Fractional Gaussian Filters (FGFs)

• Fractional Gaussian Filters are a family of signal processing constructs that generalize the classical Gaussian kernel using fractional-order derivatives or integrals.
• They accurately model stochastic fields and colored noise through techniques like Karhunen–Loève expansions and fractional Laplacian discretizations.
• In deep learning, FGFs reduce convolutional parameters while providing tunable frequency selectivity, enhancing CNN efficiency and performance.

Fractional Gaussian Filters (FGFs) denote a family of signal processing and stochastic modeling constructs in which the classical Gaussian kernel is generalized by the application of fractional-order differential or integral operators. This yields filters, random fields, or convolution kernels exhibiting tunable frequency or regularity properties. FGFs arise in both the theory of stochastic processes (especially fractional Gaussian fields as covariance structures on domains and boundaries), and as parameter-efficient architectures for convolutional neural networks (CNNs), as well as in exact representations of colored noise via linear fractional systems.

1. Mathematical Foundations of Fractional Gaussian Filters

FGFs in the continuous-time setting are characterized by the action of fractional derivatives or integrals on the Gaussian kernel, or, in the stochastic case, by prescribing random fields whose covariance operator is a fractional power of an elliptic operator such as the Laplacian. For instance, a general (isotropic) FGF on $\Omega \subset \mathbb{R}^d$ with order $s>0$ has covariance $C_s(x,y)$ given by the inverse of the fractional Laplacian:

$$C_s(x,y) = \mathbb{E}[\phi(x)\phi(y)] = (-\Delta)^{-s} \;\text{(as operator)},$$

where $\phi$ is a mean-zero Gaussian field. The spectral representation involves the eigenpairs $\{v_j,\lambda_j\}$ of $(-\Delta)^s$ with domain-dependent boundary conditions, yielding:

$$\phi(x) = \sum_{j=1}^\infty \lambda_j^{-1/2} X_j v_j(x), \quad X_j \sim \mathcal{N}(0,1),$$

with almost sure convergence in $H^{s'}$ for $s' < s - d/2$ (Nitti et al., 2023).

For boundary-dominated phenomena, as in acoustic impedance problems, FGFs are constructed on a Lipschitz boundary $s>0$0, with the covariance

$s>0$1

where $s>0$2 is the Laplace-Beltrami eigenbasis and $s>0$3 the fractional index (Karabash, 21 Jan 2026).

In signal processing, the deterministic FGF is conceptualized as the response of a fractional-order linear system driven by white noise, with transfer function $s>0$4—often defined via the fractional moments of the system's power spectral density (PSD) (Cottone et al., 2013).

2. FGFs in Stochastic Field Theory and Boundary Value Problems

The rigorous construction of FGFs as random fields crucially depends on the operator spectrum and the fractional index:

• On compact Lipschitz boundaries $s>0$5, the FGF of order $s>0$6 is realized via the Karhunen–Loève expansion:

$s>0$7

with $s>0$8 i.i.d. standard normals (Karabash, 21 Jan 2026).

• Almost sure convergence in Sobolev space $s>0$9 requires $C_s(x,y)$0 due to the Weyl-type eigenvalue asymptotics $C_s(x,y)$1.
• The covariance operator $C_s(x,y)$2, and locally $C_s(x,y)$3 for $C_s(x,y)$4 in an averaged sense.
• These fields generate random but well-posed acoustic boundary conditions when used as generalized impedance coefficients $C_s(x,y)$5. For $C_s(x,y)$6, the operator has compact resolvent part and discrete spectrum almost surely (Karabash, 21 Jan 2026).

In higher-dimensional domains for $C_s(x,y)$7, the scaling limits and discretization of FGFs are studied through lattice approximations of $C_s(x,y)$8. For appropriate relationships ($C_s(x,y)$9) between domain dimension and order, convergence of the maximum and tightness in Besov and Hölder topologies are proven (Nitti et al., 2023).

3. Fractional Gaussian Filters in Stationary Colored Noise and Fractional SDEs

FGFs can represent stationary colored Gaussian processes as outputs of linear fractional stochastic differential equations:

$$C_s(x,y) = \mathbb{E}[\phi(x)\phi(y)] = (-\Delta)^{-s} \;\text{(as operator)},$$0

where $$C_s(x,y) = \mathbb{E}[\phi(x)\phi(y)] = (-\Delta)^{-s} \;\text{(as operator)},$$1 is the Riesz fractional integral of order $$C_s(x,y) = \mathbb{E}[\phi(x)\phi(y)] = (-\Delta)^{-s} \;\text{(as operator)},$$2, $$C_s(x,y) = \mathbb{E}[\phi(x)\phi(y)] = (-\Delta)^{-s} \;\text{(as operator)},$$3, and $$C_s(x,y) = \mathbb{E}[\phi(x)\phi(y)] = (-\Delta)^{-s} \;\text{(as operator)},$$4 is white noise. The weights $$C_s(x,y) = \mathbb{E}[\phi(x)\phi(y)] = (-\Delta)^{-s} \;\text{(as operator)},$$5 are related via fractional moments of the target spectral density $$C_s(x,y) = \mathbb{E}[\phi(x)\phi(y)] = (-\Delta)^{-s} \;\text{(as operator)},$$6 (Cottone et al., 2013):

