1,721,055 research outputs found
Optimal Rates for Regularized Least-Squares Algorithm
No full textWe develop a theoretical analysis of the generalization perfor-mances of regularized least-squares algorithm on a reproducing kernel Hilbert space in the supervised learning setting. The presented results hold in the general framework of vector-valued functions, therefore they can be applied to multi-task problems. In particular we observe that the concept of e®ective dimension plays a central role in the de ̄nition of a criterion for the choice of the regularization parameter as a function of the number of samples. Moreover a complete minimax analysis of the problem is described, showing that the convergence rates obtained by regularized least-squares estimators are indeed optimal over a suitable class of priors de ̄ned by the considered kernel. Finallywe give an improved lower rate result describing worst asymptotic behavior on individual probability measures rather than over classes of priors
Geometrical and computational aspects of Spectral Support Estimation for novelty detection
No full textIn this paper we discuss the Spectral Support Estimation algorithm (De Vito et al., 2010) by analyzing its 27 geometrical and computational properties. The estimator is non-parametric and the model selection 28 depends on three parameters whose role is clarified by simulations on a two-dimensional space. The performance of the algorithm for novelty detection is tested and compared with its main competitors on a 30 collection of real benchmark datasets of different sizes and types
Galilei invariant wave equations
No full textWe describe the quantum states of an elementary particle invariant with respect to asymmetry group as solutions of an invariant wave equation. From the mathematical point ofview the wave equation is a set of differential operators on the space of vector-valued dis-tributions over the space-time. As an application, we consider the Galilei invariant particlesboth in 3 + 1 and in 2 + 1 dimensions
An extension of Mercer theorem to matrix-valued measurable kernels
No full textWe extend the classical Mercer theorem to reproducing kernel Hilbert spaces whose elements are functions from a measurable space X into Cn. Given a finite measure μ on X, we represent the reproducing kernel K as a convergent series in terms of the eigenfunctions of a suitable compact operator depending on K and μ. Our result holds under the mild assumption that K is measurable and the associated Hilbert space is separable. Furthermore, we show that X has a natural second countable topology with respect to which the eigenfunctions are continuous and such that the series representing K uniformly converges to K on compact subsets of X×X, provided that the support of μ is X.We extend the classical Mercer theorem to reproducing kernel Hilbert spaces whose elements are functions from a measurable space X into C-n. Given a finite measure mu on X, we represent the reproducing kernel K as a convergent series in terms of the eigenfunctions of a suitable compact operator depending on K and mu. Our result holds under the mild assumption that K is measurable and the associated Hilbert space is separable. Furthermore, we show that X has a natural second countable topology with respect to which the eigenfunctions are continuous and such that the series representing K uniformly converges to K on compact subsets of X x X, provided that the support of mu is X. (C) 2012 Elsevier Inc. All rights reserved
Adaptive Kernel Methods via the Balancing Principle
No full textThe regularization parameter choice is a fundamental problem in Learn- ing Theory since the performance of most supervised algorithms crucially depends on the choice of one or more of such parameters. In particular a main theoretical issue regards the amount of prior knowledge needed to choose the regularization pa- rameter in order to obtain good learning rates. In this paper we present a parameter choice strategy, called the balancing principle, to choose the regularization parameter without knowledge of the regularity of the target function. Such a choice adaptively achieves the best error rate. Our main result applies to regularization algorithms in reproducing kernel Hilbert space with the square loss, though we also study how a similar principle can be used in other situations
Learning sets with separating kernels
Full text linkWe consider the problem of learning a set from random samples. We show how relevant geometric and topological properties of a set can be studied analytically using concepts from the theory of reproducing kernel Hilbert spaces. A new kind of reproducing kernel, that we call separating kernel, plays a crucial role in our study and is analyzed in detail. We prove a new analytic characterization of the support of a distribution, that naturally leads to a family of regularized learning algorithms which are provably universally consistent and stable with respect to random sampling. Numerical experiments show that the proposed approach is competitive, and often better, than other state of the art techniques
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