This thesis develops the theory and practise of reproducing kernel methods.
Many functional inverse problems which arise in, for example, machine learning
and computer graphics, have been treated with practical success using
methods based on a reproducing kernel Hilbert space perspective. This perspective
is often theoretically convenient, in that many functional analysis
problems reduce to linear algebra problems in these spaces. Somewhat more
complex is the case of conditionally positive definite kernels, and we provide
an introduction to both cases, deriving in a particularly elementary manner
some key results for the conditionally positive definite case.
A common complaint of the practitioner is the long running time of these
kernel based algorithms. We provide novel ways of alleviating these problems
by essentially using a non-standard function basis which yields computational
advantages. That said, by doing so we must also forego the aforementioned
theoretical conveniences, and hence need some additional analysis
which we provide in order to make the approach practicable. We demonstrate
that the method leads to state of the art performance on the problem
of surface reconstruction from points.
We also provide some analysis of kernels invariant to transformations such
as translation and dilation, and show that this indicates the value of learning
algorithms which use conditionally positive definite kernels. Correspondingly,
we provide a few approaches for making such algorithms practicable.
We do this either by modifying the kernel, or directly solving problems with
conditionally positive definite kernels, which had previously only been solved
with positive definite kernels. We demonstrate the advantage of this approach,
in particular by attaining state of the art classification performance
with only one free parameter.