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A large number of learning algorithms, for example, spectral clustering, kernel Principal Components Analysis and many manifold methods are based on estimating eigenvalues and eigenfunctions of operators defined by a similarity function or a kernel, given empirical data. Thus for the analysis of algorithms, it is an important problem to be able to assess… (More)

In this letter, we investigate the impact of choosing different loss functions from the viewpoint of statistical learning theory. We introduce a convexity assumption, which is met by all loss functions commonly used in the literature, and study how the bound on the estimation error changes with the loss. We also derive a general result on the minimizer of… (More)

We discuss how a large class of regularization methods, collectively known as spectral regularization and originally designed for solving ill-posed inverse problems, gives rise to regularized learning algorithms. All of these algorithms are consistent kernel methods that can be easily implemented. The intuition behind their derivation is that the same… (More)

In regularized kernel methods, the solution of a learning problem is found by minimizing func-tionals consisting of the sum of a data and a complexity term. In this paper we investigate some properties of a more general form of the above functionals in which the data term corresponds to the expected risk. First, we prove a quantitative version of the… (More)

Many works related learning from examples to regularization techniques for inverse problems, emphasizing the strong algorithmic and conceptual analogy of certain learning algorithms with regu-larization algorithms. In particular it is well known that regularization schemes such as Tikhonov regularization can be effectively used in the context of learning… (More)

We investigate the problem of model selection for learning algorithms depending on a continuous parameter. We propose a model selection procedure based on a worst case analysis and data-independent choice of the parameter. For regularized least-squares algorithm we bound the generalization error of the solution by a quantity depending on few known constants… (More)

Within the framework of statistical learning theory we analyze in detail the so-called elastic-net regularization scheme proposed by Zou and Hastie [45] for the selection of groups of correlated variables. To investigate on the statistical properties of this scheme and in particular on its consistency properties, we set up a suitable mathematical framework.… (More)

© 2 0 0 5 m a s s a c h u s e t t s i n s t i t u t e o f t e c h n o l o g y, c a m b r i d g e , m a 0 213 9 u s a — w w w. c s a i l. m i t. e d u massachusetts institute of technology — computer science and artificial intelligence laboratory Abstract We develop a theoretical analysis of generalization performances of regularized least-squares on… (More)

- Ernesto De, Vito, Lorenzo Rosasco, Alessandro Verri, Ernesto De Vito
- 2006

In this paper we show that a large class of regularization methods designed for solving ill-posed inverse problems gives rise to novel learning algorithms. All these algorithms are consistent kernel methods which can be easily implemented. The intuition behind our approach is that, by looking at regularization from a filter function perspective, filtering… (More)