Axial Symmetry in Multivariate Distributions

• Axial symmetry in multivariate distributions is the property where a random vector’s distribution remains invariant under reflection across a fixed axis, bridging central and spherical symmetry.
• This symmetry concept underpins nonparametric inference, directional statistics, and morphometric analysis by offering group invariance properties that guide testing and estimation methods.
• Recent statistical methodologies—including covariance-based tests, spiked alternatives, and optimal transport frameworks—provide robust techniques for the consistent identification and estimation of symmetry axes.

Axial symmetry in multivariate distributions is the property that a random vector is invariant in distribution under reflection through a fixed axis. This construct mediates between central symmetry (invariance under inversion about a point) and spherical symmetry (invariance under all orthogonal transformations), and induces specific group invariance properties. Axial symmetry finds direct applications in nonparametric inference, directional statistics, and morphometric analysis, and prompts both theoretical and computational study on identifiability, testing, and estimation of symmetry directions.

1. Formal Definition and Group-Theoretic Structure

Let $X$ be a random vector in $\mathbb{R}^d$ with finite mean $\mu = \mathbb{E}[X]$. For any unit vector $u \in \mathbb{S}^{d-1}$, define the reflection operator

$R_u = 2uu^\top - I_d,$

which reflects $X$ across the hyperplane orthogonal to $u$. The distribution of $X$ is axially symmetric about the axis spanned by $u$ if

$X - \mu \overset{d}{=} R_u(X - \mu).$

Spherical symmetry corresponds to invariance under all orthogonal matrices, i.e., $AX \overset{d}{=} X$ for all $A \in O(d)$, and thus implies axial symmetry for every direction. Group-theoretically, axial symmetry is invariance under the two-point subgroup $\{I, R_u\}$ of $O(d)$, while spherical symmetry is invariance under the full orthogonal group (Cholaquidis et al., 24 Dec 2025, Huang et al., 2023).

On the unit sphere $\mathbb{S}^{p-1}$, "axial symmetry" is often referred to as antipodal symmetry: $$P(X \in A) = P(X \in -A) \quad \forall A \subset \mathbb{S}^{p-1}.$$ Rotational axial symmetry further requires invariance under all $O \in O(p)$ fixing some pole direction $\theta$ (Cutting et al., 2019).

2. Identifiability of Axes via Random Projections

For $d = 2$, if $X$ is not spherically symmetric and satisfies a Carleman condition ensuring projections are moment-determined, the axes of axial symmetry can be identified as follows. For independent random directions $h_1, h_2 \sim \mathrm{Uniform}(\mathbb{S}^1)$, consider the system: $$h_j^\top X \overset{d}{=} h_j^\top R_u X, \quad j=1,2.$$ The set of $u$ satisfying both equations with probability one recovers the true symmetry axes $\mathcal{U}$: $$\left\{ u \in \mathbb{S}^1 : h_j^\top X \overset{d}{=} h_j^\top R_u X \text{ for } j=1,2 \right\} = \mathcal{U}.$$ Each single projection rules out almost all "false" axes, and two generic directions suffice to isolate the true axes almost surely (Cholaquidis et al., 24 Dec 2025).

A conjectured extension to $d$-dimensional space posits that agreement on $d$ independent random projections likewise identifies $\mathcal{U}$, but technical obstacles arise when the indexation of false axes becomes uncountable, limiting direct application of Fubini's theorem.

3. Statistical Testing and Distribution-Free Frameworks

Statistical methodology for axial symmetry is well-developed for observations on $\mathbb{S}^{p-1}$ and in $\mathbb{R}^d$:

• Covariance-based axial tests: Test construction often exploits the sample covariance matrix $S_n$. For specified axis $\theta$, optimal Le Cam tests for uniformity involve the central sequence

$$\Delta_n = \sqrt{n} (p\,\theta^\top S_n \theta - 1), \quad \Gamma_p = \frac{2(p-1)}{p+2},$$

yielding a test statistic

$T_n = \Delta_n / \sqrt{\Gamma_p},$

which is asymptotically normal under the null (Cutting et al., 2019).

