text: 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.
  @cite_8: We present a new bound for suprema of a special type of chaos process indexed by a set of matrices, which is based on a chaining method. As applications we show significantly improved estimates for the restricted isometry constants of partial random circulant matrices and time-frequency structured random matrices. In both cases the required condition on the number m of rows in terms of the sparsity s and the vector length n is m ≳ s log2 s log2 n. © 2014 Wiley Periodicals, Inc.
  @cite_9: In this expository note, we give a modern proof of Hanson-Wright inequality for quadratic forms in sub-gaussian random variables.We deduce a useful concentration inequality for sub-gaussian random vectors.Two examples are given to illustrate these results: a concentration of distances between random vectors and subspaces, and a bound on the norms of products of random and deterministic matrices.
  @cite_6: We discuss the use of random convolutions for Compressed Sensing applications. In particular, we will show that after convolving anN -dimensional,s-sparse signal with a Rademacher or Steinhaus sequence, it can be recovered via l1-minimization using only m & s log(N=") arbitrary chosen samples with probability at least 1 ".
  @cite_3: This paper demonstrates that convolution with random waveform followed by random time-domain subsampling is a universally efficient compressive sensing strategy. We show that an @math -dimensional signal which is @math -sparse in any fixed orthonormal representation can be recovered from @math samples from its convolution with a pulse whose Fourier transform has unit magnitude and random phase at all frequencies. The time-domain subsampling can be done in one of two ways: in the first, we simply observe @math samples of the random convolution; in the second, we break the random convolution into @math blocks and summarize each with a single randomized sum. We also discuss several imaging applications where convolution with a random pulse allows us to superresolve fine-scale features, allowing us to recover high-resolution signals from low-resolution measurements.
  @cite_16: 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.
  @cite_10: 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.
  @cite_11: 
summary: The situation that @math is a subsampled random circulant matrix (see below for a formal definition) has been analyzed in several contributions @cite_3 @cite_16 @cite_11 @cite_8 @cite_6 @cite_10 . The best available result states @cite_8 that a properly normalized (deterministically) subsampled random circulant matrix (generated by a Gaussian random vector) satisfies @math with probability at least @math if [ m ^ -2 s ( ^2(s) ^2(N) + (1 )). ] The original contribution @cite_3 by Romberg uses random subsampling of a circulant matrix and requires slightly more logarithmic factors, but is able to treat sparsity with respect to an arbitrary basis. In the case of randomly subsampled random convolutions and sparsity with respect to the standard basis, stable and robust @math -sparse recovery via @math -minimization could recently be shown via the null space property @cite_9 in @cite_10 in a small sparsity regime @math under the optimal condition We note that the proof in @cite_10 provides the lower RIP-bound in and may be extended to show the lower @math RIP bound in below under condition .