Preprint
New duality in choices of feature spaces via kernel analysis
ArXiV.org
Cornell University
01/19/2025
DOI: 10.48550/arxiv.2501.11206
Abstract
We present a systematic study of the family of positive definite (p.d.)
kernels with the use of their associated feature maps and feature spaces. For a
fixed set $X$, generalizing Loewner, we make precise the corresponding
partially ordered set $Pos\left(X\right)$ of all p.d. kernels on $X$, as well
as a study of its global properties. This new analysis includes both results
dealing with applications and concrete examples, including such general notions
for $Pos\left(X\right)$ as the structure of its partial order, its products,
sums, and limits; as well as their Hilbert space-theoretic counterparts. For
this purpose, we introduce a new duality for feature spaces, feature
selections, and feature mappings. For our analysis, we further introduce a
general notion of dual pairs of p.d. kernels. Three special classes of kernels
are studied in detail: (a) the case when the reproducing kernel Hilbert spaces
(RKHSs) may be chosen as Hilbert spaces of analytic functions, (b) when they
are realized in spaces of Schwartz-distributions, and (c) arise as fractal
limits. We further prove inverse theorems in which we derive results for the
analysis of $Pos\left(X\right)$ from the operator theory of specified
counterpart-feature spaces. We present constructions of new p.d. kernels in two
ways: (i) as limits of monotone families in $Pos\left(X\right)$, and (ii) as
p.d. kernels which model fractal limits, i.e., are invariant with respect to
certain iterated function systems (IFS)-transformations.
Details
- Title: Subtitle
- New duality in choices of feature spaces via kernel analysis
- Creators
- Palle E. T JorgensenJames Tian
- Resource Type
- Preprint
- Publication Details
- ArXiV.org
- DOI
- 10.48550/arxiv.2501.11206
- ISSN
- 2331-8422
- Publisher
- Cornell University; Ithaca, New York
- Language
- English
- Date posted
- 01/19/2025
- Academic Unit
- Mathematics
- Record Identifier
- 9984775261802771
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