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 @math . We also extend this analysis to allow for weighted sparse expansions. Consequently, we arrive at compressive sensing recovery guarantees for structured measurements and redundant dictionaries, opening the door to a wide array of practical applications.
Due to the abundance of relevant applications, a number of works have studied compressive sensing for overcomplete frames. The first work on this topic aimed to recover the coefficient vector @math directly, and thus required strong incoherence assumptions on the dictionary @math @cite_45 . More recently, it was noted that if one instead aims to recover @math rather than @math , recovery guarantees can be obtained under weaker assumptions. Namely, one only needs that the measurement matrix @math respects the norms of signals which are sparse in the dictionary @math . To quantify this, Cand @cite_25 define the @math -restricted isometry property ( @math -RIP in short, see Definition below). For measurement matrices that have this property, a number of algorithms have been shown to guarantee recovery under certain assumptions. Optimization approaches such as @math -analysis @cite_2 @cite_25 @cite_14 @cite_52 @cite_0 @cite_35 @cite_28 and greedy approaches @cite_56 @cite_14 @cite_11 @cite_71 @cite_69 have been studied.
@cite_35 @cite_69 @cite_14 @cite_28 @cite_52 @cite_56 @cite_0 @cite_45 @cite_71 @cite_2 @cite_25 @cite_11
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Consider the recovery of an unknown signal @math from quantized linear measurements. In the one-bit compressive sensing setting, one typically assumes that @math is sparse, and that the measurements are of the form @math . Since such measurements give no information on the norm of @math , recovery methods typically assume that @math . We show that if one allows more generally for quantized affine measurements of the form @math , and if the vectors @math are random, an appropriate choice of the affine shifts @math allows norm recovery to be easily incorporated into existing methods for one-bit compressive sensing. In addition, we show that for arbitrary fixed @math in the annulus @math , one may estimate the norm @math up to additive error @math from @math such binary measurements through a single evaluation of the inverse Gaussian error function. Finally, all of our recovery guarantees can be made universal over sparse vectors in the sense that with high probability, one set of measurements and thresholds can successfully estimate all sparse vectors @math in a Euclidean ball of known radius. Consider the recovery of an unknown signal @math from quantized linear measurements. In the one-bit compressive sensing setting, one typically assumes that @math is sparse, and that the measurements are of the form @math . Since such measurements give no information on the norm of @math , recovery methods from such measurements typically assume that @math . We show that if one allows more generally for quantized affine measurements of the form @math , and if the vectors @math are random, an appropriate choice of the affine shifts @math allows norm recovery to be easily incorporated into existing methods for one-bit compressive sensing. Additionally, we show that for arbitrary fixed @math in the annulus @math , one may estimate the norm @math up to additive error @math from @math such binary measurements through a single evaluation of the inverse Gaussian error function. Finally, all of our recovery guarantees can be made universal over sparse vectors, in the sense that with high probability, one set of measurements and thresholds can successfully estimate all sparse vectors @math within a Euclidean ball of known radius. 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 $ ... This article presents novel results concerning the recovery of signals from undersampled data in the common situation where such signals are not sparse in an orthonormal basis or incoherent dictionary, but in a truly redundant dictionary. This work thus bridges a gap in the literature and shows not only that compressed sensing is viable in this context, but also that accurate recovery is possible via an ‘1-analysis optimization problem. We introduce a condition on the measurement sensing matrix, which is a natural generalization of the now well-known restricted isometry property, and which guarantees accurate recovery of signals that are nearly sparse in (possibly) highly overcomplete and coherent dictionaries. This condition imposes no incoherence restriction on the dictionary and our results may be the rst of this kind. We discuss practical examples and the implications of our results on those applications, and complement our study by demonstrating the potential of ‘1-analysis for such problems.