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
2032291279 2000150201
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. 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 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 $ ... We study the recovery of sparse vectors from subsampled random convolutions via @math -minimization. We consider the setup in which both the subsampling locations as well as the generating vector are chosen at random. For a subgaussian generator with independent entries, we improve previously known estimates: if the sparsity @math is small enough, i.e., @math , we show that @math measurements are sufficient to recover @math -sparse vectors in dimension @math with high probability, matching the well-known condition for recovery from standard Gaussian measurements. If @math is larger, then essentially @math measurements are sufficient, again improving over previous estimates. Our results are shown via the so-called robust null space property which is weaker than the standard restricted isometry property. Our method of proof involves a novel combination of small ball estimates with chaining techniques which should be of independent interest.