In this paper we consider memoryless one-bit compressed sensing with randomly subsampled Gaussian circulant matrices. We show that in a small sparsity regime and for small enough accuracy @math , @math measurements suffice to reconstruct the direction of any @math -sparse vector up to accuracy @math via an efficient program. We derive this result by proving that partial Gaussian circulant matrices satisfy an @math RIP-property. Under a slightly worse dependence on @math , we establish stability with respect to approximate sparsity, as well as full vector recovery results.
Standard compressive sensing with partial circulant matrices. In standard (unquantized) compressive sensing, the task is to recover an (approximately) sparse vector @math from measurements @math , where @math with @math . A number of reconstruction algorithms have been introduced, most notably @math -minimization which computes the minimizer of [ z ^N |z |_1 subject to A z = Ax. ] The ( @math -)restricted isometry property is a classical way of analyzing the performance of compressive sensing @cite_9 . The restricted isometry constant @math is defined as the smallest constant @math such that If @math then all @math -sparse signals can be reconstructed via @math -minimization exactly, see e.g. @cite_17 @cite_9 . Stability under noise and sparsity defect can be shown as well and similar guarantees also hold for other reconstruction algorithms @cite_9 . It is well-known that Gaussian random matrices satisfy @math with probability at least @math if @math [Chapter 9] fora13 .
@cite_9 @cite_17
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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. 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.