In this paper, we propose a novel algorithm for analysis-based sparsity reconstruction. It can solve the generalized problem by structured sparsity regularization with an orthogonal basis and total variation regularization. The proposed algorithm is based on the iterative reweighted least squares (IRLS) model, which is further accelerated by the preconditioned conjugate gradient method. The convergence rate of the proposed algorithm is almost the same as that of the traditional IRLS algorithms, that is, exponentially fast. Moreover, with the specifically devised preconditioner, the computational cost for each iteration is significantly less than that of traditional IRLS algorithms, which enables our approach to handle large scale problems. In addition to the fast convergence, it is straightforward to apply our method to standard sparsity, group sparsity, overlapping group sparsity and TV based problems. Experiments are conducted on a practical application: compressive sensing magnetic resonance imaging. Extensive results demonstrate that the proposed algorithm achieves superior performance over 14 state-of-the-art algorithms in terms of both accuracy and computational cost.
The conventional IRLS algorithms solve the standard sparse problem in this constrained form: In practice, the @math norm is replaced by a reweighted @math norm @cite_25 : The diagonal weight matrix @math in the @math -th iteration is computed from the solution of the current iteration @math , in particular, the diagonal elements @math . With current weights @math , we can derive the closed form solution for @math : The algorithm can be summarized in Algorithm . It has been proven that the IRLS algorithm converges exponentially fast under mild conditions @cite_30 : where @math is a fixed constant with @math . However, this algorithm is rarely used in compressive sensing applications especially for large scale problems. That is because the inverse of @math takes @math if @math is a @math sampling matrix. Even with higher convergence rate, traditional IRLS still cannot compete with the fastest first-order algorithms such as FISTA @cite_41 (some results have been shown in @cite_18 ). Moreover, none of existing IRLS methods @cite_25 @cite_30 @cite_27 could solve the overlapping group sparsity problems, which significantly limits the usage.
@cite_30 @cite_18 @cite_41 @cite_27 @cite_25
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The problem of minimizing the rank of a matrix subject to affine constraints has applications in several areas including machine learning, and is known to be NP-hard. A tractable relaxation for this problem is nuclear norm (or trace norm) minimization, which is guaranteed to find the minimum rank matrix under suitable assumptions. In this paper, we propose a family of Iterative Reweighted Least Squares algorithms IRLS-p (with 0 ≤ p ≤ 1), as a computationally efficient way to improve over the performance of nuclear norm minimization. The algorithms can be viewed as (locally) minimizing certain smooth approximations to the rank function. When p = 1, we give theoretical guarantees similar to those for nuclear norm minimization, that is, recovery of low-rank matrices under certain assumptions on the operator defining the constraints. For p < 1, IRLS-p shows better empirical performance in terms of recovering low-rank matrices than nuclear norm minimization. We provide an efficient implementation for IRLS-p, and also present a related family of algorithms, sIRLS-p. These algorithms exhibit competitive run times and improved recovery when compared to existing algorithms for random instances of the matrix completion problem, as well as on the MovieLens movie recommendation data set. In this paper we study general @math l p regularized unconstrained minimization problems. In particular, we derive lower bounds for nonzero entries of the first- and second-order stationary points and hence also of local minimizers of the @math l p minimization problems. We extend some existing iterative reweighted @math l 1 ( @math IRL 1 ) and @math l 2 ( @math IRL 2 ) minimization methods to solve these problems and propose new variants for them in which each subproblem has a closed-form solution. Also, we provide a unified convergence analysis for these methods. In addition, we propose a novel Lipschitz continuous @math ∈ -approximation to @math ? x ? p p . Using this result, we develop new @math IRL 1 methods for the @math l p minimization problems and show that any accumulation point of the sequence generated by these methods is a first-order stationary point, provided that the approximation parameter @math ∈ is below a computable threshold value. This is a remarkable result since all existing iterative reweighted minimization methods require that @math ∈ be dynamically updated and approach zero. Our computational results demonstrate that the new @math IRL 1 method and the new variants generally outperform the existing @math IRL 1 methods (Chen and Zhou in 2012; Foucart and Lai in Appl Comput Harmon Anal 26:395---407, 2009). Under certain conditions (known as the restricted isometry property, or RIP) on the mN matrix ˆ (where m<N ), vectors x 2 R N that are sparse (i.e., have most of their entries equal to 0) can be recovered exactly from y WDx even though ˆ � 1 .y is typically an .Nm -dimensional hyperplane; in addition, x is then equal to the element in ˆ � 1 .y of minimal 1-norm. This minimal element can be identified via linear programming algorithms. We study an alternative method of determining x, as the limit of an iteratively reweighted least squares (IRLS) algorithm. The main step of this IRLS finds, for a given weight vector w, the element in ˆ � 1 .y with smallest 2.w -norm. If x .n is the solution at itera- tion step n, then the new weight w .n is defined by w .n WD Œjx .n j 2 C " 2� � 1=2 , i D 1; :::;N , for a decreasing sequence of adaptively defined "n; this updated weight is then used to obtain x .nC1 and the process is repeated. We prove that whensatisfies the RIP conditions, the sequence x .n converges for all y, re- gardless of whether ˆ � 1 .y contains a sparse vector. If there is a sparse vector in ˆ � 1 .y , then the limit is this sparse vector, and when x .n is sufficiently close to the limit, the remaining steps of the algorithm converge exponentially fast (linear convergence in the terminology of numerical optimization). The same al- gorithm with the "heavier" weight w .n D Œjx .n This paper introduces a parallel and distributed algorithm for solving the following minimization problem with linear constraints: @math minimizef1(x1)+ź+fN(xN)subject toA1x1+ź+ANxN=c,x1źX1,ź,xNźXN,where @math Nź2, @math fi are convex functions, @math Ai are matrices, and @math Xi are feasible sets for variable @math xi. Our algorithm extends the alternating direction method of multipliers (ADMM) and decomposes the original problem into N smaller subproblems and solves them in parallel at each iteration. This paper shows that the classic ADMM can be extended to the N-block Jacobi fashion and preserve convergence in the following two cases: (i) matrices @math Ai are mutually near-orthogonal and have full column-rank, or (ii) proximal terms are added to the N subproblems (but without any assumption on matrices @math Ai). In the latter case, certain proximal terms can let the subproblem be solved in more flexible and efficient ways. We show that @math źxk+1-xkźM2 converges at a rate of o(1 k) where M is a symmetric positive semi-definte matrix. Since the parameters used in the convergence analysis are conservative, we introduce a strategy for automatically tuning the parameters to substantially accelerate our algorithm in practice. We implemented our algorithm (for the case ii above) on Amazon EC2 and tested it on basis pursuit problems with >300 GB of distributed data. This is the first time that successfully solving a compressive sensing problem of such a large scale is reported.