The performance of Orthogonal Matching Pursuit (OMP) for variable selection is analyzed for random designs. When contrasted with the deterministic case, since the performance is here measured after averaging over the distribution of the design matrix, one can have far less stringent sparsity constraints on the coefficient vector. We demonstrate that for exact sparse vectors, the performance of the OMP is similar to known results on the Lasso algorithm [ (2009) 2183--2202]. Moreover, variable selection under a more relaxed sparsity assumption on the coefficient vector, whereby one has only control on the @math norm of the smaller coefficients, is also analyzed. As a consequence of these results, we also show that the coefficient estimate satisfies strong oracle type inequalities.
A common sufficient condition on @math for this type of recovery is the , which requires that the the inner product between distinct columns be small. In particular, letting @math , for all @math , it is assumed that (X) = 1 n j j' |X_j^ X_ j' | is @math . Another related criterion is the @cite_14 , @cite_23 , which assumes, for all subset @math of size @math , that |(X_ T ^ X_ T )^ -1 X_ T ^ X_j |_1 < 1, for all j J - T . Here @math denotes the @math norm.
@cite_14 @cite_23
1605194072 2963583445 2168831133 2536849891
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. 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. The goal of (stable) sparse recovery is to recover a @math -sparse approximation @math of a vector @math from linear measurements of @math . Specifically, the goal is to recover @math such that @math for some constant @math and norm parameters @math and @math . It is known that, for @math or @math , this task can be accomplished using @math non-adaptive measurements CRT06:Stable-Signal and that this bound is tight DIPW, FPRU, PW11 . In this paper we show that if one is allowed to perform measurements that are adaptive , then the number of measurements can be considerably reduced. Specifically, for @math and @math we show A scheme with @math measurements that uses @math rounds. This is a significant improvement over the best possible non-adaptive bound. A scheme with @math measurements that uses two rounds. This improves over the best possible non-adaptive bound. To the best of our knowledge, these are the first results of this type. 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.