Symmetry-Aware Toolkit for Finite Mixtures

• The paper introduces a symmetry-aware finite mixtures approach that incorporates group invariance and geometric alignment to improve mixture estimation.
• It employs invariant feature extraction, bispectral embedding, and optimal transport methods to rigorously compare and test mixture models.
• The framework enhances hypothesis testing by controlling type-I error and boosting power against skew alternatives in both parametric and semiparametric settings.

A symmetry-aware toolkit for finite mixtures comprises statistical and computational methods that explicitly recognize and exploit group or geometric symmetries in the modeling and inference of mixture distributions. Such toolkits offer principled means of hypothesis testing, parameter estimation, and geometric alignment in mixture models by incorporating invariance properties, group actions, and symmetry-constrained estimation criteria. Their scope covers parametric and semiparametric mixture modeling, distributional testing for symmetry or skewness, invariant optimal transport for dataset comparison, symmetry-aware moments and quotient-space estimation, and nonparametric shape-constrained methods. The following sections detail core principles, canonical models, inference algorithms, theoretical guarantees, optimization recipes, and practical implementation for symmetry-aware finite mixture analysis.

1. Symmetry-Constrained Parametric Mixture Models

Univariate finite mixtures can be formulated so that component weights and locations obey prescribed symmetries. For mixtures of equispaced normals, as in Bacci and Bartolucci (Bacci et al., 2012), consider a $k$-component mixture

$$f(x; \theta, \sigma, \boldsymbol\pi) = \sum_{j=1}^k \pi_j\,\phi\left(x ;\theta + \sigma \delta_j,\,\sigma^2\right),$$

where each mean is located on an equispaced grid $\delta_j$ on $[-1,1]$, with $\delta_j = -1 + 2(j-1)/(k-1)$. The unconstrained model allows arbitrary $\boldsymbol\pi$, while the symmetry-constrained (“null”) model imposes $\pi_j = \pi_{k+1-j}$ for $j = 1,\ldots, \lfloor k/2 \rfloor$, i.e., pairing weights for components mirrored about the putative center $\theta$. The symmetry constraint induces a nested hypothesis test structure, enabling likelihood-based assessment of symmetry versus skewness in the generative distribution.

2. Group-Invariant Feature Extraction and Transport for Mixtures

For mixtures composed of structured data objects subject to group actions (e.g., images under rotations, molecular graphs under permutations), symmetry can be accommodated by embedding each mixture atom using group invariants. The bispectrum, computed via group Fourier transforms and irreducible representation theory, provides a complete invariant under the group action. Each signal $f:G\to\mathbb{C}$ yields

$$B_{\rho_i,\rho_j} = \hat f_{\rho_i}\,\hat f_{\rho_j}\,\hat f_{\rho_i\rho_j}^{\dagger},$$

where $\rho_i, \rho_j$ index irreps of $G$ and $\hat f_\rho$ denotes the GFT coefficients. Dataset comparison is performed in the space of bispectral features, which are indifferent to nuisance variability but discriminative of intrinsic structure.

Symmetry-aware optimal transport aligns two mixtures $\mu = \sum_i p_i\,\delta_{x^{(i)}}$, $\nu = \sum_j q_j\,\delta_{y^{(j)}}$ by solving the Kantorovich program over bispectrum-induced costs: $$\min_{\Gamma\geq 0} \langle C, \Gamma\rangle \quad\text{s.t.}\quad \Gamma \mathbf{1}_m = p,\quad \Gamma^\top \mathbf{1}_n = q,$$ with $C_{ij} = d(\varphi_i,\psi_j)$ for bispectral embeddings $\varphi_i, \psi_j$. Entropic regularization ($\epsilon$-Sinkhorn) yields scalable solutions in high dimensions (Ma et al., 25 Sep 2025).

3. Symmetry-Aware Statistical Inference and Hypothesis Testing

Testing for symmetry in mixture data hinges on constrained vs. unconstrained likelihood maximization. Given monotonic log-concave or symmetric error components, semiparametric EM (SEM) and nonparametric maximum likelihood estimation (NPMLE) routines can be adapted to restrict component shapes accordingly (Pu et al., 2017). For equispaced normal mixtures, compute the log-likelihoods $\ell_1$, $\ell_0$ for unconstrained and symmetry-constrained parameter sets via Expectation-Maximisation. The likelihood ratio statistic

$\Lambda = -2\left[\ell_0 - \ell_1\right]$

is asymptotically $\chi^2$-distributed under the null hypothesis, with degrees of freedom equal to the number of independent symmetry constraints.

