The problem of reconstructing signals and images from degraded ones is considered in this paper. The latter problem is formulated as a linear system whose coefficient matrix models the unknown point spread function and the right hand side represents the observed image. Moreover, the coefficient matrix is very ill-conditioned, requiring an additional regularization term. Different boundary condi...

Standard regularization methods that are used to compute solutions to ill-posed inverse problems require knowledge of the forward model. In many real-life applications, the forward model is not known, but training data is readily available. In this paper, we develop a new framework that uses training data, as a substitute for knowledge of the forward model, to compute an optimal low-rank regula...

This work characterizes the generalization ability of algorithms whose predictions are linear in the input vector. To this end, we provide sharp bounds for Rademacher and Gaussian complexities of (constrained) linear classes, which directly lead to a number of generalization bounds. This derivation provides simplified proofs of a number of corollaries including: risk bounds for linear predictio...

We propose a recursive algorithm for estimating time-varying signals from a few linear measurements. The signals are assumed sparse, with unknown support, and are described by a dynamical model. In each iteration, the algorithm solves an l1-l1 minimization problem and estimates the number of measurements that it has to take at the next iteration. These estimates are computed based on recent the...

We consider a random matrix whose entries are independent Gaussian variables taking values in the field of quaternions with variance 1/n. Using logarithmic potential theory, we prove the almost sure convergence, as the dimension n goes to infinity, of the empirical distribution of the right eigenvalues towards some measure supported on the unit ball of the quaternions field. Some comments on mo...

Abstract. We consider the problem of estimating an unknown vector θ from the noisy data Y = Aθ + ǫ, where A is a known m × n matrix and ǫ is a white Gaussian noise. It is assumed that n is large and A is ill-posed. Therefore in order to estimate θ, a spectral regularization method is used and our goal is to choose a spectral regularization parameter with the help of the data Y . We study data-d...

Experts classifying data are often imprecise. Recently, several models have been proposed to train classifiers using the noisy labels generated by these experts. How to choose between these models? In such situations, the true labels are unavailable. Thus, one cannot perform model selection using the standard versions of methods such as empirical risk minimization and cross validation. In order...

In this paper, we study the stability and generalization properties of penalized empirical-risk minimization algorithms. We propose a set of properties of the penalty term that is sufficient to ensure uniform β-stability: we show that if the penalty function satisfies a suitable convexity property, then the induced regularization algorithm is uniformly β-stable. In particular, our results imply...

We consider differentially private algorithms for convex empirical risk minimization (ERM). Differential privacy (Dwork et al., 2006b) is a recently introduced notion of privacy which guarantees that an algorithm’s output does not depend on the data of any individual in the dataset. This is crucial in fields that handle sensitive data, such as genomics, collaborative filtering, and economics. O...

Data warehouses (DWs) can become inconsistent when some dimensional constraints are not satisfied by the dimension instances. In this paper, we present preliminary results about the effects of the violation of partitioning constraints in homogeneous dimension instances over aggregation queries, and in particular over the summarizability property (SUMM) of the DWs. We are interested in finding w...