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New duality in choices of feature spaces via kernel analysis

Palle E. T. Jorgensen, James Tian

TL;DR

This work develops a systematic duality framework for feature spaces in kernel analysis, linking choices of feature mappings and selections to families of positive definite kernels via RKHS theory. It introduces $H_S(K)$ as a concrete realization set, establishes a dual construction for feature selections, and analyzes how kernel operations (tensor products, sums) and order relations translate into feature-space relationships. It extends the framework to Hilbert spaces of Schwartz distributions and analytic RKHSs, and proposes operator-theoretic tools such as the $K$-transform and $K^{-1}$ to study kernel inverses and dualities, with detailed treatment of Dirac masses and dual spaces. These results provide principled mechanisms for kernel construction, comparison, and inverse problems, with implications for kernel-based learning, regularization, and fractal/IFS-limit kernels.

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.

New duality in choices of feature spaces via kernel analysis

TL;DR

This work develops a systematic duality framework for feature spaces in kernel analysis, linking choices of feature mappings and selections to families of positive definite kernels via RKHS theory. It introduces as a concrete realization set, establishes a dual construction for feature selections, and analyzes how kernel operations (tensor products, sums) and order relations translate into feature-space relationships. It extends the framework to Hilbert spaces of Schwartz distributions and analytic RKHSs, and proposes operator-theoretic tools such as the -transform and to study kernel inverses and dualities, with detailed treatment of Dirac masses and dual spaces. These results provide principled mechanisms for kernel construction, comparison, and inverse problems, with implications for kernel-based learning, regularization, and fractal/IFS-limit kernels.

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 , generalizing Loewner, we make precise the corresponding partially ordered set of all p.d. kernels on , 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 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 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 , and (ii) as p.d. kernels which model fractal limits, i.e., are invariant with respect to certain iterated function systems (IFS)-transformations.
Paper Structure (13 sections, 23 theorems, 117 equations, 2 figures)

This paper contains 13 sections, 23 theorems, 117 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Feature space realizations
  4. A duality for feature selections
  5. Operations
  6. Hilbert space of distributions
  7. Ordering of kernels and RKHSs
  8. RKHS of analytic functions
  9. Three kernels of $H_{2}$-Hardy spaces
  10. Realization using tensor product of Hilbert spaces
  11. $K$-duality via the RKHS $\mathscr{H}_{K}$
  12. Dirac-masses and $\mathscr{H}_{K}$
  13. A $K$-transform and $K^{-1}$

Key Result

Lemma 2.2

Let $\mathscr{H}$ be a Hilbert space, and $\left\{ f_{n}\right\} _{n\in\mathbb{N}}\subset\mathscr{H}$. Suppose Then $\left\{ f_{n}\right\}$ is an orthonormal basis (ONB) if and only if $\left\Vert f_{n}\right\Vert _{\mathscr{H}}=1$ for all $n\in\mathbb{N}$.

Figures (2)

  • Figure 5.1: $g_{n}\left(x\right)=T^{n}f\left(x\right)$, $n=0,1,\cdots,5$.
  • Figure 5.2: $K_{n}$, $n=0,1,2,3$.

Theorems & Definitions (63)

  • Definition 2.1: Positive definite
  • Lemma 2.2: Parseval frame
  • proof
  • Lemma 2.3: Kernel representation
  • proof
  • Proposition 2.4: Products of p.d. kernels
  • proof
  • Definition 2.5: Loewner order, see e.g., MR0486556MR24487
  • Lemma 2.6
  • proof
  • ...and 53 more