Radial Cost Functions in Theory & Applications

• Radial cost functions are defined as functions that depend only on the distance between points, capturing isotropy and facilitating analysis in diverse mathematical and computational settings.
• They yield improved functional inequalities and regularity estimates in PDEs by leveraging radial symmetry to relax weight constraints and enhance embedding results.
• In optimal transport and numerical methods, radial cost functions enable efficient modeling and error analysis, driving innovations in statistical estimation, machine learning, and neural network approximations.

A radial cost function is a real-valued function on Euclidean space or a more general metric space that depends only on the distance between points—that is, for $x, y$ in $\mathbb{R}^d$, a radial cost is of the form $c(x, y) = h(|x - y|)$ for some function $h: [0, \infty) \to [0, \infty)$. These cost structures arise across optimal transport, analysis, probability, harmonic analysis, and computational sciences, enabling modeling and estimation in settings where rotational invariance or underlying isotropy is present.

1. Mathematical Formulation and Function Space Structure

Radial cost functions are defined via $c(x, y) = h(|x-y|)$, where $h$ captures the cost growth with spatial separation. Common choices include $h(r) = r^p$ (Wasserstein-$p$ cost), but the function $h$ may admit more general forms, such as subpolynomial, polynomial, or even superpolynomial growth (e.g., $h(r) = \exp(a r^p) - 1$ for $a > 0$) (Larsson et al., 2023). Such costs inherit radial symmetry, being invariant under orthogonal transformations.

The mathematical treatment of radial functions requires refined function space theory. For example, function spaces tailored for radial functions (or mode-$k$ angular Fourier components) are built using adapted Sobolev and Hankel norms. Specifically, for a function $f_k(r,z)$ corresponding to the $k^\text{th}$ Fourier mode, radial Sobolev spaces $H_{(k)}^m((0,R)\times\mathbb{R};X)$ and Hankel spaces $B_{(k)}^s((0, \infty); X)$ are defined, with smoothness characterized via Bessel-type differentiation and transformation (Groves et al., 14 Mar 2024). Such structure permits the translation of regularity, boundary-value, and distributional properties from Euclidean to radial contexts.

2. Inequalities, Regularity, and Improved Estimates for Radial Functions

Analysis of PDEs and variational problems with radial cost structures benefits from improved functional inequalities. The Caffarelli-Kohn-Nirenberg and weighted trace inequalities, central in regularity theory, admit strictly larger ranges of admissible power weights when restricted to the class of radially symmetric functions (Nápoli et al., 2010). Explicitly, consider the first-order interpolation inequality: $$\|\,|x|^{\gamma} u\|_{L^r(\mathbb{R}^n)} \leq C \|\,|x|^{\alpha} \nabla u\|_{L^p(\mathbb{R}^n)}$$ For general $u$, admissibility of $(\alpha, \gamma, p, r)$ is governed by scaling and positivity constraints on the weight difference $\alpha - \gamma$. For radial $u$, the conditions can be relaxed to

$$\gamma_0 - \alpha \geq (n-1)\left(\frac{1}{q} - \frac{1}{r}\right)$$

permitting some negative weight differences otherwise forbidden in the nonradial case. This provides critical flexibility for estimates in radial models, especially near singularities or at infinity. The gain is leveraged through convolution and fractional integral representation methods, exploiting the radial symmetry and associated Young's inequality on $(0,\infty)$ equipped with Haar measure (Nápoli et al., 2010).

Weighted trace inequalities—and their sharp versions for radial data—similarly improve, leading to enhanced regularity and embedding results for solutions to boundary-value and obstacle problems with radial cost terms.

3. Radial Cost Functions in Optimal Transport Theory and Concentration

In optimal transport, the cost $c(x, y) = h(|x - y|)$ defines the geometry of the transport problem and the associated Kantorovich-Wasserstein distances. For empirical measures $\mu_N$ approximating a target distribution $\mu$, concentration properties for the optimal transport cost under radial cost functions have been developed (Larsson et al., 2023, Wiesel, 7 Oct 2025).

When $h$ exhibits polynomial local growth and possibly superpolynomial global growth, and $\mu$ satisfies appropriate moment or exponential tail conditions, sharp nonasymptotic concentration inequalities govern the deviation probability for the empirical transport cost $D_f(\mu, \mu_N)$: $$\mathbb{P}(D_f(\mu, \mu_N) > Fx) \leq C \exp(-cN\phi_{\eta}(x)) \mathbf{1}_{\{x \leq A_0\}} + \mathbb{P}(M_1(\mu_N;G) - M_1(\mu;G) > x)$$ where $\phi_\eta(x)$ encodes the rate, $F$ aggregates moment/tail parameters, and $M_1$ are associated moment functionals (Larsson et al., 2023). Annular decomposition of $\mathbb{R}^d$ and empirical process techniques are key technical innovations, enabling extension of results beyond compactly supported measures and to non-polynomial costs.

