Charles A. Micchelli

On Learning Vector-Valued Functions

In this letter, we provide a study of learning in a Hilbert space of vector-valued functions. We motivate the need for extending learning theory of scalar-valued functions by practical considerations and establish some basic results for learning vector-valued functions that should prove useful in applications. Specifically, we allow an output space Y to be a Hilbert space, and we consider a reproducing kernel Hilbert space of functions whose values lie in Y. In this setting, we derive the form of the minimal norm interpolant to a finite set of data and apply it to study some regularization functionals that are important in learning theory. We consider specific examples of such functionals corresponding to multiple-output regularization networks and support vector machines, for both regression and classification. Finally, we provide classes of operator-valued kernels of the dot product and translation-invariant type.

POLYNOMIAL PRECONDITIONERS FOR CONJUGATE GRADIENT CALCULATIONS

Dubois, Greenbaum and Rodrigue proposed using a truncated Neumann series as an approximation to the inverse of a matrix A for the purpose of preconditioning conjugate gradient iterative approximations to

Total positivity and its applications

Ax = b

Proximity algorithms for image models: denoising

. If we assume that A has been symmetrically scaled to have unit diagonal and is thus of the form

Optimal Estimation of Linear Operators in Hilbert Spaces from Inaccurate Data

(I - G)

Learning convex combinations of continuously parameterized basic kernels

, then the Neumann series is a power series in G with unit coefficients. The incomplete inverse was thought of as a replacement of the incomplete Cholesky decomposition suggested by Meijerink and van der Vorst in the family of methods ICCG

A DC-programming algorithm for kernel selection

(n)

Computation of Curves and Surfaces

. The motivation for the replacement was the desire to have a preconditioned conjugate gradient method which only involved vector operations and which utilized long vectors.We here suggest parameterizing the incomplete inverse to form a preconditioning matrix whose inverse is a polynomial in G. We then show how to select the parameters to minimize the condition number of the product of the polynomial and

Fast Collocation Methods for Second Kind Integral Equations

(I - G)

Feature space perspectives for learning the kernel

. Theoretically the resulting algorithm...