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- Maximilian Nickel, Lorenzo Rosasco, Tomaso A. Poggio
- AAAI
- 2016

Learning embeddings of entities and relations is an efficient and versatile method to perform machine learning on rela-tional data such as knowledge graphs. In this work, we propose holographic embeddings (HOLE) to learn compositional vector space representations of entire knowledge graphs. The proposed method is related to holographic models of… (More)

In this paper, we study a family of gradient descent algorithms to approximate the regression function from Reproducing Kernel Hilbert Spaces (RKHSs), the family being characterized by a polynomial decreasing rate of step sizes (or learning rate). By solving a bias-variance trade-off we obtain an early stopping rule and some probabilistic upper bounds for… (More)

- Mauricio A. Álvarez, Lorenzo Rosasco, Neil D. Lawrence
- Foundations and Trends in Machine Learning
- 2012

Kernel methods are among the most popular techniques in machine learning. From a frequentist/discriminative perspective they play a central role in regularization theory as they provide a natural choice for the hypotheses space and the regularization functional through the notion of reproducing kernel Hilbert spaces. From a Bayesian/generative perspective… (More)

- Frank Bauer, Sergei V. Pereverzyev, Lorenzo Rosasco
- J. Complexity
- 2007

In this paper we discuss a relation between Learning Theory and Regularization of linear ill-posed inverse problems. It is well known that Tikhonov regularization can be profitably used in the context of supervised learning, where it usually goes under the name of regularized least-squares algorithm. Moreover the gradient descent algorithm was studied… (More)

- Lorenzo Rosasco, Mikhail Belkin, Ernesto De Vito
- Journal of Machine Learning Research
- 2010

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)

- Fabio Anselmi, Lorenzo Rosasco, Tomaso A. Poggio
- ArXiv
- 2015

We discuss data representation which can be learned automatically from data, are invariant to transformations, and at the same time selective, in the sense that two points have the same representation only if they are one the transformation of the other. The mathematical results here sharpen some of the key claims of i-theory, a recent theory of feedforward… (More)

The present phase of Machine Learning is characterized by supervised learning algorithms relying on large sets of labeled examples (n → ∞). The next phase is likely to focus on algorithms capable of learning from very few labeled examples (n → 1), like humans seem able to do. We propose an approach to this problem and describe the underlying theory, based… (More)

We study Nyström type subsampling approaches to large scale kernel methods, and prove learning bounds in the statistical learning setting, where random sampling and high probability estimates are considered. In particular, we prove that these approaches can achieve optimal learning bounds, provided the subsampling level is suitably chosen. These results… (More)

In this paper we discuss a novel framework for multiclass learning, defined by a suitable coding/decoding strategy, namely the simplex coding, that allows us to generalize to multiple classes a relaxation approach commonly used in binary classification. In this framework, we develop a relaxation error analysis that avoids constraints on the considered… (More)

- Lorenzo Rosasco, Ernesto De Vito, Andrea Caponnetto, Michele Piana, Alessandro Verri
- Neural Computation
- 2004

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)