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.
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 on @math as a sufficient condition to obtain consistency of M-estimators under Model I. As described earlier, this requirement of strong convexity on @math 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.
@cite_0
1874232560 2151825876 2950190315 1549918636
Many statistical M-estimators are based on convex optimization problems formed by the weighted sum of a loss function with a norm-based regularizes We analyze the convergence rates of first-order gradient methods for solving such problems within a high-dimensional framework that allows the data dimension d to grow with (and possibly exceed) the sample size n. This high-dimensional structure precludes the usual global assumptions— namely, strong convexity and smoothness conditions—that underlie classical optimization analysis. We define appropriately restricted versions of these conditions, and show that they are satisfied with high probability for various statistical models. Under these conditions, our theory guarantees that Nesterov's first-order method [12] has a globally geometric rate of convergence up to the statistical precision of the model, meaning the typical Euclidean distance between the true unknown parameter θ* and the optimal solution ^θ. This globally linear rate is substantially faster than previous analyses of global convergence for specific methods that yielded only sublinear rates. Our analysis applies to a wide range of M-estimators and statistical models, including sparse linear regression using Lasso (l1-regularized regression), group Lasso, block sparsity, and low-rank matrix recovery using nuclear norm regularization. Overall, this result reveals an interesting connection between statistical precision and computational efficiency in high-dimensional estimation. 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. 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 @math , a line of recent work has studied models with various types of low-dimensional structure, including sparse vectors, sparse and structured matrices, low-rank matrices and combinations thereof. In such settings, a general approach to estimation is to solve a regularized optimization problem, which combines a loss function measuring how well the model fits the data with some regularization function that encourages the assumed structure. This paper provides 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 some existing results, and also to obtain a number of new results on consistency and convergence rates, in both @math -error and related norms. Our analysis also identifies two key properties of loss and regularization functions, referred to as restricted strong convexity and decomposability, that ensure corresponding regularized M-estimators have fast convergence rates and which are optimal in many well-studied cases. The formulation @math minx,yf(x)+g(y),subjecttoAx+By=b,where f and g are extended-value convex functions, arises in many application areas such as signal processing, imaging and image processing, statistics, and machine learning either naturally or after variable splitting. In many common problems, one of the two objective functions is strictly convex and has Lipschitz continuous gradient. On this kind of problem, a very effective approach is the alternating direction method of multipliers (ADM or ADMM), which solves a sequence of f g-decoupled subproblems. However, its effectiveness has not been matched by a provably fast rate of convergence; only sublinear rates such as O(1 k) and @math O(1 k2) were recently established in the literature, though the O(1 k) rates do not require strong convexity. This paper shows that global linear convergence can be guaranteed under the assumptions of strong convexity and Lipschitz gradient on one of the two functions, along with certain rank assumptions on A and B. The result applies to various generalizations of ADM that allow the subproblems to be solved faster and less exactly in certain manners. The derived rate of convergence also provides some theoretical guidance for optimizing the ADM parameters. In addition, this paper makes meaningful extensions to the existing global convergence theory of ADM generalizations.