Operator-valued Kernels

• Operator-valued kernels are symmetric, positive-definite functions mapping pairs of inputs to bounded linear operators, forming the basis of reproducing kernel Hilbert spaces.
• They extend classical kernel methods to handle multi-output and functional responses by enabling spectral decompositions and efficient learning algorithms.
• Their applications span functional data analysis, Gaussian processes, control theory, and quantum information, bridging operator theory with modern machine learning.

An operator-valued kernel is a symmetric, positive-definite function $K : X \times X \to \mathcal{L}(H)$ mapping pairs of inputs in a set $X$ to bounded linear operators on a separable Hilbert space $H$. Operator-valued kernels generalize classical scalar- and vector-valued kernels and serve as the foundational objects for constructing reproducing kernel Hilbert spaces (RKHS) of $H$-valued functions. Their noncommutative structure, feature map factorizations, spectral decomposition, and applications in learning, operator theory, Gaussian processes, and stochastic analysis make them central in modern functional data analysis, multitask learning, system theory, and quantum information.

1. Formal Definition and Fundamental Properties

Let $X$ be a non-empty set, $H$ a (real or complex) separable Hilbert space, and $\mathcal{L}(H)$ the algebra of bounded linear operators on $H$. An operator-valued kernel is a map $K: X \times X \to \mathcal{L}(H)$ satisfying:

• Hermitian symmetry: $K(x, x') = K(x', x)^*$ for all $x, x' \in X$.
• Positive-definiteness: For any finite collection $(x_i, h_i)_{i=1}^n \subset X \times H$,

$$\sum_{i,j=1}^n \langle h_i, K(x_i, x_j) h_j \rangle_H \geq 0.$$

These conditions guarantee that, for each $n$, the block operator matrix $[K(x_i, x_j)]_{i,j=1}^n$ is positive semidefinite in operator theory. Such kernels naturally appear when modeling structured outputs, multi-task regression/classification, and functional responses, as well as in operator-theoretic extensions of classical probability and analysis (Kadri et al., 2015, Kadri et al., 2013, Jorgensen et al., 2024).

2. Reproducing Kernel Hilbert Spaces of Operator-Valued Functions

To every operator-valued kernel $K$, one associates a unique reproducing kernel Hilbert space $\mathcal{H}_K$ of $H$-valued functions $f: X \to H$, constructed as follows:

• Construction: Start with the pre-Hilbert space of finite linear combinations $f = \sum_{i=1}^n K(\cdot, x_i) h_i$, $x_i \in X$, $h_i \in H$, and set inner product

$$\left\langle \sum_{i=1}^n K(\cdot, x_i) h_i,\; \sum_{j=1}^m K(\cdot, y_j) k_j \right\rangle_{\mathcal{H}_K} = \sum_{i=1}^n \sum_{j=1}^m \langle h_i, K(x_i, y_j) k_j \rangle_H.$$

• Completion: Take the Hilbert space completion, obtaining the RKHS $\mathcal{H}_K \subset H^X$.
• Reproducing property: For any $f \in \mathcal{H}_K$, $x \in X$, $h \in H$,

$$\langle f, K(\cdot, x) h \rangle_{\mathcal{H}_K} = \langle f(x), h \rangle_H.$$

This property ensures point evaluation is a bounded linear map, and further induces a bijective correspondence: $K$ is the reproducing kernel of $\mathcal{H}_K$ if and only if $K$ is positive-definite in the sense above (Kadri et al., 2015, Jorgensen et al., 2024, Sababe, 31 Oct 2025). This construction generalizes the classical scalar RKHS theory (Jorgensen et al., 2024).

3. Kolmogorov Decomposition and Feature Map Realization

A central structural result for operator-valued kernels is the Kolmogorov decomposition (also “feature map representation”):

Given $K: X \times X \to \mathcal{L}(H)$ positive-definite, there exists a Hilbert space $\mathcal{G}$ and a map

$V: X \to \mathcal{L}(H, \mathcal{G})$

such that $K(x,y) = V(x)^* V(y)$ for all $x, y \in X$. The minimal such $\mathcal{G}$ is the closed span of the images $V(x)h$, $x \in X$, $h \in H$. This universal factorization is unique up to Hilbert space isomorphism and encodes the geometry of the kernel.

• The canonical choice is to use $\mathcal{G}$ as the RKHS of the scalar-valued kernel $$\widetilde{K}((x, h), (y, k)) := \langle h, K(x, y) k \rangle_H$$ on $X \times H$, with $V(x) h := \widetilde{K}(\cdot, (x, h))$ (Jorgensen et al., 2024, Kadri et al., 2015).
• If $K(x, y) = k(x, y) I_H$ with $k$ scalar-valued, this recovers the classical feature-map case.

This decomposition provides the foundation for spectral analysis, minimization algorithms, and connections to operator-theoretic dilation (e.g., Stinespring theory for CP maps) (Jorgensen et al., 2024, Sababe, 31 Oct 2025).

