Periodic Scaled Korobov Kernel (PSKK) Method

• The PSKK method is a nonparametric density estimator that leverages periodic high-order kernels in scaled Korobov spaces to model smooth, exponentially decaying densities on ℝ^d.
• It utilizes periodization via modulo operations combined with kernel ridge regression, thereby generalizing traditional periodic kernel methods to unbounded domains.
• PSKK offers enhanced convergence rates and computational efficiency in moderate to high dimensions, outperforming classical KDE by mitigating the curse of dimensionality.

The Periodic Scaled Korobov Kernel (PSKK) Method is a nonparametric statistical estimation technique designed for density estimation on unbounded domains, particularly $\mathbb{R}^d$, using periodic, high-order kernel functions constructed from scaled Korobov spaces. The PSKK approach generalizes earlier kernel methods—previously restricted to periodic or compactly supported densities—by enabling their application on general, non-periodic, exponentially decaying densities without loss of theoretical convergence guarantees.

1. Foundational Principles and Motivation

The PSKK method addresses key limitations of classic nonparametric approaches such as kernel density estimation (KDE), whose convergence rates deteriorate rapidly with increasing dimension due to the curse of dimensionality. Recent kernel-based alternatives—principally those leveraging Korobov RKHS and lattice-based approximation—were shown to attain optimal or near-optimal mean integrated squared error (MISE) rates, but only under the assumption that the target density is periodic and supported on a bounded domain (e.g., $[0,1]^d$ or its scaled boxes).

The central innovation of PSKK is to remove the necessity for periodicity of the estimated density. This is accomplished by periodizing the empirical sample using a modulo operation into a hypercube, and then performing kernel ridge regression (KRR) in a scaled Korobov space on this wrapped domain. This extension allows theoretically optimal kernel methods to be used for a much broader class of densities—namely, any sufficiently smooth density decaying fast enough at infinity—thus allowing for accurate nonparametric estimation in applications where support periodicity and compactness cannot be assumed.

2. Mathematical Structure and Estimator Formulation

a. Periodization of Densities

Given a sample $Y_1, \dots, Y_M$ from a density $f$ on $\mathbb{R}^d$, the method first maps each sample into the periodic domain $[-a, a)^d$ by applying the coordinate-wise modulo operation: $$\widetilde{Y}_{m,j} = (Y_{m,j} \bmod 2a) - a, \quad 1 \le j \le d.$$ The induced wrapped (periodic) density is

$$\widetilde{f}(\bm{x}) = \sum_{\bm{k} \in \mathbb{Z}^d} f(\bm{x} + 2a\bm{k}), \qquad \bm{x} \in [-a, a)^d.$$

For densities $f$ with exponential decay, $\widetilde{f}$ is periodic and converges to $f$ as $a \to \infty$ on any compact set.

b. Scaled Korobov Kernel and RKHS

Define the scaled periodic Korobov kernel $K_{\alpha,a,d}$ on $[-a, a]^d$. The associated scaled Korobov space $\mathcal{H}(K_{\alpha,a,d})$ is a reproducing kernel Hilbert space (RKHS) of periodic, smooth functions, with regularity parameter $\alpha > 1$ controlling smoothness and approximation order.

c. Kernel Ridge Regression in Scaled Korobov Space

Choose $N$ preselected centers $\bm{x}_1, \ldots, \bm{x}_N$ in $[-a, a]^d$. The estimator $\widetilde{f}^{\lambda}_{\widetilde{\bm{Y}}}$ is a linear combination of kernel basis functions: $$\widetilde{f}^{\lambda}_{\widetilde{\bm{Y}}}(\bm{x}) = \sum_{k=1}^N c_k K_{\alpha,a,d}(\bm{x}_k, \bm{x}),$$ where the coefficients $\bm{c}$ solve the regularized least squares problem

$$\langle \widetilde{f}^{\lambda}_{\widetilde{\bm{Y}}}, v \rangle_{L^2([-a,a]^d)} + \lambda \langle \widetilde{f}^{\lambda}_{\widetilde{\bm{Y}}}, v \rangle_{K_{\alpha,a,d}} = \frac{1}{M} \sum_{m=1}^{M} v(\widetilde{Y}_m), \qquad \forall v \in V_N,$$

with $$V_N = \operatorname{span}\{ K_{\alpha,a,d}(\bm{x}_k, \cdot) \}$$.

The final density estimator on $\mathbb{R}^d$ is defined as

$$\bar{f}(\bm{x}) = \begin{cases} \max\{\widetilde{f}^{\lambda}_{\widetilde{\bm{Y}}}(\bm{x}),\, 0\}, & \bm{x} \in [-a, a]^d, \ 0, & \text{otherwise}. \end{cases}$$

3. Theoretical Guarantees: MISE Convergence and Conditions

The PSKK estimator achieves the following mean integrated squared error (MISE) bound for densities $f$ of smoothness order $\alpha$ satisfying an exponential decay condition: $$\mathbb{E}\left[ \int_{\mathbb{R}^d} \left| \bar{f}(\bm{x}) - f(\bm{x}) \right|^2 d\bm{x} \right] = \mathcal{O}\left( M^{-1 / (1 + 1/(2\alpha) + \epsilon)} \right)$$ for any $\epsilon > 0$, uniformly over all such $f$.

