Riemannian Geometry for Compound Gaussian Models

• Riemannian geometry for compound Gaussian distributions is a framework that uses Fisher information metrics to define smooth parameter manifolds for models with sample-dependent scaling.
• It enables closed-form geodesic paths, efficient distance computations, and effective gradient-based optimization for change detection and classification tasks.
• The approach leverages matrix and vector exponentials to manage complex invariances and decoupling between scatter shape and texture in signal processing applications.

Riemannian geometry for compound Gaussian distributions establishes a differential-geometric framework for the statistical analysis, optimization, and inference with compound Gaussian and related non-centered mixtures of scaled Gaussian models. These families incorporate heteroscedasticity through sample-dependent scaling (or “texture” variables), producing parameter spaces that are smooth manifolds equipped with natural Riemannian metrics derived from Fisher information. This geometry enables the definition of geodesics, distance functions, barycenters, and recursive estimation algorithms, which are utilized in change detection, classification, and other signal processing tasks (Bouchard et al., 2020, Collas et al., 2022).

1. Compound Gaussian Family and Parameter Manifold

A $p$-dimensional complex compound Gaussian model draws each observation by scaling a zero-mean complex Gaussian vector with a positive random or deterministic texture variable: $$\mathbf x_i \sim \mathcal{CN}(0, \tau_i \Sigma),\qquad \tau_i > 0, \quad \Sigma \in \mathcal H_p^{++}$$ where $\mathcal H_p^{++}$ denotes the cone of $p \times p$ Hermitian positive-definite matrices. Identifiability is enforced by the unit-determinant constraint

$$\Sigma \in \mathcal{SH}_p^{++} = \{\Sigma \in \mathcal H_p^{++}: \det\Sigma = 1\},$$

leading to the parameter manifold

$$\mathcal M_{p,n} = \mathcal{SH}_p^{++} \times \mathbb R_{++}^n$$

with real dimension $(p(p+1)/2 - 1) + n$ (Bouchard et al., 2020).

In the “non-centered mixture of scaled Gaussian” (NC-MSG) variant,

$$x_i = \mu + \sqrt{\tau_i} y_i,\qquad y_i \sim \mathcal N(0, \Sigma),$$

with $\mu \in \mathbb R^p$, $\Sigma \in \mathrm{Sym}_p^+$, and $\prod_i \tau_i = 1$ for identifiability, providing a manifold

$$\Theta = \mathbb R^p \times \mathrm{Sym}_p^+ \times \{\tau \in \mathbb R_{++}^n: \prod_{i=1}^n \tau_i = 1\}$$

of dimension $p + \frac{p(p+1)}{2} + (n-1)$ (Collas et al., 2022).

2. Fisher Information Metric Structure

The Fisher information defines a Riemannian metric on the parameter manifold of any parametric statistical model. For the compound Gaussian,

$$\langle \xi, \eta \rangle_\theta = \frac{1}{p} \langle \xi_\Sigma, \eta_\Sigma \rangle_{\Sigma} + \frac{1}{n} \langle \xi_\tau, \eta_\tau \rangle_{\tau}$$

with

$$\langle \xi_\Sigma, \eta_\Sigma \rangle_{\Sigma} = \mathrm{tr}(\Sigma^{-1} \xi_\Sigma \Sigma^{-1} \eta_\Sigma),\qquad \langle \xi_\tau, \eta_\tau \rangle_\tau = (\xi_\tau \odot \tau^{-1})^\mathrm{T} (\eta_\tau \odot \tau^{-1})$$

where $\odot$ denotes entrywise product (Bouchard et al., 2020).

For NC-MSG, the Fisher–Rao metric is

$$g_\theta((\delta\mu,\delta\Sigma,\delta\tau), (\delta\mu',\delta\Sigma',\delta\tau')) = \sum_{i=1}^n \frac{1}{\tau_i} \delta\mu^\mathrm{T} \Sigma^{-1} \delta\mu' + \frac{n}{2} \mathrm{Tr}(\Sigma^{-1} \delta\Sigma\, \Sigma^{-1} \delta\Sigma') + \frac{p}{2}\sum_{i=1}^n \frac{\delta\tau_i}{\tau_i} \frac{\delta\tau_i'}{\tau_i}$$

with tangent spaces projected to enforce the unit-product constraint (Collas et al., 2022).

3. Geodesics and Distance Computation

For the centered CG manifold $\mathcal{M}_{p,n}$, geodesics decouple across factors:

• Closed-form (initial velocity):

$$\Sigma(t) = \Sigma\, \exp(t\,\Sigma^{-1}\xi_\Sigma),\qquad \tau(t) = \tau \odot \exp(t\,\tau^{-1} \odot \xi_\tau)$$

• Closed-form (endpoint, for constant-speed geodesic):

$$\Sigma(t) = \Sigma_0^{1/2} \big(\Sigma_0^{-1/2} \Sigma_1 \Sigma_0^{-1/2} \big)^t \Sigma_0^{1/2},\qquad \tau(t) = \tau_0^{\odot(1-t)}\odot \tau_1^{\odot t}$$

These curves satisfy $\nabla_{\dot\gamma}\dot\gamma=0$ for the product Levi-Civita connection (Bouchard et al., 2020).

