Transported RKHS Geometries

• Transported RKHS geometries are spaces derived by pulling back standard RKHS structures through maps and group actions, ensuring preservation of metric, spectral, and algebraic invariants.
• They employ pullback constructions and unitary transport operators to maintain isometry and spectral properties, allowing consistent analysis across varied domains.
• This framework underpins methods in equivariant estimation, optimal transport, and adaptive geometry, impacting nonparametric statistics and infinite-dimensional analysis.

Transported geometries in reproducing kernel Hilbert spaces (RKHSs) comprise a family of metric and function space structures arising when standard RKHSs are moved, deformed, or pulled back via group actions, maps, or functorial constructions. This viewpoint captures how all relevant analytic, spectral, and metric features—kernel functions, Hilbert norms, algebraic coproducts, and geometric invariants—are systematically transferred between models while preserving key structures. Such frameworks underpin equivariant estimation, information geometry, optimal transport, and spectral methods, enabling a unified analysis across domains from abstract harmonic analysis to nonparametric statistics and infinite-dimensional differential geometry.

1. Canonical Transport by Pullback and Group Actions

A central mechanism for transporting RKHS structures is the pullback along maps, as well as group actions as formalized via unitary transport operators.

Pullback Construction: Given a map $f: Y \to X$ and an RKHS $\Hcal_X$ (with reproducing kernel $K_X$), the pullback $\Hcal_Y$ is defined as the set $\{h \circ f : h \in \Hcal_X\} \subset L(Y)$, with the induced inner product $\langle h_1\circ f, h_2 \circ f \rangle_{\Hcal_Y} = \langle h_1, h_2 \rangle_{\Hcal_X}$ for minimal extensions. The resulting kernel is transported as $K_Y(y, y') = K_X(f(y), f(y'))$ (Giannakis et al., 2 Jan 2024).

Group Actions and Unitary Transport: For a group $G$ acting measurably on a set $E$, the action induces unitary operators $U_g$ on $L^2(\mu)$ via $(U_g f)(x) = J_g(x)^{1/2} f(g^{-1}\cdot x)$, where $J_g$ is the Radon–Nikodym derivative. Twin kernel spaces arise as the orbit of the base kernel under this action:

$$k_g(x, y) = J_g(x)^{1/2} J_g(y)^{1/2} k_e(g^{-1} \cdot x, g^{-1} \cdot y)$$

yielding a family of unitarily isomorphic RKHSs $\Hcal_{k_g}$ (Nembé, 15 Dec 2025).

2. Spectral Equivariance and Unitary Transport

Transported RKHS geometries preserve the spectral structure by conjugating integral operators with unitary maps.

Spectral Equivariance Theorem: Let $k_e(x, y) = \sum_i \lambda_i \phi_i(x) \phi_i(y)$ be a kernel with eigenfunctions $\phi_i$. For any $g \in G$, the transformed eigenfunctions $\phi_{i,g}(x) = U_g \phi_i(x)$ remain orthonormal, and the kernel $k_g(x, y)$ inherits the same eigenvalues:

$$k_g(x, y) = \sum_i \lambda_i \phi_{i,g}(x) \phi_{i,g}(y)$$

$T_g = U_g T_e U_g^{-1}$, so all minimax rates, error trade-offs, and orthogonality relations are invariant under transport (Nembé, 15 Dec 2025).

3. Transport of Metric and Geometric Structures

RKHSs manifest a canonical metric on their indexing sets, which is functorially transported by pullback or group action.

Metric Transport: The canonical pseudometric on $X$ is $d_X(x, x') = \|k^X_x - k^X_{x'}\|_{\Hcal_X}$. For the pullback, this yields $d_Y(y, y') = d_X(f(y), f(y'))$, so $f: (Y, d_Y) \to (X, d_X)$ is isometric, and all geometric features (e.g., distances, curvatures) on $X$ are inherited by $Y$ (Giannakis et al., 2 Jan 2024).

Hermitian Metrics and Holomorphic Equivariance: For domains $\Omega \subset \C^n$, Hermitian metrics can be constructed by pulling back via the evaluation map from an RKHS or its dual. If $K$ is projectively invariant under the automorphism group of $\Omega$, all pulled-back metrics—such as the Bergman metric—are invariant under the group, reflecting equivariant geometric transport (Bilokopytov, 2017).

4. Algebraic and Category-Theoretic Closure

Transported RKHSs accommodate algebraic structures, such as comultiplications, in ways that are preserved under pullback.