Model selection for $k$ (number of components) is performed by minimizing penalized log-likelihoods such as AIC or BIC: $$\mathrm{AIC} = -2\ell + 2p, \quad \mathrm{BIC} = -2\ell + p\log n,$$ where $p$ enumerates free parameters in each candidate model. Simulation studies confirm that BIC-based mixture symmetry tests maintain type-I error near nominal levels and exceed moment-based skewness tests in empirical power, especially under moderate skew alternatives (Bacci et al., 2012).

4. Quotient-Space Estimation and Orbit-Matching

When mixture components are defined only up to a finite symmetry group $G$, the correct estimand is the multiset of orbits in the quotient $\Theta/G$, rather than ordered raw parameters (Mallik, 6 Nov 2025). The Reynolds projector offers a systematic means to build $G$-invariant tensor moments: $$R_m[T] = \frac{1}{|G|}\sum_{g\in G} (g\cdot)^{\otimes m}T$$ which parameterize mixtures as convex combinations in $G$-invariant coordinates. The estimation problem is then cast as multiset assignment under the Hausdorff or bottleneck metrics: $$d_H(\mathcal{A},\mathcal{B}) = \max\big(\max_i\min_j d(\alpha_i, \beta_j), \max_j\min_i d(\alpha_i,\beta_j)\big),$$ and solved via alternating convex programming and combinatorial assignment algorithms.

Generalized method of moments (GMM) estimation proceeds in two steps: empirical projection of sample moments into invariant coordinates, and minimization of Euclidean distance between the empirical invariant stack and convex mixtures. Theoretical guarantees include global and local identifiability, asymptotic normality on the quotient, and minimax-optimal Poly-LAN rates $n^{-1/D}$, where $D$ is the lowest order of nonzero invariant curvature.

5. Semiparametric Estimation with Symmetric, Log-Concave and Shape-Constrained Errors

Component symmetry is especially relevant when the error densities are nonparametric but believed symmetric and unimodal. For mixtures of log-concave symmetric densities,

$g(x) = \sum_{j=1}^K \pi_j\,g_j(x-\mu_j),$

with each $g_j(u) = g_j(-u)$ and $\log g_j$ concave. NPMLE for a single component $h$ on $\mathbb{R}_+$ is formulated as

$$\hat\psi_m = \arg\min_{\psi\,\text{concave,nonincreasing}}\left(-\sum_{i=1}^m\psi(y_i) + m\int_0^\infty e^{\psi(x)}dx \right),$$

yielding piecewise-linear monotone estimators. The semiparametric EM algorithm alternates between E-steps updating posterior probabilities and M-steps updating $\pi_j$, $\mu_j$, and $h_j$ via weighted NPMLE on absolute residuals. This has monotonic log-likelihood convergence, and, under regularity assumptions, consistent and asymptotically efficient estimation (Pu et al., 2017).

6. Practical Guidelines, Implementation Recipes, and Performance Analysis

Recommended best practices include:

• Fit unconstrained and symmetry-constrained models for candidate odd $k$ using EM; record likelihoods and penalized criteria.
• Select $k$ using BIC for strict type-I error control; AIC for higher power (with inflation of type-I error).
• For group-invariant mixture comparisons, embed atoms via bispectral features, normalize, and solve OT using Sinkhorn with tailored regularization.
• Apply robust means and contamination-tolerant methods in orbit-space estimation to mitigate the impact of outliers or distributional contamination (Mallik, 6 Nov 2025).
• For model selection in orbit-invariant settings, threshold empirical residuals to recover the true $K$ under finite-sample concentration.
• In empirical studies, symmetry-aware mixture tests outperform classical skewness-based methods under skew alternatives and retain correct size under symmetry (Bacci et al., 2012).
• For semiparametric settings, initialization via Gaussian mixture parameters and shape regularization on mixing weights improves numerical stability.

Tables summarizing method comparison (mixture-based LR test, classical skewness, BIC/AIC model selection), error rates, and simulation results appear in the cited literature and confirm the empirical advantages of symmetry-aware approaches.

Method	Control of Type-I Error	Power Under Skewness
BIC-based mixture LR	Near-nominal	Superior to $b_1$
AIC-based mixture LR	Inflated	Highest
Classical skewness ($b_1$)	Near-nominal	Least

This framework enables a rigorous, symmetry-respecting statistical treatment of finite mixture models across a wide range of parametric, semiparametric, and structured-data domains, with proven theoretical, algorithmic, and empirical support.