Orthogonal Matching Pursuit (OMP) is a canonical greedy pursuit algorithm for sparse approximation. Previous studies of OMP have considered the recovery of a sparse signal through Φ and y = Φx + b, where is a matrix with more columns than rows and denotes the measurement noise. In this paper, based on Restricted Isometry Property (RIP), the performance of OMP is analyzed under general perturbations, which means both y and Φ are perturbed. Though the exact recovery of an almost sparse signal x is no longer feasible, the main contribution reveals that the support set of the best k-term approximation of x can be recovered under reasonable conditions. The error bound between x and the estimation of OMP is also derived. By constructing an example it is also demonstrated that the sufficient conditions for support recovery of the best k-term approximation of are rather tight. When x is strong-decaying, it is proved that the sufficient conditions for support recovery of the best k-term approximation of x can be relaxed, and the support can even be recovered in the order of the entries' magnitude. Our results are also compared in detail with some related previous ones.
Second, consider the sufficient conditions for successful support recovery of the two results. Direct comparison between ) and ) is difficult since as far as we know, there is no clear comparison between @math and @math for arbitrary sensing matrix. For simplicity, consider the scenario that the sensing matrix is Gaussian, and @math , @math , and @math increase in a proportional manner, i.e. @math and @math as @math , where @math are two constants. Results in @cite_36 show that there exists a constant @math such that @math with high probability. Another result in @cite_8 reveals that @math holds with high probability where @math is a constant. Thus @math with high probability. Inequality ) implies that and the following inequality implies ). Since the bound in ) decreases with a higher order than that in ) as @math increases, the sufficient condition ) is more relaxed in this sense.
@cite_36 @cite_8
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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. In sparse principal component analysis we are given noisy observations of a low-rank matrix of dimension @math and seek to reconstruct it under additional sparsity assumptions. In particular, we assume here each of the principal components @math has at most @math non-zero entries. We are particularly interested in the high dimensional regime wherein @math is comparable to, or even much larger than @math . In an influential paper, johnstone2004sparse introduced a simple algorithm that estimates the support of the principal vectors @math by the largest entries in the diagonal of the empirical covariance. This method can be shown to identify the correct support with high probability if @math , and to fail with high probability if @math for two constants @math . Despite a considerable amount of work over the last ten years, no practical algorithm exists with provably better support recovery guarantees. Here we analyze a covariance thresholding algorithm that was recently proposed by KrauthgamerSPCA . On the basis of numerical simulations (for the rank-one case), these authors conjectured that covariance thresholding correctly recover the support with high probability for @math (assuming @math of the same order as @math ). We prove this conjecture, and in fact establish a more general guarantee including higher-rank as well as @math much smaller than @math . Recent lower bounds berthet2013computational, ma2015sum suggest that no polynomial time algorithm can do significantly better. The key technical component of our analysis develops new bounds on the norm of kernel random matrices, in regimes that were not considered before. Consider @math sensors placed randomly and independently with the uniform distribution in a unit square. The sensors have identical sensing range equal to @math , for some @math . We are interested in moving the sensors from their initial positions to new positions so as to ensure that the unit square is completely covered, i.e., every point in the squarei¾?is within the range of a sensor. If the @math -th sensor is displaced a distance @math , what is a displacement of minimum cost? As cost measure for the displacement of the team of sensors we consider thei¾? @math -total movement defined as the sum @math , for some constant @math . We assume that @math and @math are chosen so as to allow full coverage of the square and @math . The main contribution of the paper is to show the existence of a tradeoff between the square sensing radius and @math -total movement and can be summarized as follows:1.If the square sensing radius is equal to @math and @math is the square of a natural number we present an algorithm and show that in expectation the @math -total movement is in @math .2.If the square sensing radius is greater than @math and @math is natural number then we present an algorithm and show that in expectation the @math -total movement is in @math . Therefore this sharp decrease from @math to @math in the @math -total movement of the sensors to attain complete coverage of the square indicates the presence of an interesting threshold on the square sensing radius when it increases from @math to @math . In addition, we simulate our algorithms above and discuss the results of our simulations.