In Compressive Sensing theory and its applications, quantization of signal measurements, as integrated into any realistic sensing model, impacts the quality of signal reconstruction. In fact, there even exist incompatible combinations of quantization functions (e.g., the 1-bit sign function) and sensing matrices (e.g., Bernoulli) that cannot lead to an arbitrarily low reconstruction error when the number of observations increases. This work shows that, for a scalar and uniform quantization, provided that a uniform random vector, or "random dithering", is added to the compressive measurements of a low-complexity signal (e.g., a sparse or compressible signal, or a low-rank matrix) before quantization, a large class of random matrix constructions known to respect the restricted isometry property (RIP) are made "compatible" with this quantizer. This compatibility is demonstrated by the existence of (at least) one signal reconstruction method, the "projected back projection" (PBP), whose reconstruction error is proved to decay when the number of quantized measurements increases. Despite the simplicity of PBP, which amounts to projecting the back projection of the compressive observations (obtained from their multiplication by the adjoint sensing matrix) onto the low-complexity set containing the observed signal, we also prove that given a RIP matrix and for a single realization of the dithering, this reconstruction error decay is also achievable uniformly for the sensing of all signals in the considered low-complexity set. We finally confirm empirically these observations in several sensing contexts involving sparse signals, low-rank matrices, and compressible signals, with various RIP matrix constructions such as sub-Gaussian random matrices and random partial Discrete Cosine Transform (DCT) matrices.
), Kolmogorov entropy (KE, Sec. ), union of low-dimensional subspaces (ULS), low-rank models (LR), second order cone programming (SOCP), Gaussian random matrices (GRM), sub-Gaussian random matrices (SGRM), and subsampled Gaussian circulant matrix (SGCM) @cite_40 . The concept of multiple of @math '' is defined in Sec. . We now provide a comparison of our work with the most relevant literature. This one is connected to 1-bit CS, QCS or other non-linear sensing models, when these are applied componentwise (as for scalar quantization) on compressive sensing measurements, possibly with a random or adaptive pre-quantization dithering for certain studies, with algorithms similar or related to PBP. Most of the works presented below are summarized in Table , reporting there, amongst other aspects, the sensing model, the algorithm, the type of admissible sensing matrices and the low-complexity sets chosen in each of the referenced works.
@cite_40
2130345277 2963583445 2123372478 2160051394
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. Abstract We study the recovery of Hermitian low rank matrices X ∈ C n × n from undersampled measurements via nuclear norm minimization. We consider the particular scenario where the measurements are Frobenius inner products with random rank-one matrices of the form a j a j ⁎ for some measurement vectors a 1 , … , a m , i.e., the measurements are given by b j = tr ( X a j a j ⁎ ) . The case where the matrix X = x x ⁎ to be recovered is of rank one reduces to the problem of phaseless estimation (from measurements b j = | 〈 x , a j 〉 | 2 ) via the PhaseLift approach, which has been introduced recently. We derive bounds for the number m of measurements that guarantee successful uniform recovery of Hermitian rank r matrices, either for the vectors a j , j = 1 , … , m , being chosen independently at random according to a standard Gaussian distribution, or a j being sampled independently from an (approximate) complex projective t-design with t = 4 . In the Gaussian case, we require m ≥ C r n measurements, while in the case of 4-designs we need m ≥ Cr n log ⁡ ( n ) . Our results are uniform in the sense that one random choice of the measurement vectors a j guarantees recovery of all rank r-matrices simultaneously with high probability. Moreover, we prove robustness of recovery under perturbation of the measurements by noise. The result for approximate 4-designs generalizes and improves a recent bound on phase retrieval due to Gross, Krahmer and Kueng. In addition, it has applications in quantum state tomography. Our proofs employ the so-called bowling scheme which is based on recent ideas by Mendelson and Koltchinskii. Hoeffding's U-statistics model combinatorial-type matrix parameters (appearing in CS theory) in a natural way. This paper proposes using these statistics for analyzing random compressed sensing matrices, in the non-asymptotic regime (relevant to practice). The aim is to address certain pessimisms of worst-case restricted isometry analyses, as observed by both Blanchard and Dossal, We show how U-statistics can obtain average-case analyses, by relating to statistical restricted isometry property (StRIP) type recovery guarantees. However unlike standard StRIP, random signal models are not required; the analysis used here holds in the almost sure (probabilistic) sense. For Gaussian bounded entry matrices, we show that both @math -minimization and LASSO essentially require on the order of @math measurements to respectively recover at least @math fraction, and @math fraction, of the signals. Noisy conditions are considered. Empirical evidence suggests our analysis to compare well to Donoho and Tanner's recent large deviation bounds for @math -equivalence, in the regime of block lengths @math with high undersampling ( @math measurements); similar system sizes are found in recent CS implementation. In this work, it is assumed throughout that matrix columns are independently sampled. Consider a Bernoulli-Gaussian complex n-vector whose components are Vi = XiBi, with Xi C N(0, Px) and binary Bi mutually independent and iid across i. This random q-sparse vector is multiplied by a square random matrix U, and a randomly chosen subset, of average size n p, p ∈ [0,1], of the resulting vector components is then observed in additive Gaussian noise. We extend the scope of conventional noisy compressive sampling models where U is typically a matrix with iid components, to allow U satisfying a certain freeness condition. This class of matrices encompasses Haar matrices and other unitarily invariant matrices. We use the replica method and the decoupling principle of Guo and Verdu, as well as a number of information-theoretic bounds, to study the input-output mutual information and the support recovery error rate in the limit of n → ∞. We also extend the scope of the large deviation approach of Rangan and characterize the performance of a class of estimators encompassing thresholded linear MMSE and l1 relaxation.