Lorentzian Kernel Approximation

• Lorentzian kernel approximation is a technique using positive-definite kernels with Lorentzian symmetry, enabling efficient spectral decompositions and finite-dimensional subspace constructions.
• It exploits rapid eigenvalue decay and optimal subspace selection via Mercer expansion, greedy algorithms, and Taylor-series techniques to minimize L2 approximation errors.
• Advanced methods including random Fourier features and hybrid rational-kernel models maintain computational efficiency and structural invariance in both real- and complex-valued function spaces.

Lorentzian kernel approximation is a central technique in statistical learning, numerical analysis, signal processing, and applied mathematics, enabling efficient representations, computations, and interpolations using positive-definite kernels that respect Lorentzian or related symmetry properties. The approximation leverages spectral decompositions, greedy numerical algorithms, random feature constructions, and new analytic insights into eigenfunction growth and regularization for both real- and complex-valued function spaces. Recent advances rigorously connect optimality, computational efficiency, and invariance in the approximation process.

1. Mercer Expansion and the Structure of Lorentzian Kernels

Any continuous, positive-definite Lorentzian kernel $K$ defined on a compact domain $\Omega$ has a Mercer (spectral) expansion: $$K(x, y) = \sum_{j=1}^\infty \lambda_j\, \varphi_j(x) \varphi_j(y)$$ where $\{\lambda_j\}$ are non-increasing, positive eigenvalues and $\{\varphi_j\}$ form an orthonormal system in $L_2(\Omega)$, also orthogonal in the kernel's native Hilbert space $\mathcal{H}$ (Santin et al., 2014, Bagchi, 2020). This expansion encapsulates all mapping properties of $K$ from $\mathcal{H}$ into $L_2(\Omega)$, with eigenfunctions satisfying: $$T f(x) = \int_{\Omega} K(x, y) f(y) dy = \lambda f(x)$$ For practical applications—including those involving the archetype Lorentzian kernel

$k(x, y) = \frac{1}{1 + \|x-y\|^2/\sigma^2}$

—the Mercer expansion enables both theoretical analysis and explicit constructions of finite-dimensional approximating subspaces.

2. Eigenvalue Decay, Optimal Subspaces, and Approximation Error Bounds

The efficacy of Lorentzian kernel approximation is fundamentally governed by the decay of eigenvalues $\lambda_j$ in the Mercer expansion. For smooth radial kernels (which subsume the Lorentzian), eigenvalues often decay nearly exponentially: $$\lambda_i(K) \leq \sqrt{\kappa}\; C' \exp(-C i^{1/d})$$ where constants depend only on the kernel and ambient dimension $d$ (Belkin, 2018). This decay leads to an effective low-dimensionality: a small number of leading eigenfunctions accounts for most of the kernel's mapping behavior.

Approximation errors in the $L_2$ norm are directly given by Kolmogorov $n$-widths: $$d_n(S(\mathcal{H}); L_2(\Omega)) = \sqrt{\lambda_{n+1}}$$ This quantifies the minimal achievable worst-case error when using any $n$-dimensional subspace. The subspace

$$E_n = \mathrm{span}\{\sqrt{\lambda_1}\varphi_1, \ldots, \sqrt{\lambda_n}\varphi_n\}$$

is optimal: no other $n$-dimensional subspace yields uniformly smaller errors (Santin et al., 2014). Pointwise approximation errors are controlled by the Power Function via: $$\|P_{E_n}\|_{L_2} = \sqrt{\sum_{j=n+1}^\infty \lambda_j}$$ decaying rapidly with eigenvalue tails.

3. Computational Methods: Greedy Point Selection, Random Fourier Features, and Taylor Expansions

When analytic computation of the eigenbasis is infeasible, numerical approximations are constructed by selecting subspaces spanned by kernel translates $K(\cdot, x_i)$ at chosen centers $x_i$. Greedy point selection strategies exploit the Power Function to identify centers that maximize the residual norm at each step, yielding nearly optimal subspaces with well-conditioned Gram matrices (Santin et al., 2014).

Alternatively, for translation-invariant kernels (such as the Lorentzian), the Random Fourier Features (RFF) method operates via Bochner’s theorem, expressing the kernel as a Fourier transform: $$k(x, y) = \int_{\mathbb{R}^d} e^{i w^T(x-y)} p(w) dw$$ where for the Lorentzian, $p(w) = C e^{-\gamma \|w\|}$. Kernel evaluation is then approximated via an empirical average using random samples from $p(w)$, with concentration inequalities (Hoeffding, Matrix Bernstein) bounding the deviation: $$P(|\hat{k}(x, y) - k(x, y)| > \varepsilon) \leq \delta$$ as long as the number of features $D$ scales with $\varepsilon^{-2} \log(n/\delta)$ (Bagchi, 2020).

