N-D Anisotropic Beta Kernels

• N-dimensional anisotropic Beta kernels are mathematical constructs that integrate directional sensitivity with adaptive scaling for high-dimensional analysis.
• They enhance multivariate density estimation and nonlocal operator analysis by incorporating anisotropic bandwidth structures and spectral asymptotics.
• Their versatile applications span explicit radiance field rendering, fractional PDEs, and advanced statistical methodologies in computational science.

N-dimensional anisotropic Beta kernels are a class of mathematical constructs arising in operator theory, statistics, nonlocal analysis, and radiance field modeling, characterized by directional sensitivity and flexible frequency adaptation. Their central feature is the incorporation of anisotropy—differential scaling or behavior along distinct dimensions—into the Beta kernel formalism, enabling fine-grained control over approximation, estimation, and physical modeling in high-dimensional settings.

1. Mathematical Structure of Anisotropic Beta Kernels

The prototypical N-dimensional anisotropic Beta kernel arises either as an explicit density or as the kernel of a compact integral operator. Generally, these kernels possess the form

$$K_\beta(x, y) = \frac{1}{\pi} \frac{1}{1 + |x-y|^2 + \beta^2 \Theta(x, y)}$$

where %%%%1%%%%, $\beta > 0$ is a small, positive perturbation parameter, and $\Theta(x, y)$ is a symmetric, nonnegative, homogeneous function of degree $\gamma$ (commonly $\gamma \geq 1$), that encodes the anisotropic contribution (Mityagin et al., 2011).

In multivariate density estimation, the Beta kernel is adapted to bounded supports and incorporates a shape or dispersion parameter that can be directionally tuned; for instance, in estimator form:

$$\widehat{f}_n(\mathbf{x}) = \frac{1}{n}\sum_{i=1}^{n} K_{\theta(\mathbf{x},\mathbf{H})}(\mathbf{X}_i)$$

with $K_{\theta(\mathbf{x},\mathbf{H})}$ constructed to match the mode and dispersion to $\mathbf x$ and a full, possibly anisotropic, bandwidth matrix $\mathbf H$ (Kokonendji et al., 2015).

Recent radiance field methods (DBS, UBS) employ N-dimensional anisotropic Beta kernels via reparameterized Beta shape parameters in explicit rendering primitives. The general expression for the Beta kernel in UBS is

$$\sigma(x,q) = B(x, q; \mu, \Sigma, b), \quad \beta_i = 4\exp(b_i)$$

where $\mu \in \mathbb{R}^N$, $\Sigma \in \mathbb{R}^{N \times N}$, $b \in \mathbb{R}^{N-2}$ are learnable parameters, with one shared spatial beta and per-coordinate non-spatial betas (e.g., view direction, time) (Liu et al., 30 Sep 2025).

2. Spectral and Asymptotic Properties

Spectral analysis of operators with anisotropic Beta kernels reveals nontrivial eigenvalue behavior determined by the interplay between the isotropic component and the anisotropic perturbation. In the canonical integral operator setting with small $\beta$,

$$M_\beta = 1 - \lambda_1 \beta^{\frac{2}{\gamma+1}} + o\left(\beta^{\frac{2}{\gamma+1}}\right)$$

where $\lambda_1$ is the lowest eigenvalue of a model pseudo-differential operator $A = |D_x| + \Theta(x,x)/2$. The exponent $\frac{2}{\gamma+1}$ directly reflects the anisotropic scaling imposed by $\Theta$. These operators are positivity improving, guaranteeing a unique, strictly positive principal eigenfunction (Mityagin et al., 2011).

In heat kernel analysis, solutions to anisotropic equations admit moment-based expansions with directionally dependent moment orders. For mixed-order diffusions,

$$u(x, y, t) \approx \sum_{|\beta| + 2|\gamma| \leq k} \frac{(-1)^{|\beta| + |\gamma|}}{\beta! \gamma!} M_{\beta, \gamma}(f) D_x^\beta D_y^\gamma G_t(x, y)$$

where mixed moments reflect the anisotropic scaling of derivatives in $x$ (order-2) and $y$ (order-4) (Ignat et al., 2012).

3. Statistical Estimation and Bandwidth Structures

Multivariate associated Beta kernel estimators allow for adaptive smoothing on bounded domains. The estimator’s kernel depends on the evaluation point and the smoothing (bandwidth) matrix:

• Diagonal bandwidths: independent smoothing, no correlation.
• Full bandwidths: capture scale and cross-dimensional correlation, encode full anisotropy.
• Scott bandwidths: scalar scaling of a pilot matrix, offering a trade-off between performance and computational complexity.

