l 1 norm is not flexible enough to handle certain signal structures that are represented using a few groups of coefficients. Formulations that make use of mixed norms provide an alternative that can handle such signals by forcing sparsity on a group level and allowing non-sparse distributions within the groups. However, conventional mixed norms allow only non-overlapping groups - a restriction that leads to characteristics unlikely for natural signals. In this paper, we investigate mixed norms with overlapping groups. We consider a simple denoising formulation that gives a convex optimization problem and provide an algorithm that solves the problem. We use the algorithm to evaluate the performance of mixed norms with overlapping groups as signal priors. This paper provides novel results for the recovery of signals from undersampled measurements based on analysis @math -minimization, when the analysis operator is given by a frame. We both provide so-called uniform and nonuniform recovery guarantees for cosparse (analysis-sparse) signals using Gaussian random measurement matrices. The nonuniform result relies on a recovery condition via tangent cones and the uniform recovery guarantee is based on an analysis version of the null space property. Examining these conditions for Gaussian random matrices leads to precise bounds on the number of measurements required for successful recovery. In the special case of standard sparsity, our result improves a bound due to Rudelson and Vershynin concerning the exact reconstruction of sparse signals from Gaussian measurements with respect to the constant and extends it to stability under passing to approximately sparse signals and to robustness under noise on the measurements.