• Bingham and single-spiked tests: For unspecified symmetry axes, the Bingham test computes the quadratic form $$Q_n = np(p+2)/2 \cdot (\operatorname{tr}[S_n^2] - 1/p)$$, which is omnibus for general antipodal asymmetry but suboptimal against single-spiked alternatives. Eigenvalue-based "spiked" tests examine the extremal eigenvalues of $S_n$, yielding asymptotically optimal power when the alternative is concentrated along one axis.
• Optimal transport-based distribution-free tests: The generalized sign and signed-rank (GWSR) tests leverage group actions and OT to produce finite-sample distribution-free statistics for axial symmetry in any dimension. For axis $u$,

$$T_n^{\rm sign} = \frac{1}{\sqrt{n}} \sum_{i=1}^n \mathrm{sign}(u^\top X_i)$$

is exactly distribution-free under the null $H_0: X \eqd R_u X$. The GWSR test

$$T_n^{\rm GWSR} = \frac{1}{\sqrt{n}} \sum_{i=1}^n \mathrm{sign}(u^\top X_i)\,u^\top R_i$$

achieves Pitman efficiency up to the benchmark Hotelling $T^2$ test, with no loss under normal alternatives (Huang et al., 2023).

4. Consistent Estimation of Symmetry Directions

Nonparametric estimation of symmetry axes in the plane uses empirical approximation of projection discrepancies:

1. Split the data into two equal halves.
2. Sample $k$ random directions $\{h_1, ..., h_k\}$.
3. For each candidate axis $u$, compute plug-in discrepancies

$$\widehat g_{n,h}(u) = \sup_t \left| F^n_{\langle X, h \rangle}(t) - F^n_{\langle R_u X, h \rangle}(t) \right|,$$

averaging over $h_j$.

1. Estimate the symmetry set as

$$\widehat{\mathcal U}_n = \{ u \in \mathbb{S}^1 : \widehat g_n(u) < \epsilon_n \},$$

with $\epsilon_n \to 0$ at a suitable logarithmic rate.

Uniform convergence and Hausdorff consistency follow from empirical process theory. Numerical optimization can be performed by discretizing the sphere and applying peak-finding algorithms such as AMPD (Cholaquidis et al., 24 Dec 2025).

5. Interplay Between the "Size" of Symmetry Sets and Spherical Symmetry

The cardinality or measure of symmetry directions strongly constrains the overall symmetry class:

• In $\mathbb{R}^2$, if infinitely many projections yield identical distributions, the distribution must be spherically symmetric.
• If the set of symmetry axes $\mathcal{U} \subset \mathbb{S}^{d-1}$ has positive surface measure, spherical symmetry ensues automatically. Key structural lemmas include closedness of $\mathcal{U}$ and closure under conjugation, facilitating percolation arguments proving the uniqueness of spherical symmetry in "large" axial symmetry sets (Cholaquidis et al., 24 Dec 2025).

6. Numerical Illustrations and Applications

Simulation studies for bivariate normal and uniform distributions demonstrate rapid and accurate identification of symmetry axes, with detection rates approaching 100% as sample size and number of projections increase. In medical imaging, estimation of axial symmetry in anatomical structures via projection methods effectively recovers physical symmetry axes (e.g., in chest X-ray lung masks) (Cholaquidis et al., 24 Dec 2025).

On spheres, Monte Carlo evaluation of the covariance-based and spiked tests reveals that single-spiked tests outperform omnibus Bingham tests for alternatives concentrated along one axis, especially in higher dimensions (Cutting et al., 2019). OT-based tests provide a distribution-free alternative, with exact level and tractable power in finite samples (Huang et al., 2023).

7. Extensions and Open Directions

Extending projection-based identifiability and consistent estimation to higher dimensions ($d > 2$) remains nontrivial, with measure-theoretic obstacles related to the topology of null sets. Alternative projection modalities (e.g., projections onto subspaces, nonlinear feature maps) may capture more general group-invariance patterns. Optimization procedures for axis estimation on spheres in high dimensions may benefit from gradient-based and manifold-search techniques. OT-based confidence cones for the axis direction offer distribution-free inference regions for symmetry axes (Cholaquidis et al., 24 Dec 2025, Huang et al., 2023).

A plausible implication is that generalizations to other symmetry groups using similar random-projection or transport-based methodology could unify symmetry testing and estimation across diverse domains.