Abstract. The linear inverse source and scattering problems are studied from the perspective of compressed sensing. By introducing the sensor as well as target ensembles, the maximum number of recoverable targets is proved to be at least proportional to the number of measurement data modulo a log-square factor with overwhelming probability. Important contributions include the discoveries of the threshold aperture, consistent with the classical Rayleigh criterion, and the incoherence effect induced by random antenna locations. The prediction of theorems are confirmed by numerical simulations. 1.

Identification of defective members of large populations has been widely studied in the statistics community under the name of group testing. It involves grouping subsets of items into different pools and detecting defective members based on the set of test results obtained for each pool. In a classical noiseless group testing setup, it is assumed that the sampling procedure is fully known to the reconstruction algorithm, in the sense that the existence of a defective member in a pool results in the test outcome of that pool to be positive. However, this may not be always a valid assumption in some cases of interest. In particular, we consider the case where the defective items in a pool can become independently inactive with a certain probability. Hence, one may obtain a negative test result in a pool despite containing some defective items. As a result, any sampling and reconstruction method should be able to cope with two different types of uncertainty, i.e., the unknown set of defective items and the partially unknown, probabilistic testing procedure. In this work, motivated by the application of detecting infected people in viral epidemics, we design non-adaptive sampling procedures that allow successful identification of the defective items through a set of probabilistic tests. Our design requires only a small number of tests to single out the defective items.

Abstract. Applications of compressed sensing motivate the possibility of using different operators to encode and decode a signal of interest. Since it is clear that the operators cannot be too different, we can view the discrepancy between the two matrices as a perturbation. The stability of ℓ1-minimization and greedy algorithms to recover the signal in the presence of additive noise is by now well-known. Recently however, work has been done to analyze these methods with noise in the measurement matrix, which generates a multiplicative noise term. This new framework of generalized perturbations (i.e., both additive and multiplicative noise) extends the prior work on stable signal recovery from incomplete and inaccurate measurements of Candès, Romberg and Tao using Basis Pursuit (BP), and of Needell and Tropp using Compressive Sampling Matching Pursuit (CoSaMP). We show, under reasonable assumptions, that the stability of the reconstructed signal by both BP and CoSaMP is limited by the noise level in the observation. Our analysis extends easily to arbitrary greedy methods. 1.

Abstract—Over the past three decades we have steadily increased our knowledge on the genetic basis of many severe disorders. Nevertheless, there are still great challenges in applying this knowledge routinely in the clinic, mainly due to the relatively tedious and expensive process of genotyping. Since the genetic variations that underlie the disorders are relatively rare in the population, they can be thought of as a sparse signal. Using methods and ideas from compressed sensing and group testing, we have developed a cost-effective genotyping protocol to detect carriers for severe genetic disorders. In particular, we have adapted our scheme to a recently developed class of high throughput DNA sequencing technologies. The mathematical framework presented here has some important distinctions from the ’traditional ’ compressed sensing and group testing frameworks in order to address biological and technical constraints of our setting. Index Terms—compressed sensing, DNA, genotyping, group testing

Abstract. The compressed sensing problem for redundant dictionaries aims to use a small number of linear measurements to represent signals that are sparse with respect to a general dictionary. Under an appropriate restricted isometry property for a dictionary, reconstruction methods based on ℓ q minimization are known to provide an effective signal recovery tool in this setting. This note explores conditions under which ℓ q minimization is robust to measurement noise, and stable with respect to perturbations of the sensing matrix A and the dictionary D. We propose a new condition, the D null space property, which guarantees that ℓ q minimization produces solutions that are robust and stable against perturbations of A and D. We also show that ℓ q minimization is jointly stable with respect to imprecise knowledge of the measurement matrix A and the dictionary D when A satisfies the restricted isometry property. 1.

We consider the problem of calibrating a compressed sensing measurement system under the assumption that the decalibration consists in unknown gains on each measure. We focus on blind calibration, using measures performed on a few unknown (but sparse) signals. A naive formulation of this blind calibration problem, using ℓ1 minimization, is reminiscent of blind source separation and dictionary learning, which are known to be highly non-convex and riddled with local minima. In the considered context, we show that in fact this formulation can be exactly expressed as a convex optimization problem, and can be solved using off-the-shelf algorithms. Numerical simulations demonstrate the effectiveness of the approach even for highly uncalibrated measures, when a sufficient number of (unknown, but sparse) calibrating signals is provided. We observe that the success/failure of the approach seems to obey sharp phase transitions. Index Terms — compressed sensing; calibration; dictionary learning; blind signal separation; sparse recovery. 1.

In additive component model estimation problems, the number of additive components (model order) and values of the model parameters in each of the additive components are estimated. Traditional methods typically estimate parameters for a set of models with fixed order; parameter estimation is performed over a continuous space when parameters are not discrete. The model order is estimated as the minimizer, over the set of fixed model orders, of a cost function compromising between signal fit to measurements and model complexity. This dissertation explores dictionary-based estimation methods for joint model order and parameter estimation. In dictionary estimation, the continuous parameter space is discretized, forming a dictionary. Each column of the dictionary is a model component at a sampled parameter value, and a linear combination of a subset of columns is used to represent the model. It is assumed that the model consists of a small number of components, and a sparse reconstruction algorithm is used to select a sparse superposition of columns to represent the signal. The number of columns selected is the estimated model order, and the parameters of each column are the

Abstract—In this work, we consider a general form of noisy compressive sensing (CS) when there is uncertainty in the measurement matrix as well as in the measurements. Matrix uncertainty is motivated by practical cases in which there are imperfections or unknown calibration parameters in the signal acquisition hardware. While previous work has focused on analyzing and extending classical CS algorithms like the LASSO and Dantzig selector for this problem setting, we propose a new algorithm whose goal is either minimization of mean-squared error or maximization of posterior probability in the presence of these uncertainties. In particular, we extend the Approximate Message Passing (AMP) approach originally proposed by Donoho, Maleki, and Montanari, and recently generalized by Rangan, to the case of probabilistic uncertainties in the elements of the measurement matrix. Empirically, we show that our approach performs near oracle bounds. We then show that our matrix-uncertain AMP can be applied in an alternating fashion to learn both the unknown measurement matrix and signal vector. We also present a simple analysis showing that, for suitably large systems, it suffices to treat uniform matrix uncertainty as additive white Gaussian noise. I.

underdetermined linear regression y = Ax + w