In bivariate (and higher-dimensional) cases, the Sarmanov construction enables explicit modeling of dependence:

$$BS_{\theta(\mathbf{x}, \mathbf{H})}(v_1, v_2) = \text{Beta kernel marginals} \times \left(1 + h_{12} \dots \right)$$

with shape and correlation terms parameterized by the target $\mathbf{x}$ and $\mathbf{H}$; optimal bandwidth is chosen via Least Squares Cross-Validation (LSCV) (Kokonendji et al., 2015). Empirical findings demonstrate that full and Scott bandwidths markedly outperform diagonal ones when the target density exhibits correlation or multimodality.

4. Nonlocal, Fractional, and Physical Models

Nonlocal operators with singular anisotropic Beta kernels generate processes whose jump behavior is directionally split:

$$\mu(x, dy) = \sum_{k=1}^{d} \alpha_k(2-\alpha_k) |x_k - y_k|^{-1-\alpha_k} dy_k \prod_{i \ne k} \delta_{\{x_i\}}(dy_i)$$

yielding operators

$$\mathcal{L} \approx -\sum_{k=1}^{d} (-\partial_{kk})^{\alpha_k / 2}$$

each dimension is governed by its own order $\alpha_k$ (Chaker et al., 2018). This framework allows for the derivation of regularity results (weak Harnack inequalities, Hölder continuity), Sobolev embeddings, and anisotropic Poincaré inequalities, foundational for the theory of anisotropic fractional PDEs.

In nonlocal capillarity, anisotropic Beta kernels appear as interaction energy densities of the form

$K(\zeta) = \frac{a(\omega)}{|\zeta|^{n+s}}$

with directionality encoded in $a(\omega)$. Two fractional exponents $s_1, s_2$ may model different physical regimes (container vs. environment). The nonlocal Young's law for contact angle is formulated as a cancellation of singular integrals over anisotropic sectors (Luca et al., 2022). Unique solvability is ensured under monotonicity and integrability constraints, and equilibrium shapes reflect the anisotropic kernel.

5. Functional Inequalities with Anisotropic Weights

Anisotropic Hardy and Caffarelli–Kohn–Nirenberg inequalities integrate spatial weights with nontrivial directional dependencies:

$$\| |x|^\beta |x'|^{\alpha + 1} \nabla u \|_{L^p} \geq C \| |x|^\beta |x'|^\alpha u \|_{L^p}$$

for $x' = (x_1, ..., x_{n-1}, 0)$ (Huang et al., 19 Nov 2024). Best constants are computed via weighted integration by parts, parameter optimization in spherical coordinates, and explicit solution of accompanying variational equations. These inequalities, when sharpened, provide essential tools for PDE theory, especially in the stability analysis of axisymmetric solutions and nonlocal operator estimates.

6. Radiance Field Rendering: Deformable and Universal Beta Kernels

Recent advances in explicit radiance field rendering introduce deformable and universal Beta splatting methods (DBS, UBS), leveraging N-dimensional anisotropic Beta kernels for geometry and color modeling. Key formulation:

$$\mathscr{B}(x; b) = (1 - x)^{\beta(b)}, \quad \beta(b) = 4e^b$$

with $x \in [0, 1]$ and $b$ a learnable shape parameter controlling frequency content (Liu et al., 27 Jan 2025, Liu et al., 30 Sep 2025). These primitives permit:

• Adaptive smoothing (flat to sharp features).
• Bounded support, eliminating ad hoc Gaussian truncation.
• Per-dimension control to decouple spatial, angular, and temporal correlations.

The UBS framework introduces hierarchical conditioning via per-dimension Beta modulation, explicit spatial-orthogonal Cholesky parameterization, and ensures backward compatibility with Gaussian kernels (by setting all $b_i=0$). Universal Beta Splatting outperforms prior Gaussian-splatting methods across static, view-dependent, and dynamic scene benchmarks, with parameter-efficiency and interpretable decomposition of scene properties (surface, texture, specularity, dynamics) (Liu et al., 30 Sep 2025).

7. Applications, Implications, and Future Directions

Anisotropic Beta kernels underpin a wide spectrum of advanced mathematical and computational techniques:

• Compact and adaptive density estimation on bounded domains, with efficient modeling of multimodal or correlated data.
• Spectral analysis and eigenvalue asymptotics in operator theory, particularly relevant for problems in superconductivity and non-Fermi liquids.
• Nonlocal, fractional PDEs where directional behavior governs dynamics, regularity, and physical interpretation.
• Explicit radiance field rendering in computer graphics and vision, enabling scalable, interpretable, and real-time photorealism.

The literature indicates that the flexibility and generality of the N-dimensional anisotropic Beta kernel framework afford expressive modeling power and numerical efficiency, while their mathematical properties continue to guide developments across both pure and applied contexts. Interpretability (e.g., decomposition of learned kernels into spatial/angular/temporal features), consistency guarantees (e.g., kernel-agnostic MCMC via opacity regularization), and empirical superiority position Beta kernels as a foundational tool in modern computational science.