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

Comment on "Support Vector Machines with Applications"

Comment on ``Support Vector Machines with Applications'' [math.ST/0612817]Comment: Published at http://dx.doi.org/10.1214/088342306000000484 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Rejoinder to "Support Vector Machines with Applications"

Rejoinder to ``Support Vector Machines with Applications'' [math.ST/0612817]Comment: Published at http://dx.doi.org/10.1214/088342306000000501 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Kernel methods in machine learning

We review machine learning methods employing positive definite kernels. These methods formulate learning and estimation problems in a reproducing kernel Hilbert space (RKHS) of functions defined on the data domain, expanded in terms of a kernel. Working in linear spaces of function has the benefit of facilitating the construction and analysis of learning algorithms while at the same time allowing large classes of functions. The latter include nonlinear functions as well as functions defined on nonvectorial data. We cover a wide range of methods, ranging from binary classifiers to sophisticated methods for estimation with structured data.Comment: Published in at http://dx.doi.org/10.1214/009053607000000677 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

SVMs for Automatic Speech Recognition: a Survey

Hidden Markov Models (HMMs) are, undoubtedly, the most employed core technique for Automatic Speech Recognition (ASR). Nevertheless, we are still far from achieving high-performance ASR systems. Some alternative approaches, most of them based on Artificial Neural Networks (ANNs), were proposed during the late eighties and early nineties. Some of them tackled the ASR problem using predictive ANNs, while others proposed hybrid HMM/ANN systems. However, despite some achievements, nowadays, the preponderance of Markov Models is a fact. During the last decade, however, a new tool appeared in the field of machine learning that has proved to be able to cope with hard classification problems in several fields of application: the Support Vector Machines (SVMs). The SVMs are effective discriminative classifiers with several outstanding characteristics, namely: their solution is that with maximum margin; they are capable to deal with samples of a very higher dimensionality; and their convergence to the minimum of the associated cost function is guaranteed. These characteristics have made SVMs very popular and successful. In this chapter we discuss their strengths and weakness in the ASR context and make a review of the current state-of-the-art techniques. We organize the contributions in two parts: isolated-word recognition and continuous speech recognition. Within the first part we review several techniques to produce the fixed-dimension vectors needed for original SVMs. Afterwards we explore more sophisticated techniques based on the use of kernels capable to deal with sequences of different length. Among them is the DTAK kernel, simple and effective, which rescues an old technique of speech recognition: Dynamic Time Warping (DTW). Within the second part, we describe some recent approaches to tackle more complex tasks like connected digit recognition or continuous speech recognition using SVMs. Finally we draw some conclusions and outline several ongoing lines of research

Recurrent kernel machines : computing with infinite echo state networks

Echo state networks (ESNs) are large, random recurrent neural networks with a single trained linear readout layer. Despite the untrained nature of the recurrent weights, they are capable of performing universal computations on temporal input data, which makes them interesting for both theoretical research and practical applications. The key to their success lies in the fact that the network computes a broad set of nonlinear, spatiotemporal mappings of the input data, on which linear regression or classification can easily be performed. One could consider the reservoir as a spatiotemporal kernel, in which the mapping to a high-dimensional space is computed explicitly. In this letter, we build on this idea and extend the concept of ESNs to infinite-sized recurrent neural networks, which can be considered recursive kernels that subsequently can be used to create recursive support vector machines. We present the theoretical framework, provide several practical examples of recursive kernels, and apply them to typical temporal tasks

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