q-Gaussian Kernels: Theory & Applications

• q-Gaussian kernels are kernel functions derived from the q-Gaussian distribution that generalize traditional Gaussian kernels with tunable tail properties.
• They are widely used in nonextensive statistics, machine learning, physics, and signal processing to flexibly model heavy-tailed, compact, or intermediate regimes.
• q-Gaussian kernels support robust stochastic optimization and edge detection through adaptive smoothing, enhanced uncertainty control, and efficient numerical transformations.

A q-Gaussian kernel is a function derived from the q-Gaussian distribution—a parametric deformation of the classical Gaussian obtained by replacing the exponential function with a power function of exponent $1/(1-q)$ (%%%%1%%%%)—that generalizes traditional kernel structures to reflect nonextensive statistical and entropy properties. q-Gaussian kernels are central to diverse areas, including noncommutative probability, stochastic optimization, statistical mechanics, signal and image processing, quantum mechanics, and machine learning. The kernels are typically parameterized by a real-valued $q$ parameter that controls the shape, support, and tail behavior, enabling the modeling of compact, heavy-tailed, or intermediate regimes. Their properties and analytical structure subsume and generalize several classical kernels, such as Gaussian, uniform, and Cauchy, and imbue new flexibility in algorithms and physical models.

1. Mathematical Foundations of the q-Gaussian Kernel

The q-Gaussian kernel, %%%%3%%%%, is defined by the formula

$$G_q(x; \mu, \sigma) = \frac{1}{Z_q \sigma} \left[ 1 - (1-q) \frac{(x-\mu)^2}{\sigma^2} \right]_+^{1/(1-q)}$$

where $Z_q$ is a normalization constant, $\mu$ is the mean, $\sigma$ is the scale, and $[\,\cdot\,]_+$ restricts to nonnegative arguments, enforcing compact support for $q < 1$ and heavy tails for $q > 1$ (Lima et al., 2017).

When $q = 1$, the kernel reduces to the standard Gaussian

$$G_1(x) = \frac{1}{\sqrt{2\pi}\sigma} \exp\left( -\frac{(x-\mu)^2}{2\sigma^2} \right)$$

For $1 < q < 3$, the tails decay polynomially as a Student-t distribution, and for $q < 1$, the support is strictly bounded (Matsuzoe et al., 2020, Lima et al., 2017). The q-exponential, $\exp_q(x) = [1 + (1-q)x]^{1/(1-q)}$, underpins the functional form and admits generalization to higher dimensions and covariance structures.

2. Free Probability, Orthogonal Polynomials, and Divisibility

In noncommutative probability theory, q-Gaussians arise as distributions of self-adjoint operators $\{X = c(f) + c^*(f)\}$ on q-deformed Fock spaces, where creation and annihilation operators obey the q-canonical commutation relation,

$c(f) c^*(g) - q c^*(g) c(f) = (f|g)_q \cdot 1$

The spectral measure is characterized by q-Hermite polynomials obeying

$$x H_n(x|q) = H_{n+1}(x|q) + \frac{1-q^n}{1-q} H_{n-1}(x|q)$$

(Anshelevich et al., 2010). The density on $[-2, 2]$ can be written via Chebyshev polynomials of the second kind:

$$f_q(x) = \frac{1}{2\pi} \sqrt{4 - x^2} \sum_{k=1}^{\infty} (-1)^{k-1} q^{k(k-1)/2} \mathcal{U}_{2k-2}(x/2)$$

Critically, for $q\in[0, 1]$, all q-Gaussian measures are freely infinitely divisible, meaning their distribution supports the existence of free convolution semigroups and Lévy processes (Anshelevich et al., 2010).

3. Entropy Maximization and Statistical Mechanics

q-Gaussian distributions are maximizers of the nonextensive Tsallis entropy,

$$S_q[p(x)] = \frac{1}{q-1} \left( 1 - \int p(x)^q dx \right)$$

subject to normalization and a fixed generalized variance or expectation. This produces the canonical q-Gaussian kernel under constraints and introduces tunable deviation from the Boltzmann–Gibbs-Shannon regime (Lima et al., 2017, Vignat et al., 2010).

The information geometric structure of the q-Gaussian family is formalized via escort expectations and refined q-logarithmic functions,

$\ln_q(t) = \frac{t^{1-q} - 1}{1-q}$

with the Riemannian metric on the statistical manifold given by

$g_{q,1}(\xi) = \frac{1}{\sigma^2}$

and modifications for non-trivial escort gauges, leading to gauge freedom in entropy definitions and relative entropies (Matsuzoe et al., 2020).

4. Analytical and Numerical Properties: Fourier Analysis and Kernel Transformations

The q-Gaussian kernel supports analytical and numerical evaluation of its Fourier transform:

• For $q = 1$, the transform is Gaussian.
• For $q < 1$, it involves confluent hypergeometric functions or beta distributions.
• For $q > 1$, the transform is given in terms of Whittaker functions or modified Bessel functions.

The Heisenberg uncertainty relationship generalizes to q-Gaussian kernels,

$$4\pi \|x G_{1,q}\|_2 \cdot \|y \mathcal{F}(G_{1,q})\|_2 \geq \|G_{1,q}\|_2^2$$

where the space window increases and the cutoff frequency decreases with increasing $q$ (Rodrigues et al., 2016). In higher dimensions or anisotropic settings, only numerical approximation is tractable.

