Regularization is commonly used in classifier design, to assure good generalization. Classical regularization enforces a cost on classifier complexity, by constraining parameters. This is usually combined with a margin loss, which favors large-margin decision rules. A novel and unified view of this architecture is proposed, by showing that margin losses act as regularizers of posterior class pr...

We introduce and study a mathematical framework for broad class of regularization functionals ill-posed inverse problems: Regularization Graphs. graphs allow to construct using as building blocks linear operators convex functionals, assembled by means that can be seen generalizations classical infimal convolution operators. This exhaustively covers existing approaches it is flexible enough craf...

Dropout is a simple but effective technique for learning in neural networks and other settings. A sound theoretical understanding of dropout is needed to determine when dropout should be applied and how to use it most effectively. In this paper we continue the exploration of dropout as a regularizer pioneered by Wager et al. We focus on linear classification where a convex proxy to the misclass...

Manifold regularization is an approach which exploits the geometry of the marginal distribution. The main goal of this paper is to analyze the convergence issues of such regularization algorithms in learning theory. We propose a more general multi-penalty framework and establish the optimal convergence rates under the general smoothness assumption. We study a theoretical analysis of the perform...

Regularization approaches based on spectral filtering can be highly effective in solving ill-posed inverse problems. These methods, however, require computing the singular value decomposition (SVD) and choosing appropriate regularization parameters. These tasks can be prohibitively expensive for large-scale problems. In this paper, we present a framework that uses operator approximations to eff...

We examine several zero-range potentials in non-relativistic quantum mechanics. The study of such potentials requires regularization and renormalization. We contrast physical results obtained using dimensional regularization and cutoff schemes and show explicitly that in certain cases dimensional regularization fails to reproduce the results obtained using cutoff regularization. First we consid...

The area of mathematical inverse problems is quite broad and involves the qualitative and quantitative analysis of a wide variety of physical models. Applications include, for example, the problem of inverse heat conduction, image reconstruction, tomography, the inverse scattering problem, and the determination of unknown coefficients or boundary parameters appearing in partial differential equ...

Instead of the Tikhonov regularization method which with a scalar being the regularization parameter, Liu et al. [1] have proposed a novel regularization method with a vector as being the regularization parameter. As a continuation we further propose an optimally scaled vector regularization method (OSVRM) to solve the ill-posed linear problems, which is better than the Tikhonov regularization ...

The effectiveness of regularization to improve SNR in parallel imaging techniques has been reported in previous works [1-2], but how to optimize the regularization parameter remains a problem. The regularization parameter controls the degree of regularization and thereby determines the compromise between SNR and artifacts. Over-regularization causes high level of artifact, while under-regulariz...