Improving A*OMP: Theoretical and Empirical Analyses With a Novel Dynamic Cost Model

Best-first search has been recently utilized for compressed sensing (CS) by the A* orthogonal matching pursuit (A*OMP) algorithm. In this work, we concentrate on theoretical and empirical analyses of A*OMP. We present a restricted isometry property (RIP) based general condition for exact recovery of sparse signals via A*OMP. In addition, we develop online guarantees which promise improved recovery performance with the residue-based termination instead of the sparsity-based one. We demonstrate the recovery capabilities of A*OMP with extensive recovery simulations using the adaptive-multiplicative (AMul) cost model, which effectively compensates for the path length differences in the search tree. The presented results, involving phase transitions for different nonzero element distributions as well as recovery rates and average error, reveal not only the superior recovery accuracy of A*OMP, but also the improvements with the residue-based termination and the AMul cost model. Comparison of the run times indicate the speed up by the AMul cost model. We also demonstrate a hybrid of OMP and A?OMP to accelerate the search further. Finally, we run A*OMP on a sparse image to illustrate its recovery performance for more realistic coefcient distributions

Search-based methods for the sparse signal recovery problem in compressed

The sparse signal recovery, which appears not only in compressed sensing but also in other related problems such as sparse overcomplete representations, denoising, sparse learning, etc. has drawn a large attraction in the last decade. The literature contains a vast number of recovery methods, which have been analysed in theoretical and empirical aspects. This dissertation presents novel search-based sparse signal recovery methods. First, we discuss theoretical analysis of the orthogonal matching pursuit algorithm with more iterations than the number of nonzero elements of the underlying sparse signal. Second, best-first tree search is incorporated for sparse recovery by a novel method, whose tractability follows from the properly defined cost models and pruning techniques. The proposed method is evaluated by both theoretical and empirical analyses, which clearly emphasize the improvements in the recovery accuracy. Next, we introduce an iterative two stage thresholding algorithm, where the forward step adds a larger number of nonzero elements to the sparse representation than the backward one removes. The presented simulation results reveal not only the recovery abilities of the proposed method, but also illustrate optimal choices for the step sizes. Finally, we propose a new mixed integer linear programming formulation for sparse recovery. Due to the equivalency of this formulation to the original problem, the solution is guaranteed to be correct when it can be solved in reasonable time. The simulation results indicate that the solution can be found easily under some reasonable assumptions, especially for signals with constant amplitude nonzero elements

A* Orthogonal Matching Pursuit: Best-First Search for Compressed Sensing Signal Recovery

Compressed sensing is a developing field aiming at reconstruction of sparse signals acquired in reduced dimensions, which make the recovery process under-determined. The required solution is the one with minimum $\ell_0$ norm due to sparsity, however it is not practical to solve the $\ell_0$ minimization problem. Commonly used techniques include $\ell_1$ minimization, such as Basis Pursuit (BP) and greedy pursuit algorithms such as Orthogonal Matching Pursuit (OMP) and Subspace Pursuit (SP). This manuscript proposes a novel semi-greedy recovery approach, namely A* Orthogonal Matching Pursuit (A*OMP). A*OMP performs A* search to look for the sparsest solution on a tree whose paths grow similar to the Orthogonal Matching Pursuit (OMP) algorithm. Paths on the tree are evaluated according to a cost function, which should compensate for different path lengths. For this purpose, three different auxiliary structures are defined, including novel dynamic ones. A*OMP also incorporates pruning techniques which enable practical applications of the algorithm. Moreover, the adjustable search parameters provide means for a complexity-accuracy trade-off. We demonstrate the reconstruction ability of the proposed scheme on both synthetically generated data and images using Gaussian and Bernoulli observation matrices, where A*OMP yields less reconstruction error and higher exact recovery frequency than BP, OMP and SP. Results also indicate that novel dynamic cost functions provide improved results as compared to a conventional choice.Comment: accepted for publication in Digital Signal Processin

Sıkıştırmalı algılama işaret geri kazanma probleminde A* aramasının karmaşıklık ve doğruluk analizi (Analysis of accuracy and complexity of A* search for compressed sensing signal recovery)

A* Orthogonal Matching Pursuit (A*OMP) utilizes best-first search for recovery of sparse signals from reduced dimensions. It applies A* search on a tree whose branches are extended similar to Orthogonal Matching Pursuit (OMP). A*OMP makes a tractable tree search possible via effective complexity-accuracy trade-off parameters. Here, we concentrate on the effects of these parameters on the recovery performance. Via empirical comparison of complexity and performance, we demonstrate the effects of the search parameters on search size and recovery. We also compare A*OMP with well-known compressed sensing (CS) recovery techniques to reveal the improvement in the reconstruction

Compressed sensing signal recovery via A* orthogonal matching pursuit

Reconstruction of sparse signals acquired in reduced dimensions requires the solution with minimum ℓ0 norm. As solving the ℓ0 minimization directly is unpractical, a number of algorithms have appeared for finding an indirect solution. A semi-greedy approach, A* Orthogonal Matching Pursuit (A*OMP), is proposed in [1] where the solution is searched on several paths of a search tree. Paths of the tree are evaluated and extended according to some cost function, for which novel dynamic auxiliary cost functions are suggested. This paper describes the A*OMP algorithm and the proposed cost functions briefly. The novel dynamic auxiliary cost functions are shown to provide improved results as compared to a conventional choice. Reconstruction performance is illustrated on both synthetically generated data and real images, which show that the proposed scheme outperforms well-known CS reconstruction methods

A mixed integer linear programming formulation for the sparse recovery problem in compressed sensing

We propose a new mixed integer linear programming (MILP) formulation of the sparse signal recovery problem in compressed sensing (CS). This formulation is obtained by introduction of an auxiliary binary vector, where ones locate the recovered nonzero indices. Joint optimization for finding this auxiliary vector together with the underlying sparse vector leads to the proposed MILP formulation. By addition of a few appropriate constraints, this problem can be solved by existing MILP solvers. In contrast to other methods, this formulation is not an approximation of the sparse optimization problem, but is its equivalent. Hence, its solution is exactly equal to the optimal solution of the original sparse recovery problem, once it is feasible. We demonstrate this by recovery simulations involving different sparse signal types. The proposed scheme improves recovery over the mainstream CS recovery methods especially when the underlying sparse signals have constant amplitude nonzero elements

Compressed sensing signal recovery via forward–backward pursuit

Recovery of sparse signals from compressed measurements constitutes an ℓ0 norm minimization problem, which is unpractical to solve. A number of sparse recovery approaches have appeared in the literature, including ℓ1 minimization techniques, greedy pursuit algorithms, Bayesian methods and nonconvex optimization techniques among others. This manuscript introduces a novel two stage greedy approach, called the Forward–Backward Pursuit (FBP). FBP is an iterative approach where each iteration consists of consecutive forward and backward stages. The forward step first expands the support estimate by the forward step size, while the following backward step shrinks it by the backward step size. The forward step size is larger than the backward step size, hence the initially empty support estimate is expanded at the end of each iteration. Forward and backward steps are iterated until the residual power of the observation vector falls below a threshold. This structure of FBP does not necessitate the sparsity level to be known a priori in contrast to the Subspace Pursuit or Compressive Sampling Matching Pursuit algorithms. FBP recovery performance is demonstrated via simulations including recovery of random sparse signals with different nonzero coefficient distributions in noisy and noise-free scenarios in addition to the recovery of a sparse image