Positive Definite Kernels and Boundary Spaces
نویسندگان
چکیده
We consider a kernel based harmonic analysis of “boundary,” and boundary representations. Our setting is general: certain classes of positive definite kernels. Our theorems extend (and are motivated by) results and notions from classical harmonic analysis on the disk. Our positive definite kernels include those defined on infinite discrete sets, for example sets of vertices in electrical networks, or discrete sets which arise from sampling operations performed on positive definite kernels in a continuous setting. Below we give a summary of main conclusions in the paper: Starting with a given positive definite kernel K we make precise generalized boundaries for K. They are measure theoretic “boundaries.” Using the theory of Gaussian processes, we show that there is always such a generalized boundary for any positive definite kernel.
منابع مشابه
Conditionally Positive Definite Kernels and Pontryagin Spaces
Conditionally positive definite kernels provide a powerful tool for scattered data approximation. Many nice properties of such methods follow from an underlying reproducing kernel structure. While the connection between positive definite kernels and reproducing kernel Hilbert spaces is well understood, the analog relation between conditionally positive definite kernels and reproducing kernel Po...
متن کاملOpen problem : kernel methods on manifolds and metric spaces
Radial kernels are well-suited for machine learning over general geodesic metric spaces, where pairwise distances are often the only computable quantity available. We have recently shown that geodesic exponential kernels are only positive definite for all bandwidths when the input space has strong linear properties. This negative result hints that radial kernel are perhaps not suitable over geo...
متن کاملDiscovering Domain-Specific Composite Kernels
Kernel-based data mining algorithms, such as Support Vector Machines, project data into high-dimensional feature spaces, wherein linear decision surfaces correspond to non-linear decision surfaces in the original feature space. Choosing a kernel amounts to choosing a high-dimensional feature space, and is thus a crucial step in the data mining process. Despite this fact, and as a result of the ...
متن کاملKernels, Associated Structures and Generalizations
This paper gives a survey of results in the mathematical literature on positive definite kernels and their associated structures. We concentrate on properties which seem potentially relevant for Machine Learning and try to clarify some results that have been misused in the literature. Moreover we consider different lines of generalizations of positive definite kernels. Namely we deal with opera...
متن کاملOpen Problem: Kernel methods on manifolds and metric spaces. What is the probability of a positive definite geodesic exponential kernel?
Radial kernels are well-suited for machine learning over general geodesic metric spaces, where pairwise distances are often the only computable quantity available. We have recently shown that geodesic exponential kernels are only positive definite for all bandwidths when the input space has strong linear properties. This negative result hints that radial kernel are perhaps not suitable over geo...
متن کامل