A compressed sensing method consists of a rectangular measurement matrix, @math with @math , together with an associated recovery algorithm, @math . Compressed sensing methods aim to construct a high quality approximation to any given input vector @math using only @math as input. In particular, we focus herein on instance optimal nonlinear approximation error bounds for @math and @math of the form @math for @math , where @math is the best possible @math -term approximation to @math . In this paper we develop a compressed sensing method whose associated recovery algorithm, @math , runs in @math -time, matching a lower bound up to a @math factor. This runtime is obtained by using a new class of sparse binary compressed sensing matrices of near optimal size in combination with sublinear-time recovery techniques motivated by sketching algorithms for high-volume data streams. The new class of matrices is constructed by randomly subsampling rows from well-chosen incoherent matrix constructions which already have a sub-linear number of rows. As a consequence, fewer random bits than previously required are needed in order to select the rows utilized by the fast reconstruction algorithms considered herein.
Perhaps the measurement matrices considered herein are most similar to previous compressed sensing matrices based on unbalanced expander graphs (see, e.g., @cite_20 @cite_23 @cite_9 ). Indeed, the measurement matrices used in this paper are created by randomly sampling rows from larger binary matrices that are, in fact, the adjacency matrices of a subclass of unbalanced expander graphs. However, unlike previous approaches which use the properties of general unbalanced expanders, we use different combinatorial techniques which allow us to develop @math -time recovery algorithms. To the best of our knowledge, the runtimes we obtain by doing so are the best known for any such method having @math '' error guarantees.
@cite_9 @cite_23 @cite_20
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We investigate the problem of reconstructing a high-dimensional nonnegative sparse vector from lower-dimensional linear measurements. While much work has focused on dense measurement matrices, sparse measurement schemes can be more efficient both with respect to signal sensing as well as reconstruction complexity. Known constructions use the adjacency matrices of expander graphs, which often lead to recovery algorithms which are much more efficient than minimization. However, prior constructions of sparse measurement matrices rely on expander graphs with very high expansion coefficients which make the construction of such graphs difficult and the size of the recoverable sets very small. In this paper, we introduce sparse measurement matrices for the recovery of nonnegative vectors, using perturbations of the adjacency matrices of expander graphs requiring much smaller expansion coefficients, hereby referred to as minimal expanders. We show that when minimization is used as the reconstruction method, these constructions allow the recovery of signals that are almost three orders of magnitude larger compared to the existing theoretical results for sparse measurement matrices. We provide for the first time tight upper bounds for the so called weak and strong recovery thresholds when minimization is used. We further show that the success of optimization is equivalent to the existence of a “unique” vector in the set of solutions to the linear equations, which enables alternative algorithms for minimization. We further show that the defined minimal expansion property is necessary for all measurement matrices for compressive sensing, (even when the non-negativity assumption is removed) therefore implying that our construction is tight. We finally present a novel recovery algorithm that exploits expansion and is much more computationally efficient compared to minimization. We investigate the sparse recovery problem of reconstructing a high-dimensional non-negative sparse vector from lower dimensional linear measurements. While much work has focused on dense measurement matrices, sparse measurement schemes are crucial in applications, such as DNA microarrays and sensor networks, where dense measurements are not practically feasible. One possible construction uses the adjacency matrices of expander graphs, which often leads to recovery algorithms much more efficient than @math minimization. However, to date, constructions based on expanders have required very high expansion coefficients which can potentially make the construction of such graphs difficult and the size of the recoverable sets small. In this paper, we construct sparse measurement matrices for the recovery of non-negative vectors, using perturbations of the adjacency matrix of an expander graph with much smaller expansion coefficient. We present a necessary and sufficient condition for @math optimization to successfully recover the unknown vector and obtain expressions for the recovery threshold. For certain classes of measurement matrices, this necessary and sufficient condition is further equivalent to the existence of a "unique" vector in the constraint set, which opens the door to alternative algorithms to @math minimization. We further show that the minimal expansion we use is necessary for any graph for which sparse recovery is possible and that therefore our construction is tight. We finally present a novel recovery algorithm that exploits expansion and is much faster than @math optimization. Finally, we demonstrate through theoretical bounds, as well as simulation, that our method is robust to noise and approximate sparsity. Compressed sensing seeks to recover a sparse vector from a small number of linear and non-adaptive measurements. While most work so far focuses on Gaussian or Bernoulli random measurements we investigate the use of partial random circulant and Toeplitz matrices in connection with recovery by 1-minization. In contrast to recent work in this direction we allow the use of an arbitrary subset of rows of a circulant and Toeplitz matrix. Our recovery result predicts that the necessary number of measurements to ensure sparse reconstruction by 1-minimization with random partial circulant or Toeplitz matrices scales linearly in the sparsity up to a log-factor in the ambient dimension. This represents a significant improvement over previous recovery results for such matrices. As a main tool for the proofs we use a new version of the non-commutative Khintchine inequality. The task of compressed sensing is to recover a sparse vector from a small number of linear and non-adaptive measurements, and the problem of finding a suitable measurement matrix is very important in this field. While most recent works focused on random matrices with entries drawn independently from certain probability distributions, in this paper we show that a partial random symmetric Bernoulli matrix whose entries are not independent, can be used to recover signal from observations successfully with high probability. The experimental results also show that the proposed matrix is a suitable measurement matrix.