In this paper, we analyze the minimization of seminorms ∥L · ∥ on R^n under the constraint of a bounded I-divergence D(b,H·) for rather general linear operators H and L. The I-divergence is also known as Kullback–Leibler divergence and appears in many models in imaging science, in particular when dealing with Poisson data but also in the case of multiplicative Gamma noise. Often H represents, e.g., a linear blur operator and L is some discrete derivative or frame analysis operator. A central part of this paper consists in proving relations between the parameters of I-divergence constrained and penalized problems. To solve the I-divergence constrained problem, we consider various first-order primal–dual algorithms which reduce the problem to the solution of certain proximal minimization problems in each iteration step. One of these proximation problems is an I-divergence constrained least-squares problem which can be solved based on Morozov’s discrepancy principle by a Newton method. We prove that these algorithms produce not only a sequence of vectors which converges to a minimizer of the constrained problem but also a sequence of parameters which converges to a regularization parameter so that the corresponding penalized problem has the same solution. Furthermore, we derive a rule for automatically setting the constraint parameter for data corrupted by multiplicative Gamma noise. The performance of the various algorithms is finally demonstrated for different image restoration tasks both for images corrupted by Poisson noise and multiplicative Gamma noise.

Many statistical models seek relationship between variables via subspaces of reduced dimensions. For instance, in factor models, variables are roughly distributed around a low dimensional subspace determined by the loading matrix; in mixed linear regression models, the coefficient vectors for different mixtures form a subspace that captures all regression functions; in multiple index models, the effect of covariates is summarized by the effective dimension reduction space.

Blocking is commonly used in randomized experiments to increase efficiency of estimation. A generalization of blocking removes allocations with imbalance in covariate distributions between treated and control units, and then randomizes within the remaining set of allocations with balance. This idea of rerandomization was formalized by Morgan and Rubin (Annals of Statistics, 2012, 40, 1263–1282), who suggested using Mahalanobis distance between treated and control covariate means as the criterion for removing unbalanced allocations. Kallus (Journal of the Royal Statistical Society, Series B: Statistical Methodology, 2018, 80, 85–112) proposed reducing the set of balanced allocations to the minimum. Here we discuss the implication of such an ‘optimal’ rerandomization design for inferences to the units in the sample and to the population from which the units in the sample were randomly drawn. We argue that, in general, it is a bad idea to seek the optimal design for an inference because that inference typically only reflects uncertainty from the random sampling of units, which is usually hypothetical, and not the randomization of units to treatment versus control.

This article is concerned with screening features in ultrahigh-dimensional data analysis, which has become increasingly important in diverse scientific fields. We develop a sure independence screening procedure based on the distance correlation (DC-SIS). The DC-SIS can be implemented as easily as the sure independence screening (SIS) procedure based on the Pearson correlation proposed by Fan and Lv. However, the DC-SIS can significantly improve the SIS. Fan and Lv established the sure screening property for the SIS based on linear models, but the sure screening property is valid for the DC-SIS under more general settings, including linear models. Furthermore, the implementation of the DC-SIS does not require model specification (e.g., linear model or generalized linear model) for responses or predictors. This is a very appealing property in ultrahigh-dimensional data analysis. Moreover, the DC-SIS can be used directly to screen grouped predictor variables and multivariate response variables. We establish the sure screening property for the DC-SIS, and conduct simulations to examine its finite sample performance. A numerical comparison indicates that the DC-SIS performs much better than the SIS in various models. We also illustrate the DC-SIS through a real-data example.

No abstract uploaded!