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.
More recently, in the context of @math and noise-shaping quantization, @cite_44 has extended initial works restricted to the use of Gaussian random matrices @cite_27 by proving that if @math is either a bounded orthogonal ensemble ( a random partial Fourier matrices) or a partial circulant ensemble (also known as subsampled circulant matrix), a convex program proposed in the paper achieves a uniform reconstruction of any @math with an error decaying polynomially fast in @math for the @math quantization, or exponentially fast in @math for the noise-shaping quantization. However, the analysis of @cite_44 is limited for the estimation of sparse signals, although they characterize binary embeddings of finite and general low-complexity sets from the same quantization schemes.
@cite_44 @cite_27
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This paper deals with two related problems, namely distance-preserving binary embeddings and quantization for compressed sensing . First, we propose fast methods to replace points from a subset @math , associated with the Euclidean metric, with points in the cube @math and we associate the cube with a pseudo-metric that approximates Euclidean distance among points in @math . Our methods rely on quantizing fast Johnson-Lindenstrauss embeddings based on bounded orthonormal systems and partial circulant ensembles, both of which admit fast transforms. Our quantization methods utilize noise-shaping, and include Sigma-Delta schemes and distributed noise-shaping schemes. The resulting approximation errors decay polynomially and exponentially fast in @math , depending on the embedding method. This dramatically outperforms the current decay rates associated with binary embeddings and Hamming distances. Additionally, it is the first such binary embedding result that applies to fast Johnson-Lindenstrauss maps while preserving @math norms. Second, we again consider noise-shaping schemes, albeit this time to quantize compressed sensing measurements arising from bounded orthonormal ensembles and partial circulant matrices. We show that these methods yield a reconstruction error that again decays with the number of measurements (and bits), when using convex optimization for reconstruction. Specifically, for Sigma-Delta schemes, the error decays polynomially in the number of measurements, and it decays exponentially for distributed noise-shaping schemes based on beta encoding. These results are near optimal and the first of their kind dealing with bounded orthonormal systems. Subsampled (or partial) Fourier matrices were originally introduced in the compressive sensing literature by Later, in papers by Candes and Tao and Rudelson and Vershynin, it was shown that (random) subsampling of the rows of many other classes of unitary matrices also yield effective sensing matrices. The key requirement is that the rows of U, the unitary matrix, must be highly incoherent with the basis in which the signal is sparse. In this paper, we consider acquisition systems that — despite sensing sparse signals in an incoherent domain — cannot randomly subsample rows from U. We consider a general class of systems in which the sensing matrix corresponds to subsampling of the rows of matrices of the form Φ = RU (instead of U), where R is typically a low-rank matrix whose structure reflects the physical technological constraints of the acquisition system. We use the term “structurally-subsampled unitary matrices” to describe such sensing matrices. We investigate the restricted isometry property of a particular class of structurally-subsampled unitary matrices that arise naturally in application areas such as multiple-antenna channel estimation and sub-nyquist sampling. In addition, we discuss an immediate application of this work in the area of wireless channel estimation, where the main results of this paper can be applied to the estimation of multiple-antenna orthogonal frequency division multiplexing channels that have sparse impulse responses. This paper provides novel results for the recovery of signals from undersampled measurements based on analysis @math -minimization, when the analysis operator is given by a frame. We both provide so-called uniform and nonuniform recovery guarantees for cosparse (analysis-sparse) signals using Gaussian random measurement matrices. The nonuniform result relies on a recovery condition via tangent cones and the uniform recovery guarantee is based on an analysis version of the null space property. Examining these conditions for Gaussian random matrices leads to precise bounds on the number of measurements required for successful recovery. In the special case of standard sparsity, our result improves a bound due to Rudelson and Vershynin concerning the exact reconstruction of sparse signals from Gaussian measurements with respect to the constant and extends it to stability under passing to approximately sparse signals and to robustness under noise on the measurements. Consider the problem of recovering an unknown signal from undersampled measurements, given the knowledge that the signal has a sparse representation in a specified dictionary @math . This problem is now understood to be well-posed and efficiently solvable under suitable assumptions on the measurements and dictionary if the number of measurements scales roughly with the sparsity level. One sufficient condition for such is the @math -restricted isometry property ( @math -RIP), which asks that the sampling matrix approximately preserve the norm of all signals which are sufficiently sparse in @math . While many classes of random matrices are known to satisfy such conditions, such matrices are not representative of the structural constraints imposed by practical sensing systems. We close this gap in the theory by demonstrating that one can subsample a fixed orthogonal matrix in such a way that the @math -RIP will hold, provided this basis is sufficiently incoherent with the sparsifying dictionary $ ...