We are motivated by problems that arise in a number of applications such as explosives detection and online Marketing, where the observations are governed by Poisson statistics. Here each observation is 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 analyse the performance of a Maximum Likelihood (ML) decoder, which for our Poisson setting is computationally tractable. We derive fundamental sample complexity bounds for sparse recovery in the high-dimensional setting. We show that when the sensing matrix satisfies the so-called Restricted Eigenvalue (RE) condition the l1 regularized ML decoder is consistent. Moreover, it converges exponentially fast in terms of number of observations. Our results apply to both deterministic and random sensing matrices and we present several results for both cases.
@cite_0: High-dimensional statistical inference deals with models in which the the number of parameters p is comparable to or larger than the sample size n. Since it is usually impossible to obtain consistent procedures unless p n → 0, a line of recent work has studied models with various types of structure (e.g., sparse vectors; block-structured matrices; low-rank matrices; Markov assumptions). In such settings, a general approach to estimation is to solve a regularized convex program (known as a regularized M-estimator) which combines a loss function (measuring how well the model fits the data) with some regularization function that encourages the assumed structure. The goal of this paper is to provide a unified framework for establishing consistency and convergence rates for such regularized M-estimators under high-dimensional scaling. We state one main theorem and show how it can be used to re-derive several existing results, and also to obtain several new results on consistency and convergence rates. Our analysis also identifies two key properties of loss and regularization functions, referred to as restricted strong convexity and decomposability, that ensure the corresponding regularized M-estimators have fast convergence rates.
More generally @cite_0 describes a unified framework for analysis of regularized @math estimators in high dimensions. They also mention extension of their framework to GLMs and describe strong convexity" of the objective function as a sufficient condition to obtain consistency of M-estimators under Model I. As we described this requirement of strong-convexity is not consistent with our model. In addition the statistical aspects in that work requires that the components of the sensing matrix be characterized by , which we do not require here.