Structure-Preserving Kernel Ridge Regression

• Structure-preserving kernel ridge regression is a nonparametric method that preserves the geometric and algebraic structures intrinsic to Hamiltonian and Poisson systems.
• It employs a reproducing kernel Hilbert space with a differential reproducing property to achieve closed-form solutions and statistical consistency in vector field regression.
• Regularization effectively resolves non-identifiability issues by constraining the solution space, ensuring recovery of true system invariants with high numerical accuracy.

Structure-preserving kernel ridge regression (SPKRR) is a nonparametric machine learning methodology that extends kernel ridge regression (KRR) to the learning of functions generating vector fields with underlying geometric or physical structure, most notably Hamiltonian and Poisson systems. Its defining characteristic is the explicit preservation of the geometric and algebraic structures intrinsic to the dynamical system—such as symplectic or Poisson brackets—during the learning process. This approach not only achieves statistical consistency and closed-form solutions but resolves fundamental identifiability issues arising from symmetries inherent in the underlying physical system.

1. Foundations and Problem Formulation

SPKRR is motivated by the problem of reconstructing an unknown scalar function—typically a Hamiltonian $H:P\to\mathbb R$ or more generally a function on a Poisson manifold—given noisy vector observations sampled from the pushforward of its differential under a bundle map such as the Poisson tensor. The canonical data model assumes observed pairs $\{(z^{(n)}, X^{(n)}_{\sigma^2})\}_{n=1}^N$ with

$$z^{(n)}\sim\mu \quad (\text{i.i.d. on } P), \qquad X^{(n)}_{\sigma^2} = \Pi^\sharp(\mathrm dH)(z^{(n)}) + \varepsilon^{(n)}, \qquad \mathbb{E}[\varepsilon^{(n)}] = 0, \; \mathrm{Var}[\varepsilon^{(n)}] = \sigma^2I,$$

where $\Pi^\sharp$ maps covectors to vectors via the Poisson tensor and $\varepsilon^{(n)}$ models independent noise (Hu et al., 18 Apr 2025, Hu et al., 15 Mar 2024).

The objective is to estimate $H$ given data generated via $X_H(z) = \Pi^\sharp[\mathrm dH(z)]$, which introduces both operator-valued observation models and identifiability challenges due to the presence of Casimir functions (null directions of $\Pi^\sharp$).

2. Structure-Preserving Kernel Ridge Regression Methodology

Functional Setting and Loss Construction

SPKRR is posed in a reproducing kernel Hilbert space (RKHS) of sufficiently smooth scalar functions $\mathcal H_K \subset C^1(P)$, endowed with a positive-definite kernel $K$. The method leverages a differential reproducing property: $$D\,h(z)\cdot v = \langle h,\; D^{(1,0)}K((z,\cdot)\cdot v) \rangle_{\mathcal H_K},$$ allowing efficient computation of functionals dependent on vector field data (Hu et al., 18 Apr 2025, Hu et al., 15 Mar 2024). The empirical risk is formulated as

$$\widehat h_{\lambda,N} =\arg\min_{h\in\mathcal H_K} \;\frac1N\sum_{n=1}^N \| \Pi^\sharp(z^{(n)})\,\nabla h(z^{(n)}) - X^{(n)}_{\sigma^2} \|^2 + \lambda\,\|h\|^2_{\mathcal H_K},$$

with Tikhonov regularization $\lambda > 0$ ensuring norm control and well-posedness.

Matrix-Level Representer Theorem and Solution

The (vector-valued) representer theorem states that the solution $\widehat h_{\lambda,N}$ admits the form

$$\widehat h_{\lambda,N}(x) = \sum_{i=1}^N \alpha_i\, K(x, z^{(i)}), \quad \boldsymbol\alpha = (G_N + \lambda N I)^{-1} \mathbf{X}_{\sigma^2,N},$$

where $G_N$ is the “differential Gram matrix”

$$[G_N]_{ij} = \langle D^{(1,0)}K(z^{(i)}, \cdot), D^{(1,0)}K(z^{(j)}, \cdot) \rangle_{\mathcal H_K},$$

and $\mathbf{X}_{\sigma^2,N}$ stacks the observed vector field data. Thus, the regression task reduces to a linear system involving derivatives of the kernel (Hu et al., 18 Apr 2025, Hu et al., 15 Mar 2024).

3. Non-Identifiability, Casimir Functions, and Regularization

For Poisson systems with degenerate $\Pi$, the existence of nontrivial Casimir functions $C$ (satisfying $\Pi^\sharp\mathrm dC=0$) renders $H$ only identifiable up to an arbitrary Casimir. The regularization term $\|h\|_{\mathcal H_K}^2$ plays a critical role by penalizing components in the Casimir subspace and constraining the solution to $(\ker\Pi^\sharp)^\perp$ in $\mathcal H_K$, thus enforcing uniqueness of the minimizer. No explicit basis for the Casimirs is required in the computational formulation; the penalty suffices to “lock out” the nullspace associated with these symmetries (Hu et al., 18 Apr 2025).