Taylor-series techniques offer another avenue, especially for radial kernels. The kernel function is approximated by matching the Taylor coefficients of its generating function. For instance, for

$k(x, y) = \phi(\|x-y\|^2)$

the minimal moment function $w_m^x$ satisfies

$\int_0^1 z^\ell w_m^x(z) dz = x^\ell$

and the error can be bounded by the Taylor series remainder, uniformly over the domain (Dommel et al., 11 Mar 2024). This yields explicit bounds on the approximation error and reveals that eigenfunctions can be controlled polynomially in their index (rather than exponentially).

4. Capacity, Regularization, and Learning-Theoretic Bounds

Lorentzian kernel approximation spaces, when equipped with norm constraints (such as $\|f\|_{\mathcal{H}} \leq R$), exhibit restricted capacity as measured by the fat shattering dimension: $V_\gamma(B) = O(\log^d(R/\gamma))$ (Belkin, 2018). This sharply limits the ability to fit arbitrary functions and explains the generalization ability of kernel methods: universal approximation is possible in principle, but constrained to balls of bounded norm.

Crucially, advanced approximation techniques and sharp eigenfunction bounds substantiate the use of much smaller regularization parameters than traditional theory, directly improving kernel approximation and learning outcomes. For low-rank methods such as Nyström, the required number of support points is shown to grow only polylogarithmically as the regularization parameter becomes small: $$\mathcal{N}_\infty(\lambda) = O((\ln(\lambda^{-1}))^{2d})$$ permitting highly efficient approximations for Lorentzian kernels (Dommel et al., 11 Mar 2024).

5. Complex-Valued Lorentzian Kernel Approximation and Hybrid Rational-Kernel Models

For approximating complex-valued functions, especially frequency response functions, new reproducing kernel Hilbert spaces have been constructed as real vector spaces of complex-valued functions. Here, a kernel pair $(k, c)$ (where $c(s, s_0) = k(s, s_0^*)$) is used for minimum norm interpolation, encoded by the system: $K_n \gamma + C_n \gamma^* = y$ yielding interpolants of the form: $$g(s) = \sum_{i=1}^n \gamma_i k(s,s_i) + \sum_{i=1}^n \gamma_i^* c(s,s_i)$$ This formulation enforces symmetry and minimal norm properties (Bect et al., 2023).

Hybrid approaches combine smooth kernel interpolants with low-order rational functions, specifically tailored to capture resonant or pole-driven features in the target functions. Adaptive model selection via leave-one-out cross-validation (with instability penalties) selects the rational order to optimize approximation (Bect et al., 2023). Numerical comparisons indicate that such hybrid schemes are competitive with, or in some cases outperform, standard rational approximation methods.

6. Lorentzian Invariance, Discrete Approximation, and Structural Models

Recent developments connect Lorentzian kernel approximation with structural approximation in algebraic and geometric models of Minkowski space-time (Zilber, 4 Aug 2025). Here, continuous Minkowski space-time together with its Lorentz group is realized as the limit of finite cyclic lattices equipped with finite quasi-Lorentz group actions. The structural approximation map $lm_{\mathcal{D}}$, constructed via ultraproducts, ensures that the discrete models preserve Lorentz symmetry, and the limit recovers the continuum metric structure: $X \mapsto M X M^\dagger$ for $M \in SL(2, A^{(2)})/C$, maintaining $\det(X)$.

This approach offers new insight into Lorentzian kernel approximation, suggesting that such kernels—and their computational approximations—can be examined through the lens of finite model theory and structural algebraic limits. Possible implications include applications in quantum field theory, numerical relativity, and the paper of propagators respecting Lorentz symmetry.

7. Summary and Perspectives

Lorentzian kernel approximation is governed by spectral optimality, exponential eigenvalue decay, and careful numerical subspace construction (greedy, random features, Taylor matching). The effectiveness depends crucially on kernel smoothness, with measure-independent bounds characteristic of high-dimensional settings. Recent advances in analytic bounds on eigenfunction growth and regularization enable efficient and accurate low-rank approximations, favoring practical adoption in statistical learning and inverse problems. Extensions to complex-valued function spaces, hybrid rational-kernel models, and structurally invariant discrete models further broaden the domain of applicability, opening new directions for rigorous computational schemes that respect underlying symmetries and analytic properties.