For entropy-regularized OT and radial costs with a local Lipschitz property (e.g., $|h(t) - h(t')| \leq C_p (t \vee t')^{p-1}|t - t'|$), sample complexity for convergence of the empirical EOT cost achieves an order $1/\sqrt{n}$ (up to logarithmic terms). The constant depends on explicit measures of the support's metric covering number, adapting the rate to the intrinsic data dimension rather than the ambient dimension. This adaptation is critical in machine learning and high-dimensional statistics, where data may concentrate on low-dimensional sets even in high-dimensional spaces (Wiesel, 7 Oct 2025).

4. Applications and Analytical Implications

Radial cost functions are central in models exhibiting spherical symmetry: astrophysical and fluid mechanical systems, controlled diffusion in radial media, and economic or geometric problems with isotropic costs. The extended admissibility range of power weights in inequalities for radial functions enables more precise control on model solutions' singular or asymptotic behavior (Nápoli et al., 2010). In evolution equations governed by Wasserstein gradient flows, generalized five gradients inequalities for non-quadratic, strictly convex, radially symmetric costs underpin bounded variation (BV) and Sobolev regularity estimates for solutions to Fokker–Planck or continuity-type PDEs, as well as JKO scheme discrete-time variational approximations (Caillet, 2022).

In statistical and computational contexts, sharper OT concentration bounds inform robust estimation, generative modeling, and high-dimensional statistical inference, allowing for precise error control in empirical distributional distance estimation (Larsson et al., 2023, Wiesel, 7 Oct 2025). Entropic regularization further mitigates the curse of dimensionality, with rates adaptive to underlying data complexity.

5. Radial Structure in Harmonic Analysis and Operator Algebras

Positive definite and conditionally negative definite functions on groups, when possessing radial symmetry with respect to combinatorial length or $\ell^p$-length, admit integral representations analogous to classical Bochner and Lévy–Khintchine formulas. For the free group, a normalized $\ell^2$-radial positive definite function $\varphi$ satisfies

$\varphi(g) = \int_{-1}^1 s^{|g|_2} d\mu(s)$

for some unique probability measure $\mu$, and a conditionally negative definite function $\psi$ with $\psi(e) = 0$ admits

$$\psi(g) = \int_{-1}^1 \frac{1 - s^{|g|_2}}{1 - s} d\nu(s)$$

(Chuah et al., 2021). Such characterizations are foundational in noncommutative harmonic analysis, operator algebras, and the paper of Laplacian-type operators on infinite graphs and groups.

These integral documents solidify the analytic structure of "radial cost" in non-Euclidean settings, informing the design of Fourier multipliers, Schur multipliers, and completely positive maps underlying quantum information and operator spectrum analysis.

6. Radial Cost Functions in Numerical Methods and Neural Approximation

Radial cost or basis functions underpin mesh-free numerical discretizations, such as in dynamic programming for optimal control, where radial basis functions and Shepard-type moving least squares approximations provide convergent, contraction-preserving discretizations for the BeLLMan operator. Error bounds scale linearly with the fill distance, and the scheme robustly handles irregular domains and higher dimensions, as shown in test problems spanning simple ODEs to complex geometry (Junge et al., 2014).

Deep convolutional neural networks exhibit provably superior approximation capacity for composite functions $f \circ Q(x) = f(|x|^2)$ (the canonical radial form) compared with shallow or generic fully connected networks in high-dimensional settings (Mao et al., 2021). Explicit parameter scaling is established: deep CNNs require $O(\varepsilon^{-2})$ parameters to achieve $L^\infty$ accuracy $\varepsilon$ for radial target functions, bypassing the $O(\varepsilon^{-(d-1)})$ scaling required for shallow alternatives. Generalization analysis reveals a bias-capacity trade-off as network depth increases.

Finally, in molecular simulation, estimators for radial distribution functions based on force sampling instead of histograms leverage the radial structure to achieve lower variance—distributing each particle pair's contribution across all relevant distances—and integrate analytically the "ideal gas" part, yielding substantial improvements in statistical efficiency (Rotenberg, 2020).

---

Radial cost functions, by reducing multidimensional problems to problems of a single variable (the radius), not only facilitate analysis and computation but also reveal structural properties essential in both theory and applications. Their paper interconnects functional inequalities, regularity and concentration in transport, harmonic analysis, algorithmic design, and modern statistical learning.