Support vector machine for functional data classification

In many applications, input data are sampled functions taking their values in infinite dimensional spaces rather than standard vectors. This fact has complex consequences on data analysis algorithms that motivate modifications of them. In fact most of the traditional data analysis tools for regression, classification and clustering have been adapted to functional inputs under the general name of functional Data Analysis (FDA). In this paper, we investigate the use of Support Vector Machines (SVMs) for functional data analysis and we focus on the problem of curves discrimination. SVMs are large margin classifier tools based on implicit non linear mappings of the considered data into high dimensional spaces thanks to kernels. We show how to define simple kernels that take into account the unctional nature of the data and lead to consistent classification. Experiments conducted on real world data emphasize the benefit of taking into account some functional aspects of the problems.Comment: 13 page

Qualitative Robustness of Support Vector Machines

Support vector machines have attracted much attention in theoretical and in applied statistics. Main topics of recent interest are consistency, learning rates and robustness. In this article, it is shown that support vector machines are qualitatively robust. Since support vector machines can be represented by a functional on the set of all probability measures, qualitative robustness is proven by showing that this functional is continuous with respect to the topology generated by weak convergence of probability measures. Combined with the existence and uniqueness of support vector machines, our results show that support vector machines are the solutions of a well-posed mathematical problem in Hadamard's sense

On the consistency of Multithreshold Entropy Linear Classifier

Multithreshold Entropy Linear Classifier (MELC) is a recent classifier idea which employs information theoretic concept in order to create a multithreshold maximum margin model. In this paper we analyze its consistency over multithreshold linear models and show that its objective function upper bounds the amount of misclassified points in a similar manner like hinge loss does in support vector machines. For further confirmation we also conduct some numerical experiments on five datasets.Comment: Presented at Theoretical Foundations of Machine Learning 2015 (http://tfml.gmum.net), final version published in Schedae Informaticae Journa

Robustness and Regularization of Support Vector Machines

We consider regularized support vector machines (SVMs) and show that they are precisely equivalent to a new robust optimization formulation. We show that this equivalence of robust optimization and regularization has implications for both algorithms, and analysis. In terms of algorithms, the equivalence suggests more general SVM-like algorithms for classification that explicitly build in protection to noise, and at the same time control overfitting. On the analysis front, the equivalence of robustness and regularization, provides a robust optimization interpretation for the success of regularized SVMs. We use the this new robustness interpretation of SVMs to give a new proof of consistency of (kernelized) SVMs, thus establishing robustness as the reason regularized SVMs generalize well

A Comparative Study of Pairwise Learning Methods based on Kernel Ridge Regression

Many machine learning problems can be formulated as predicting labels for a pair of objects. Problems of that kind are often referred to as pairwise learning, dyadic prediction or network inference problems. During the last decade kernel methods have played a dominant role in pairwise learning. They still obtain a state-of-the-art predictive performance, but a theoretical analysis of their behavior has been underexplored in the machine learning literature. In this work we review and unify existing kernel-based algorithms that are commonly used in different pairwise learning settings, ranging from matrix filtering to zero-shot learning. To this end, we focus on closed-form efficient instantiations of Kronecker kernel ridge regression. We show that independent task kernel ridge regression, two-step kernel ridge regression and a linear matrix filter arise naturally as a special case of Kronecker kernel ridge regression, implying that all these methods implicitly minimize a squared loss. In addition, we analyze universality, consistency and spectral filtering properties. Our theoretical results provide valuable insights in assessing the advantages and limitations of existing pairwise learning methods.Comment: arXiv admin note: text overlap with arXiv:1606.0427