For models with non-zero location $\mu$, such as NC-MSG, the full geodesic equations—including all variables—do not have a closed form. Instead, second-order retractions are employed: $$R_\theta(t\xi) = \big(\mu + t\,\delta\mu + \tfrac{t^2}{2} a_\mu,\; \Sigma + t\,\delta\Sigma + \tfrac{t^2}{2} a_\Sigma,\; \tau + t\,\delta\tau + \tfrac{t^2}{2} a_\tau\big)$$ with $(a_\mu, a_\Sigma, a_\tau)$ chosen to preserve manifold constraints (Collas et al., 2022).

The Riemannian squared distance for centered CG is

$$d^2(\theta_0,\theta_1) = \frac{1}{p} \|\log(\Sigma_0^{-1/2} \Sigma_1 \Sigma_0^{-1/2})\|_F^2 + \frac{1}{n} \|\log(\tau_0^{-1} \odot \tau_1)\|_2^2$$

reflecting distances in the affine-invariant metric on $\mathcal{SH}_p^{++}$ and log-Euclidean structure on textures (Bouchard et al., 2020).

4. Geometric Properties and Invariances

Both component manifolds are Riemannian symmetric spaces: $(\mathcal{SH}_p^{++}, \langle\cdot,\cdot\rangle)$ is nonpositively curved (isomorphic to $SL(p)/SU(p)$), and $(\mathbb R_{++}^n, \langle\cdot,\cdot\rangle)$ is flat. The product geometry is geodesically complete and primarily nonpositively curved. The Fisher–Rao metric is invariant under congruence transformations (induced by $SL(p)$ on $\Sigma$) and texture rescalings. The orthogonality between scatter shape and texture reflects their statistical decoupling (Bouchard et al., 2020). For NC-MSG models, explicit formulas for sectional curvature are not generally available, and even in the Gaussian case with nonzero location parameter, curvature expressions evade closed-form representation (Collas et al., 2022).

5. Computation: Algorithms and Gradient Methods

Numerical algorithms for geodesics and distances leverage matrix and vector exponentials. Each geodesic evaluation on $\mathcal{M}_{p,n}$ requires one matrix exponential (or SVD/eigen-decomposition) and one vector exponential, with log-determinant and entrywise logarithm operations for distance computation (Bouchard et al., 2020).

For optimization (e.g., recursive detection, maximum likelihood, and barycenter computation), the Riemannian geometry yields efficient gradient-descent algorithms:

• Natural (Riemannian) gradient: Given cost $h(\theta)$ on $\Theta$, the Riemannian gradient is obtained via $I(\theta)^{-1}\nabla h(\theta)$, with an orthogonal projection onto the tangent space as required by parameter constraints.
• Retraction-based updates: Parameter updates are performed via $$R_{\theta_k}(-\alpha_k\,\mathrm{grad}\,h(\theta_k))$$ with step-size $\alpha_k$ selected by Armijo backtracking (Collas et al., 2022).

For NC-MSG, the processes of regularized maximum likelihood estimation and symmetrized KL barycenter minimization both employ the Fisher–Rao Riemannian gradient and (second-order) retraction. Each iteration has complexity $O(np^2 + p^3)$, and convergence is attained in $5$–$60$ iterations, substantially outpacing naive steepest descent or ad-hoc metric approaches (Collas et al., 2022).

6. Kullback–Leibler Divergence and Statistical Barycenters

A closed-form Kullback–Leibler divergence is available for NC-MSG distributions. Given $p_1$ and $p_2$ parameterized by $(\mu_1, \Sigma_1, \tau^{(1)})$ and $(\mu_2, \Sigma_2, \tau^{(2)})$, the divergence is

$$D_{KL}(p_1 \| p_2) = \frac{1}{2} \sum_{i=1}^n \left[ \frac{\tau_i^{(1)}}{\tau_i^{(2)}} \mathrm{Tr}(\Sigma_2^{-1} \Sigma_1) + \frac{1}{\tau_i^{(2)}} (\mu_2 - \mu_1)^T \Sigma_2^{-1} (\mu_2 - \mu_1) - p \right] + \frac{1}{2} \sum_{i=1}^n \log \frac{|\Sigma_2| (\tau_i^{(2)})^p}{|\Sigma_1| (\tau_i^{(1)})^p}$$

A symmetrized version, $$\delta(\theta_1, \theta_2) = \frac{1}{2}(D_{KL}(p_1\|p_2) + D_{KL}(p_2\|p_1))$$, serves as a divergence for barycenter computation. The barycenter (Fréchet mean) is computed by minimizing the average symmetrized divergence, also using Riemannian gradient descent with retraction (Collas et al., 2022).

7. Applications and Practical Implications

Riemannian geometry for compound Gaussian models underpins recursive change detection, with algorithms leveraging the explicit geodesic and exponential map for low-complexity updates of sufficient statistics, e.g., in Constant False Alarm Rate (CFAR) detectors, where the overall per-block update cost is $O(p^3 + n)$ (Bouchard et al., 2020). For non-centered models, the KL divergence enables nearest-centroid classification that remains robust under affine transformations due to model invariances. Numerical experiments, such as on hyperspectral time series (e.g., Breizhcrops), demonstrate high accuracy and computational efficiency, with Fisher–Rao RGD converging significantly faster and using less computation per step than non-geometric or ad-hoc alternatives (Collas et al., 2022).

These geometric approaches have been integrated and contextualized within a lineage of Riemannian statistics on symmetric space models, recursive estimation, and SPD matrix analysis (e.g., Moakher, Bhatia, Smith, Zhou & Said). Ongoing research addresses existence and uniqueness criteria for regularized Riemannian objectives and seeks closed-form results for curvature and geodesics in more general parameter settings (Bouchard et al., 2020, Collas et al., 2022).