Transformations between q-Gaussians with differing $q$ parameters are governed by a groupoid structure, with probability-preserving mappings constructed via inverses of the regularized incomplete Beta function, enabling connections between normalizable kernels across the range $q \in [-\infty, 3)$ (Tateishi et al., 2013).

5. Algorithmic Applications: Stochastic Optimization and Machine Learning

q-Gaussian kernels underpin enhanced smoothed functional (SF) algorithms for stochastic optimization. As kernel elements in convolutional SF methods, they afford tunable control over tail behavior and smoothing, with the smoothed gradient estimator,

$$D_{q,\beta}[J(\theta)] = \frac{2}{\beta(3-q)} \mathbb{E}_{G_q} \left( \frac{\eta J(\theta + \beta\eta)}{1 - \frac{1-q}{3-q} \|\eta\|^2} \right)$$

ensuring convergence to scaled gradients and flexibility over escape from local minima. Two-timescale algorithms utilize one- and two-sided q-Gaussian smoothing for robust optimization, with performance dependent on $q$ and the smoothing parameter $\beta$ (Ghoshdastidar et al., 2012, Ghoshdastidar et al., 2012). Kernel methods using q-Gaussian functions generalize similarity metrics for machine learning, regression, and classification in the presence of heavy tails, compact supports, or non-Gaussian noise (Lima et al., 2017).

6. Physical Models and Quantum Systems

q-Gaussian kernels are relevant for quantum potentials, where ground state wavefunctions in $D$ dimensions take the form,

$\psi(r) = C [1 - (q-1)\beta r^2]^{1/[2(q-1)]}$

In configuration space, potentials derived to support such ground states are,

$$V(r) = \frac{\beta}{2} \frac{-D + \beta r^2 [D(q-1) + 3 - 2q]}{[1 - (q-1)\beta r^2]^2}$$

Confinement (infinite wall at a cutoff boundary) and Coulomb-like asymptotic behavior are modulated by $q$, with $q \rightarrow 1$ recovering harmonic oscillator and $q = 0$ yielding semicircular law (Vignat et al., 2010).

In continuous-variable quantum kernel machine learning, the quantum kernel is written as the product of a Gaussian and an algebraic function (polynomial of stellar rank $n$),

$k(x, x') = G(x, x') \cdot A(x, x')$

where increasing stellar rank yields higher expressivity and quantum–classical separation, matched by q-Gaussian-like behavior in practice (Henderson et al., 11 Jan 2024). Proper tuning of kernel bandwidth and stellar rank allows explicit trade-off between generalization and discrimination.

7. Signal and Image Processing, and Diffusion Models

q-Gaussian kernels generalize classical kernels such as Gaussian, Laplacian, and Cauchy in edge detection, noise reduction, and feature extraction. For instance, edge detection via Difference of Gaussians (DoG) using q-Gaussian kernels,

$$\text{DoG}_q(\sigma_1, \sigma_2, q) = G_q(\sigma_1, q) - G_q(\sigma_2, q)$$

demonstrates enhanced adaptivity and richer detail in extracted edges, benefiting from tunable tail and compactness properties (Assirati et al., 2013).

In diffusive systems, two classes of q-Gaussian distributions,

1. $P_q(x,a) = \sqrt{a} A_q e_q(-a x^2)$
2. $G_q(x,a) = B_q [e_q(-x^2)]^a$ with time-dependent standard deviation $\sigma_q(t)$, provide solutions to q-deformed diffusion and diffusion-decay equations. The width of the density saturates to a finite value, and the effective diffusion and decay coefficients depend on both position and time (Chung et al., 19 May 2025).

Table: q-Gaussian Kernel Variants and Key Applications

q-Gaussian Variant	Analytical Behavior	Application Contexts
$q < 1$ (compact support)	Finite, bounded domain	Classical processes, imaging, bounded errors
$q = 1$ (Gaussian)	Exponential tails	Traditional statistics, thermodynamics
$q > 1$ (heavy tails)	Power-law decay, infinite support	Nonextensive statistics, finance, turbulence

The choice of $q$ parameter influences localization, tail weight, and stability under convolution; it allows practitioners to select kernels appropriate for non-Gaussian, correlated, or anomalous statistics.

8. Summary and Perspectives

q-Gaussian kernels generalize classical kernels by introducing parametric control via the $q$ index, encompassing Gaussian, compactly supported, and heavy-tailed regimes. Their roots in nonextensive entropy maximization, orthogonal polynomial structures, free probability, and information geometry provide a rich analytic and probabilistic foundation. Algorithmic implementations utilize their flexible shape for robust stochastic optimization, kernel-based learning, and edge detection, while their role in physical and quantum models unifies classical and noncommutative perspectives. Ongoing work explores discrete and continuous deformations for modeling processes in finite regions, characteristic function computation for uncertainties, and transformation groupoids for connecting diverse q-Gaussian kernel instances. The theory offers a systematic framework for tuning kernel behavior across a wide spectrum of statistical, physical, and informational systems.