Reproducing Kernel Hilbert Algebras (RKHAs): An RKHS $\Hcal_X$ endowed with a bounded comultiplication $\Delta_X: \Hcal_X \to \Hcal_X \otimes \Hcal_X$ (i.e., an RKHA), remains an RKHA upon pullback. The comultiplication is $\Delta_Y = (T_f^* \otimes T_f^*) \Delta_X T_f$, with comultiplicative consistency $\Delta_Y(k^Y_y) = k^Y_y \otimes k^Y_y$ and norm bound $\|\Delta_Y\| \leq \|\Delta_X\|$ (Giannakis et al., 2 Jan 2024).

Monoidal Category Structure: The class of RKHAs is closed under Hilbert space tensor product and pullback, making the subcategory of RKHAs a monoidal category, with spectrum as a monoidal functor to topological spaces. The image of this functor includes all compact subspaces of $\R^n$ for $n > 0$ (Giannakis et al., 2 Jan 2024).

5. Applications: Spectral Estimation and Adaptive Geometry

Transported geometries enable adaption of classical estimators and adaptive models.

Orthogonal Polynomial and Kernel Estimation: In twin kernel geometries:

• Hard-truncated orthogonal polynomial estimators $\hat f_{e,K}(x)$ in a base RKHS transport to $\hat f_{g,K}(x) = U_g \hat f_{e,K}(x)$ in the twin geometry.
• Kernel smoothers correspond to soft spectral filtering with filter functions $S(\lambda)$, acting identically in every transported RKHS.
• Bias, variance, and minimax rates are preserved due to unitarity (Nembé, 15 Dec 2025).

Concrete Examples:

• Hermite polynomials with dilation group actions, yielding transported bases $$\phi_{i,\alpha}(x) = \exp(\frac14(1-\alpha^2)x^2) H_i(\alpha^{-1} x)$$.
• Legendre polynomials with affine group, producing affine-invariant bases $a^{-1/2} P_i((x-b)/a)$.
• Adaptive estimation by combining twin bases centered at different modes (Nembé, 15 Dec 2025).

Spline and Wavelet Methods: Spline smoothing and wavelet transforms emerge as spectral penalization and multiscale transport in twin Sobolev and dyadic dilation RKHSs, evidencing the universality of the transported viewpoint for nonparametric inference (Nembé, 15 Dec 2025).

6. RKHS Geometry in Optimal Transport and Information Metrics

Transported RKHS geometries underpin Riemannian and information geometric structures in probability space.

Sinkhorn Divergences and Entropic Geometry: The Hessian of the Sinkhorn divergence $S_\varepsilon$ induces a Riemannian metric on $\mathcal{P}(X)$, with the tangent space at $\mu$ modeled as an RKHS $\mathcal{H}_\mu$ of functions with the self-transport kernel. The induced path-length metric $d_\varepsilon$ is equivalent to a norm on an RKHS, metrizing weak convergence. Transport along geodesics (e.g., translation in $\R^d$) preserves geometry as in classical optimal transport, but now regularized by entropy and governed by RKHS tangents (Lavenant et al., 8 May 2024).

Equivalence and Limitations: The constructed Riemannian metric via the Hessian produces a true geodesic space, though the raw Sinkhorn divergence fails joint convexity and the triangle inequality, underscoring the importance of RKHS-based path metrics (Lavenant et al., 8 May 2024).

7. Invariance, Covariance, and Projective Transport

The invariance of transported metrics under symmetries or automorphism groups is characterized by projective invariance of kernels.

Projective Invariance: If $$K(\phi(z), \phi(w)) = \omega_\phi(z) \overline{\omega_\phi(w)} K(z, w)$$ for all automorphisms $\phi$ and holomorphic weights $\omega_\phi$, then the pulled-back metric on the domain is invariant under the group. This mechanism specifies when complex geometric invariants, such as the Bergman or Poincaré metrics, are realized by transport from abstract Hilbert space metrics (Bilokopytov, 2017).

Structural Restrictions: Attempting to realize full invariance directly on the dual or the ambient Hilbert space admits only the trivial (constant function) RKHS, confirming that transport via pullback is the correct notion for encoding symmetry in nontrivial function spaces (Bilokopytov, 2017).

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The systematic transport of RKHS structures via pullback, unitary group action, or categorical construction preserves spectral, geometric, and algebraic invariants. This enables a unified analysis and design of statistical estimators, geometric algorithms, and algebraic function spaces under deformation, symmetry, and geometric adaptation. The foundational results cited provide the explicit isometric, spectral, and algebraic mechanisms for these transports, demonstrating broad implications for spectral nonparametrics, invariant metrics, and optimal transport on infinite-dimensional spaces (Giannakis et al., 2 Jan 2024, Nembé, 15 Dec 2025, Lavenant et al., 8 May 2024, Bilokopytov, 2017).