We consider the problem of recovering a signal observed in Gaussian noise. If the set of signals is convex and compact, and can be specified beforehand, one can use classical linear estimators that achieve a risk within a constant factor of the minimax risk. However, when the set is unspecified, designing an estimator that is blind to the hidden structure of the signal remains a challenging problem. We propose a new family of estimators to recover signals observed in Gaussian noise. Instead of specifying the set where the signal lives, we assume the existence of a well-performing linear estimator. Proposed estimators enjoy exact oracle inequalities and can be efficiently computed through convex optimization. We present several numerical illustrations that show the potential of the approach.

We present a theoretical framework for adaptive estimation and prediction of signals of unknown structure in the presence of noise. The framework allows to address two intertwined challenges: (i) designing optimal statistical estimators; (ii) designing efficient numerical algorithms. In particular, we establish oracle inequalities for the performance of adaptive procedures, which rely upon convex optimization and thus can be efficiently implemented. As an application of the proposed approach, we consider denoising of harmonic oscillations.

We discuss new methods for the recovery of signals with block-sparse structure, based on l1-minimization. Our emphasis is on the efficiently computable error bounds for the recovery routines. We optimize these bounds with respect to the method parameters to construct the estimators with improved statistical properties. We justify the proposed approach with an oracle inequality which links the properties of the recovery algorithms and the best estimation performance.

We discuss two new methods of recovery of sparse signals from noisy observation based onℓ1-minimization. While they are closely related to the well-known techniques such as Lasso and Dantzig Selector, these estimators come with efficiently verifiable guaranties of performance. By optimizing these bounds with respect to the method parameters we are able to construct the estimators which possess better statistical properties than the commonly used ones. We link our performance estimations to the well known results of Compressive Sensing and justify our proposed approach with an oracle inequality which links the properties of the recovery algorithms and the best estimation performance when the signal support is known. We also show how the estimates can be computed using the Non-Euclidean Basis Pursuit algorithm.

We develop a new linear estimator for estimating an unknown vector x in a linear model, in the presence of bounded data uncertainties. The estimator is designed to minimize the worst-case regret across all bounded data vectors, namely the worst-case difference between the MSE attainable using a linear estimator that does not know the true parameters x, and the optimal MSE attained using a linear estimator that knows x. We demonstrate through several examples that the minimax regret estimator can significantly increase the performance over the conventional least-squares estimator, as well as several other least-squares alternatives.

In previous work, Elad and Bruckstein (EB) have provided a sufficient condition for replacing an l/sub 0/ optimization by linear programming minimization when searching for the unique sparse representation. We establish here that the EB condition is both sufficient and necessary.

In this paper,we present coincidence-list-ordered sets expectation-maximization (COSEM), an algorithm for iterative image reconstruction directly from list-mode coincidence acquisition data. The COSEM algorithm is based on the ordered sets EM algorithm for binned data but has several extensions that makes it suitable for rotating two planar detector tomographs. We develop the COSEM algorithm and extend it to include analytic calculation of detection probability, noise reducing iterative filtering schemes, and on-the-fly attenuation correction methods. We present an adaptation of COSEM to the Varicam\VG camera and show results from clinical and phantom studies.

The paper is devoted to spatial adaptive estimation of signals satisfying linear differential inequalities with an unknown differential operator of a given order. The classes of signals under consideration cover a wide variety of classes common to nonparametric regression. In particular, they contain the signals whose parameters of smoothness are not uniformly bounded, even locally. We develop an estimator which is optimal in order over a wide range of the classes and "discrete" global accuracy measures.