Universality and orthogonality in reproducing Kernel Hilbert spaces

Author(s)

Victor Simões Barbosa

• Document type: Master's Dissertation
• Press: São Carlos.
• Institution: Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação (ICMC/SB)
• Defense date: 2013-02-19

Examining board members

Valdir Antonio Menegatto; Jorge Tulio Mujica Ascui; Claudemir Pinheiro de Oliveira

• Advisor: Valdir Antonio Menegatto

Abstract

We analyze the role of feature maps of a positive denite kernel K acting on a Hausdorff topological space E in two specific properties: the universality of K and the orthogonality in the reproducing kernel Hilbert space of K from disjoint supports. Feature maps always exist but may not be unique. A feature map may be interpreted as a kernel based procedure that maps the data from the original input space E into a potentially higher dimensional \"feature space\" in which linear methods may then be used. Both properties, universality and orthogonality from disjoint supports, make sense under continuity of the kernel. Universality of K is equivalent to the fundamentality of {K(. ; y) : y \'IT BELONGS\' X} in the space of all continuous functions on X, with the topology of uniform convergence, for all nonempty compact subsets X of E. One of the main results in this work is a characterization of the universality of K from a similar concept for the feature map. Orthogonality from disjoint supports seeks the orthogonality of any two functions in the reproducing kernel Hilbert space of K when the functions have disjoint supports (AU)

• FAPESP's process: 10/13025-3 - Orthogonality in native spaces with the density property.
• Grantee: Victor Simões Barbosa
• Support Opportunities: Scholarships in Brazil - Master