Support Vector Machines: Training and Applications

The Support Vector Machine (SVM) is a new and very promising classification technique developed by Vapnik and his group at AT&T Bell Labs. This new learning algorithm can be seen as an alternative training technique for Polynomial, Radial Basis Function and Multi-Layer Perceptron classifiers. An interesting property of this approach is that it is an approximate implementation of the Structural Risk Minimization (SRM) induction principle. The derivation of Support Vector Machines, its relationship with SRM, and its geometrical insight, are discussed in this paper. Training a SVM is equivalent to solve a quadratic programming problem with linear and box constraints in a number of variables equal to the number of data points. When the number of data points exceeds few thousands the problem is very challenging, because the quadratic form is completely dense, so the memory needed to store the problem grows with the square of the number of data points. Therefore, training problems arising in some real applications with large data sets are impossible to load into memory, and cannot be solved using standard non-linear constrained optimization algorithms. We present a decomposition algorithm that can be used to train SVM's over large data sets. The main idea behind the decomposition is the iterative solution of sub-problems and the evaluation of, and also establish the stopping criteria for the algorithm. We present previous approaches, as well as results and important details of our implementation of the algorithm using a second-order variant of the Reduced Gradient Method as the solver of the sub-problems. As an application of SVM's, we present preliminary results we obtained applying SVM to the problem of detecting frontal human faces in real images.

A Note on Support Vector Machines Degeneracy

When training Support Vector Machines (SVMs) over non-separable data sets, one sets the threshold $b$ using any dual cost coefficient that is strictly between the bounds of $0$ and $C$. We show that there exist SVM training problems with dual optimal solutions with all coefficients at bounds, but that all such problems are degenerate in the sense that the "optimal separating hyperplane" is given by ${f w} = {f 0}$, and the resulting (degenerate) SVM will classify all future points identically (to the class that supplies more training data). We also derive necessary and sufficient conditions on the input data for this to occur. Finally, we show that an SVM training problem can always be made degenerate by the addition of a single data point belonging to a certain unboundedspolyhedron, which we characterize in terms of its extreme points and rays.

Dynamic graphics of parametrically linked multivariate methods used in compositional data analysis

Many multivariate methods that are apparently distinct can be linked by introducing one or more parameters in their definition. Methods that can be linked in this way are correspondence analysis, unweighted or weighted logratio analysis (the latter also known as "spectral mapping"), nonsymmetric correspondence analysis, principal component analysis (with and without logarithmic transformation of the data) and multidimensional scaling. In this presentation I will show how several of these methods, which are frequently used in compositional data analysis, may be linked through parametrizations such as power transformations, linear transformations and convex linear combinations. Since the methods of interest here all lead to visual maps of data, a "movie" can be made where where the linking parameter is allowed to vary in small steps: the results are recalculated "frame by frame" and one can see the smooth change from one method to another. Several of these "movies" will be shown, giving a deeper insight into the similarities and differences between these methods

Implantación de un SIG basado en software libre para el estudio de recursos hídricos y procesos hidrológicos

En este trabajo se describe la solución ideada para la implantación de un Sistema de Información Geográfica que debe dar servicio al Instituto Universitario del Agua y del Medio Ambiente de la Universidad de Murcia y al Instituto Euromediterráneo del Agua. Dada la naturaleza de ambas instituciones, se trata de una herramienta orientada fundamentalmente al estudio de recursos hídricos y procesos hidrológicos. El proceso se inició con una identificación de las necesidades de los usuarios (con perfiles y requerimiento diferentes) y el posterior desarrollo del diseño conceptual que pudiera asegurar la satisfacción de estas necesidades. Debido a que los requerimientos de los usuarios así lo demandaban, se ha tenido en cuenta tanto a usuarios que trabajan en entorno linux como a otros que lo hacen en entorno windows. Se ha optado por un sistema basado en software libre utilizando GRASS para el manejo de información raster y modelización; postgis (sobre postgreSQL) y GRASS para la gestión de información vectorial; y QGIS, gvSIG y Kosmo como interfaces gráficas de usuario. Otros programas utilizados para propósitos específicos han sido R, Mapserver o GMT

Vector Fields

Exercises and solutions about vector fields. Diagrams for the questions are all together in the support.zip file, as .eps files

Vector Calculus

Exercises and solutions about vector calculus

Vector functions and curves

Exercises and solutions about vector functions and curves.

University Logo in Vector and Raster

This is a selection of University of Southampton Logos in both vector (svg) and raster (png) formats. These are suitable for use on the web or in small documents and posters. You can open the SVG files using inkscape (http://inkscape.org/download/?lang=en) and edit them directly. The University logo should not be modified and attention should be paid to the branding guidelines found here: http://www.edshare.soton.ac.uk/10481 You must always leave a space the width of an capital O in Southampton on all 4 edges of the logo. The negative space makes it appear more prominently on the page.

University Logo in Vector and Raster (Smoother Version)

These are a range of logos created in the same way as Mr Patrick McSweeny http://www.edshare.soton.ac.uk/11157. The logo has been extracted from PDF documents and is smoother and accurate to the original logo design. Many thanks to to McSweeny for publishing the logo, in SVG originally, I struggled to find it anywhere else. Files are in Inkscape SVG, PDF and PNG. From Mr Patrick McSweeney: This is a selection of University of Southampton Logos in both vector (svg) and raster (png) formats. These are suitable for use on the web or in small documents and posters. You can open the SVG files using inkscape (http://inkscape.org/download/?lang=en) and edit them directly. The University logo should not be modified and attention should be paid to the branding guidelines found here: http://www.edshare.soton.ac.uk/10481 You must always leave a space the width of an capital O in Southampton on all 4 edges of the logo. The negative space makes it appear more prominently on the page.

School of Physics and Astronomy Logo - Vector and Raster

Logo for the school of Physics and Astronomy in Inkscape SVG, PDF and high-resolution PNG format

Un metodo para conectar la proyección octogonal de un vector sobre un subespacio de R dotado de un producto escalar

Se trata de un estudio matemático sobre proyecciones octogonales. En el se analizan las diversas posibilidades y variables y se concluye con las posibles soluciones a aplicar al nuevo modelo vectorial.

Los medios de representación gráfica en la modelización en ciencias físicas : el caso del vector fuerza.

Resumen tomado de la publicación