4. Spectral Theory, Mercer Expansion, and Regularity

When $K$ is continuous and positive-definite on a compact $X$, and $H$ is separable, the integral operator

$(Tf)(x) = \int_X K(x, y) f(y) d\mu(y)$

is compact, self-adjoint, and admits a spectral decomposition:

$$K(x, y) = \sum_{n=1}^\infty \lambda_n \phi_n(x) \otimes \phi_n(y)^*,$$

where $$\{\lambda_n\} \subset \mathbb{R}_+, \phi_n(x) \in H$$, and the convergence is absolute in operator norm (Kadri et al., 2015, Zweck et al., 2024, Sababe, 31 Oct 2025).

• Trace class and regularity: Under additional Hölder-continuity and boundedness conditions, the associated integral operator is trace class; this underpins uniform eigen-expansions and Fredholm-determinant calculations (Zweck et al., 2024).
• Mercer theorem (operator extension): Establishes criteria under which $K$ admits such expansions, paralleling the scalar classical result (Zweck et al., 2024).

This spectral representability facilitates the efficient implementation of learning algorithms and the study of the functional-analytic properties of operator RKHSs.

5. Learning Theory, Algorithms, and Empirical Risk Minimization

Operator-valued kernels enable extensions of classical kernel algorithms to vector-valued, functional, or structured outputs.

• Representer Theorem (operator-valued): Any minimizer of a regularized empirical risk

$$\min_{f \in \mathcal{H}_K} \sum_{i=1}^n \| y_i - f(x_i) \|_H^2 + \lambda \| f \|_{\mathcal{H}_K}^2$$

admits a finite expansion

$$f^*(\cdot) = \sum_{i=1}^n K(\cdot, x_i) \alpha_i, \quad \alpha_i \in H$$

(Kadri et al., 2015, Kadri et al., 2013).

• Functional response and multitask regression/classification: Algorithms such as functional kernel ridge regression and functional regularized least squares classification exploit operator-valued kernels to encode complex target structure (e.g., $L^2$-function-valued outputs, multi-output regression, or structured prediction) (Kadri et al., 2015, Kadri et al., 2012, Audiffren et al., 2013).
• Spectral solution and Kronecker structure: For “separable” kernels of the form $K(x, x') = g(x, x') T$ (with $g$ scalar kernel, $T$ positive operator), the block kernel matrix admits fast diagonalization via the Kronecker product of the Gram and operator eigenbases (Kadri et al., 2015, Kadri et al., 2013).

Operator-valued kernel frameworks enable analytic learning rates, dimension-free convergence theorems under SGD, and algorithmic tractability even in infinite-dimensional output spaces (Yang et al., 25 Apr 2025, Audiffren et al., 2013).

6. Applications: Functional Data Analysis, Stochastic Processes, and System Theory

Operator-valued kernels are foundational in multiple applied domains:

• Functional data analysis: Estimation of function-valued outputs (e.g., predicting continuous time series or curves) leverages integral operator kernels and functional-response learning (Kadri et al., 2015, Kadri et al., 2012).
• Gaussian processes with operator-valued covariance: Gaussian process priors with operator-valued covariance kernels admit precise covariance control in $H$-valued stochastic process models, including covariance structure and Karhunen–Loève expansions (Jorgensen et al., 2024, Sababe, 31 Oct 2025, Jorgensen et al., 2024).
• System and control theory: In infinite-dimensional control (e.g., LQR for PDEs), the operator-valued kernel encodes the Gramian of controlled trajectories, with explicit formulas via Riccati equations and kernel-based representer theorems for optimal control solutions (Aubin-Frankowski et al., 2022).
• Operator dilation and noncommutative extensions: Universal kernel models provide a Hilbert space framework for iterated completely positive maps, quantum channels, and operator system extensions (Tian, 17 Nov 2025).

7. Generalizations, Noncommutative and Quantum Extensions

Recent work extends operator-valued kernels into noncommutative and quantum domains:

• Entangled and quantum-inspired kernels: By leveraging partial trace and entanglement constructions, one generalizes separable kernels to entangled operator-valued forms, enabling flexible modeling of quantum channel learning, multi-output induction, and quantum state tomography (Huusari et al., 2021, Kadri et al., 4 Jun 2025).
• Multilinear, noncommutative, and Banach-space-valued kernels: Operator-valued Calderón–Zygmund theory, trace class criteria, and extension theorems for kernels on free semigroups generalize classical harmonic analysis and moment problem results to the operator setting (Plinio et al., 2019, Tian, 13 Oct 2025).
• Random feature approximations and scalable learning: Generalizations of Bochner’s theorem yield operator-valued Fourier feature maps, supporting scalable approximations and uniform convergence for high-dimensional multi-output learning (Brault et al., 2016, Minh, 2016).

These developments solidify operator-valued kernels as a central mathematical object, bridging machine learning, stochastic analysis, dynamical systems, and functional analysis.