The analysis decomposes the estimation error into:

• Approximation (projection) error: from finite-rank Galerkin subspace in $\mathcal{H}(K_{\alpha,a,d})$,
• Regularization error: from the kernel ridge regression penalty,
• Truncation (periodization) error: governed by the exponential decay of $f$ at infinity,
• Variance: from the finite sample approximation.

These results match the minimax rate obtained in periodic settings and are essentially optimal for smooth, exponentially decaying densities.

4. Comparison with Prior and Alternative Methods

Earlier work by Kazashi and Nobile demonstrated that kernel estimators in Korobov spaces on compact periodic domains can reach these optimal rates, but applicability was limited to periodic densities on the unit cube. The PSKK method’s key distinction is the rigorous extension to $\mathbb{R}^d$: through periodization via modulo and appropriate scaling, the approach is valid for all sufficiently smooth, fast-decaying densities, vastly increasing its practical range.

In direct comparison with traditional KDE:

• PSKK achieves sharper convergence rates in moderate and high dimensions, provided the density is smooth and decays sufficiently quickly.
• PSKK leverages lattice structures (e.g., circulant matrices) and is computationally efficient for large $N$.
• Traditional KDE is flexible but subject to the curse of dimensionality; its rates deteriorate as $M^{-4/(d+4)}$ under standard smoothness, while PSKK attains rates scaling as $M^{-1/(1 + 1/(2\alpha) + \epsilon)}$ (much closer to $M^{-1}$ for large smoothness).

5. Numerical Evidence and Parameter Considerations

Theoretical findings are supported by comprehensive numerical experiments across dimensions $d=2$ to $d=6$ and for various density classes (e.g., multimodal Gaussian mixtures). Results consistently show that:

• PSKK matches its predicted rate for large sample sizes ($M$),
• It significantly outperforms KDE in higher dimensions ($d \geq 4$) as $M$ increases,
• The method is robust to moderate changes in meta-parameters $(a, N, \lambda)$, although theory-guided tuning remains important.

For large-scale problems, the circulant structure of the kernel matrix enables efficient computation (e.g., fast inversion via FFT methods), making the approach feasible for $N$ in the thousands or more.

Aspect	PSKK Method	Kazashi-Nobile (2023)	KDE
Domain	$\mathbb{R}^d$ (unbounded, non-periodic)	periodic cube $[0,1]^d$	$\mathbb{R}^d$
Periodicity required	No (periodization handled via modulo)	Yes	No
MISE convergence	$\mathcal{O}(M^{-1/(1+1/(2\alpha)+\epsilon)})$	Same	$\mathcal{O}(M^{-4/(d+4)})$
Key limitation	Needs smoothness and decay, parameter tuning	Not for non-periodic	Curse of dimensionality
Scalability	Efficient (lattice/circulant; FFT)	Efficient on cube	Simple, flexible

6. Broader Applications and Implications

Applications of the PSKK method include:

• Uncertainty quantification in science and engineering, where densities may be multimodal or highly anisotropic and supported on $\mathbb{R}^d$,
• Statistical inverse problems, including Bayesian parameter identification,
• Data-driven scientific modeling, e.g., in physics-informed or simulation-based inference where periodicity is unnatural,
• Machine learning contexts such as anomaly detection, generative modeling, or Bayesian density estimation for non-compact domains.

A direct implication is that minimax optimal rates for smooth density estimation are available in practical, non-periodic, unbounded domains, provided one can exploit smoothness and decay—a substantial expansion over the classical RKHS density estimation landscape.

7. Algorithmic Implementation and Parameter Selection

Algorithmic steps for PSKK involve:

1. Mapping all data into $[-a,a]^d$ via modulo (periodization),
2. Selecting a lattice of centers for the kernel basis (circulant matrix enables fast arithmetic),
3. Solving the regularized least squares system for the expansion coefficients,
4. Defining the estimator as the non-negative part of the periodic estimator within the box, zero elsewhere.

Parameter choices $(a, N, \lambda)$ are made following theoretical guidance balancing truncation (periodization), projection, and variance errors, and must respect the exponential decay of $f$ for accurate estimation.

A plausible implication is that, in practical settings, selection of $a$ and $N$ is guided by tail decay of the data and the anticipated signal complexity.

---

In summary, the PSKK method combines advanced kernel-based approximation theory with periodization and scalable implementation to deliver near-optimal density estimation for a broad class of smooth, exponentially decaying densities on unbounded domains, offering a practical and theoretically justified alternative to classical kernel density estimation and extending kernel-based techniques far beyond their traditional periodic boundaries.