We are motivated by problems that arise in a number of applications such as Online Marketing and Explosives detection, where the observations are usually modeled using Poisson statistics. We model each observation as a Poisson random variable whose mean is a sparse linear superposition of known patterns. Unlike many conventional problems observations here are not identically distributed since they are associated with different sensing modalities. We analyze the performance of a maximum likelihood (ML) decoder, which for our Poisson setting involves a non-linear optimization but yet is computationally tractable. We derive fundamental sample complexity bounds for sparse recovery when the measurements are contaminated with Poisson noise. In contrast to the least-squares linear regression setting with Gaussian noise, we observe that in addition to sparsity, the scale of the parameters also fundamentally impacts @math error in the Poisson setting. We show that our upper bounds are tight under suitable regularity conditions. Specifically, we derive a minimax matching lower bound on the mean-squared error and show that our constrained ML decoder is minimax optimal for this regime.
We can view model I as an instance of a general class of sparse recovery problems. Indeed, @cite_9 studies the convergence behavior of @math regularized ML estimation for exponential family distributions and GLM in this context. The bounds on error for sparse recovery of the parameter are based on the RE condition. Moreover, in order to get useful bounds on estimation error of GLM, they additionally need the natural sufficient statistic of the exponential family to be sub-gaussian. This condition could clearly be violated in our setting where the data is Poisson distributed and there is no constraint on the sensing matrix to be sub-gaussian.
@cite_9
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The problem of sparse signal recovery from a relatively small number of noisy measurements has been studied extensively in the recent literature on compressed sensing. However, the focus of those studies appears to be limited to the case of linear projections disturbed by Gaussian noise, and the sparse signal reconstruction problem is treated as linear regression with l 1 -norm regularization constraint. A natural question to ask is whether one can accurately recover sparse signals under different noise assumptions. Herein, we extend the results of [13] to the more general case of exponential-family noise that includes Gaussian noise as a particular case, and yields l 1 -regularized Generalized Linear Model (GLM) regression problem. We show that, under standard restricted isometry property (RIP) assumptions on the design matrix, l 1 -minimization can provide stable recovery of a sparse signal in presence of the exponential-family noise, provided that certain sufficient conditions on the noise distribution are satisfied. This paper revisits the sparse multiple measurement vector (MMV) problem, where the aim is to recover a set of jointly sparse multichannel vectors from incomplete measurements. This problem is an extension of single channel sparse recovery, which lies at the heart of compressed sensing. Inspired by the links to array signal processing, a new family of MMV algorithms is considered that highlight the role of rank in determining the difficulty of the MMV recovery problem. The simplest such method is a discrete version of MUSIC which is guaranteed to recover the sparse vectors in the full rank MMV setting, under mild conditions. This idea is extended to a rank aware pursuit algorithm that naturally reduces to Order Recursive Matching Pursuit (ORMP) in the single measurement case while also providing guaranteed recovery in the full rank setting. In contrast, popular MMV methods such as Simultaneous Orthogonal Matching Pursuit (SOMP) and mixed norm minimization techniques are shown to be rank blind in terms of worst case analysis. Numerical simulations demonstrate that the rank aware techniques are significantly better than existing methods in dealing with multiple measurements. We investigate the problem of reconstructing a high-dimensional nonnegative sparse vector from lower-dimensional linear measurements. While much work has focused on dense measurement matrices, sparse measurement schemes can be more efficient both with respect to signal sensing as well as reconstruction complexity. Known constructions use the adjacency matrices of expander graphs, which often lead to recovery algorithms which are much more efficient than minimization. However, prior constructions of sparse measurement matrices rely on expander graphs with very high expansion coefficients which make the construction of such graphs difficult and the size of the recoverable sets very small. In this paper, we introduce sparse measurement matrices for the recovery of nonnegative vectors, using perturbations of the adjacency matrices of expander graphs requiring much smaller expansion coefficients, hereby referred to as minimal expanders. We show that when minimization is used as the reconstruction method, these constructions allow the recovery of signals that are almost three orders of magnitude larger compared to the existing theoretical results for sparse measurement matrices. We provide for the first time tight upper bounds for the so called weak and strong recovery thresholds when minimization is used. We further show that the success of optimization is equivalent to the existence of a “unique” vector in the set of solutions to the linear equations, which enables alternative algorithms for minimization. We further show that the defined minimal expansion property is necessary for all measurement matrices for compressive sensing, (even when the non-negativity assumption is removed) therefore implying that our construction is tight. We finally present a novel recovery algorithm that exploits expansion and is much more computationally efficient compared to minimization. The use of M-estimators in generalized linear regression models in high dimensional settings requires risk minimization with hard L0 constraints. Of the known methods, the class of projected gradient descent (also known as iterative hard thresholding (IHT)) methods is known to offer the fastest and most scalable solutions. However, the current state-of-the-art is only able to analyze these methods in extremely restrictive settings which do not hold in high dimensional statistical models. In this work we bridge this gap by providing the first analysis for IHT-style methods in the high dimensional statistical setting. Our bounds are tight and match known minimax lower bounds. Our results rely on a general analysis framework that enables us to analyze several popular hard thresholding style algorithms (such as HTP, CoSaMP, SP) in the high dimensional regression setting. Finally, we extend our analysis to the problem of low-rank matrix recovery.