$$C_s(x,y) = \mathbb{E}[\phi(x)\phi(y)] = (-\Delta)^{-s} \;\text{(as operator)},$$7

with $$C_s(x,y) = \mathbb{E}[\phi(x)\phi(y)] = (-\Delta)^{-s} \;\text{(as operator)},$$8. By superposing fractional Brownian motions $$C_s(x,y) = \mathbb{E}[\phi(x)\phi(y)] = (-\Delta)^{-s} \;\text{(as operator)},$$9 of varying Hurst index, the FGF realizes any prescribed PSD exactly, up to numerical error from discretization along the Mellin–contour. This approach requires neither ad-hoc optimization nor fitting as in ARMA filtering, and is fully characterized by closed-form weights.

4. Fractional Gaussian Filters in Deep Learning Architectures

Parametric Fractional Gaussian Filters provide substantial parameter reduction in CNNs by replacing standard convolutional kernels with fractional-derivative Gaussians:

• In the “FracSRF” approach, each filter is parameterized by $\phi$0 (scalar weight), fractional orders $\phi$1 and $\phi$2, and optionally scale $\phi$3. The filter is $\phi$4, with

$\phi$5

interpolating between Hermite-Gaussian derivatives at integer orders (Saldanha et al., 2021).

• The frequency response is $\phi$6, with band-pass center $\phi$7 controllable by learned parameters.
• The “FGFP” framework further reduces parameters to seven per kernel: fractional differentiation orders $\phi$8 in spatial and input-channel axes, kernel centers $\phi$9, and $\{v_j,\lambda_j\}$0. Grünwald–Letnikov fractional differences are used for efficient numerical implementation (Tu et al., 30 Jul 2025).

Empirical evidence demonstrates that FracSRF and FGFP models achieve 3× parameter reduction with ≤1% accuracy drop on benchmarks such as CIFAR-10/100 and ImageNet, with superior performance under strong compression on high-frequency tasks (Saldanha et al., 2021, Tu et al., 30 Jul 2025).

5. Numerical and Practical Implementation Strategies

The construction of FGFs in both theory and engineering involves specific numerical procedures:

• For stochastic simulation, discrete approximations of fractional Laplacians ($\{v_j,\lambda_j\}$1), mollification, and Krylov solvers using FFTs are employed, with error $\{v_j,\lambda_j\}$2 for solution regularity $\{v_j,\lambda_j\}$3 and mesh width $\{v_j,\lambda_j\}$4 (Nitti et al., 2023).
• In convolutional neural networks, Hermite polynomial recursion is used for integer-order Gaussian derivatives, augmented by on-the-fly linear interpolation for fractional orders, with gradient backpropagation stabilized by parameter clipping and regularization (Saldanha et al., 2021).
• Grünwald–Letnikov approximations reduce the computational burden of fractional-order derivatives to three terms per axis, enabling per-layer replacement of standard convolution with seven-parameter FGFs, and adaptive unstructured pruning further compresses the remaining weights (Tu et al., 30 Jul 2025).
• For colored noise systems, discretization of Mellin-inverse integrals with respect to the imaginary axis, and evaluation of fractional integrals via either Fourier multipliers or time-domain convolution, ensures precise match to target PSDs. The holomorphic properties of the integrand support rapid numerical convergence (Cottone et al., 2013).

6. Significance, Applications, and Comparative Advantages

FGFs encode a wide range of smoothness and frequency selectivity unavailable to standard kernels:

• In stochastic PDE theory, FGFs underpin randomized dissipative boundary conditions with rigorously controllable regularity, spectrum, and maximal dissipativity on nonsmooth domains (Karabash, 21 Jan 2026).
• For simulation of fractional Gaussian fields, spectral and finite-difference approaches provide efficient, accurate methods with convergence guarantees in optimal function spaces (Nitti et al., 2023).
• In deep learning, FGFs deliver crucial benefits:
  • Parameter efficiency: Reduction from $\{v_j,\lambda_j\}$5 (or higher) to 3–7 parameters per kernel (Tu et al., 30 Jul 2025, Saldanha et al., 2021).
  • Flexibility: Continuous control over filter frequencies by learning fractional orders enables precise frequency-selective convolution, outperforming integer-order fixed bases for information-rich or texture-reliant datasets.
  • Data and computational efficiency: Empirical benchmarks demonstrate substantially slower accuracy degradation with training set subsampling and state-of-the-art accuracy/compression trade-offs when combined with advanced pruning (Saldanha et al., 2021, Tu et al., 30 Jul 2025).
  • Exact spectrum shaping: In signal processing, FGFs can realize any target colored noise spectrum without iterative fitting or ARMA model selection (Cottone et al., 2013).

These features collectively establish FGFs as foundational tools in functional analysis, stochastic processes, and parameter-efficient neural network design.