4. Theoretical Guarantees and Error Analysis

SPKRR admits full statistical consistency and convergence rates under standard “source conditions” on the true $H$. Specifically, assume $H = Q^\gamma \psi$ for $\gamma > 0$ and $Q = A^*A$ the associated covariance operator. Then, provided $\lambda \sim N^{-\alpha}$, for $\alpha \in (0, 1/3)$,

$$\| \widehat h_{\lambda,N} - H \|_{\mathcal H_K} \lesssim N^{-\min\{ \alpha\,\gamma,\; \frac{1}{2}(1-3\alpha)\}},$$

which can be improved to $\alpha \in (0, 1/2)$ under coercivity conditions ($\|A h\|_{L^2(\mu)} \gtrsim \|h\|_{\mathcal H_K}$) (Hu et al., 18 Apr 2025, Hu et al., 15 Mar 2024). These rates are PAC-style and reflect the balance between estimation and approximation errors, analogous to classical kernel regression but in a structure-aware setting.

SPKRR is also equivalent to a posterior mean under a Gaussian process prior when regularization hyperparameters are suitably chosen ($\lambda = \sigma^2/N$), demonstrating that its solution inherits the minimax and probabilistic optimality properties of GP regression for function learning from vector field data (Hu et al., 15 Mar 2024).

5. Algorithmic Implementation

The solution of SPKRR involves the following steps (see table for notational summary):

Symbol	Object/function	Description
$K(x, y)$	Kernel	Reproducing kernel on $P\times P$
$D^{(1,0)}K(z, \cdot)$	Kernel differential	Partial derivative of $K$ w.r.t. first argument
$G_N$	Differential Gram matrix	$$[G_N]_{ij} = \langle D^{(1,0)}K(z^{(i)}, \cdot), D^{(1,0)}K(z^{(j)}, \cdot) \rangle_{\mathcal H_K}$$
$\mathbf{X}_{\sigma^2,N}$	Vectorized data	Stacked vector field observations, $\in \mathbb R^{dN}$

The computational core is solving $$(G_N + \lambda N I) \boldsymbol\alpha = \mathbf{X}_{\sigma^2,N}$$, followed by evaluating $\widehat h(x) = \sum_i \alpha_i K(x, z^{(i)})$. For differentiable kernels (e.g., Gaussian kernels), all required derivatives and Gram matrices are explicit and tractable.

6. Numerical Performance and Empirical Validation

Numerical experiments validate the efficacy and structure-preservation of SPKRR. For example, in a two-vortex system on $S^2 \times S^2$ with $1\,200$ samples and a Gaussian kernel, SPKRR recovers the true Hamiltonian, including singular structures and qualitative long-time dynamics. The induced flow preserves Casimirs (vortex strengths) exactly. Quantitatively, mean-squared vector-field errors reach $\approx 10^{-6}$ outside singular neighborhoods. Parameter selection (e.g., $\lambda = 10^{-2} N^{-0.4}$ after cross-validation) and high data efficiency are reported (Hu et al., 18 Apr 2025).

Comparative studies with Hamiltonian neural networks (HNNs) on standard 4D Hamiltonian systems show that SPKRR achieves superior recovery of potentials, insensitivity to nonconvexity, and closed-form training via matrix inversion, with computational runtime and accuracy advantages (Hu et al., 15 Mar 2024).

7. Relationship to Broader Kernel Methods and Interpretability

SPKRR extends and generalizes classical KRR by posing regression in function spaces where the loss is defined not on direct function values, but on images under vector bundle maps or gradients. Standard KRR with ridge penalty and RKHS structure is recovered as a particular case. Recent work on kernel method interpretability shows that in high-dimensional feature regimes, KRR predictions can be exactly re-expressed as weighted linear combinations of original features, with ridge penalties inherited in a special metric in the original space. This result holds in general for kernel losses of the form $f(\eta) + \lambda \eta^\top K^{-1} \eta$, including kernel Poisson regression, and implies that the functional regression coefficients (where present) can be interpreted directly as effect sizes on the original predictor variables (Groenen et al., 21 Aug 2025).

A plausible implication is that, in settings where the kernel’s feature map lies in the span of original predictors, structure-preserving methods may inherit interpretability features analogous to standard KRR; however, the key distinction remains the geometric structure encoded by the differential operators and regularization, which is absent in purely value-based regression.

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Structure-preserving kernel ridge regression provides a theoretically solid and computationally explicit paradigm for learning scalar functions whose gradients or other derivatives generate vector fields encoding rich geometric structure. By combining RKHS machinery, bespoke loss functionals, and principled regularization, SPKRR achieves unique, statistically consistent estimators with provable long-time preservation of fundamental invariants in high-dimensional nonlinear systems (Hu et al., 18 Apr 2025, Hu et al., 15 Mar 2024).