Extension to general covariance structures

Extend the theoretical analysis of the CPGD principle for Gaussian mixture models from diagonal covariance matrices to general covariance structures, while addressing the resulting non-diagonal metric tensor and guaranteeing positive definiteness of the estimated covariance matrices.

Background

The paper studies Conic Particle Gradient Descent with natural or Riemannian gradient updates for Gaussian mixture models whose component covariance matrices are diagonal. This restriction makes the Fisher–Rao metric tensor diagonal and permits the theoretical convergence analysis developed in the paper.

General covariance structures would produce a non-diagonal metric tensor, complicating the CPGD updates. In addition, any computational extension must preserve positive definiteness of the estimated covariance matrices. The authors explicitly identify extending the theoretical analysis to this setting as unresolved.

References

Extending this theoretical analysis to general covariance structures is still an open problem. From a computational perspective, the extension of the CPGD principle to general covariance matrices is not straightforward.

— Riemannian Gradient Descent for Gaussian Mixture Models with unknown diagonal covariances  (2609.30220 - Giard et al., 24 Sep 2026) in Remark ‘Extension to general covariance structures’, Section 4.1 (Statistical motivation)