Deep Transported Gaussian Models for the Correlation Matrices

Covariance matrices are widely used in representation learning, supported by geometric methods and statistical modeling on the symmetric positive-definite (SPD) manifold. Correlation matrices provide a scale-invariant, standardized representation of statistical dependence, yet probabilistic modeling on the correlation manifold remains comparatively underexplored. We introduce CorTG, a deep transported Gaussian model for full-rank correlation matrices. CorTG combines the off-log map with a learnable invertible transformation and models the transformed data using anisotropic Gaussian distributions. This construction yields an exactly normalized probability density, enables tractable likelihood evaluation through the Jacobian determinant, and supports sampling through the inverse transformation. The induced Riemannian metric also enables efficient computation of distances, geodesics, and Fr\'echet means. We further derive spectral bounds for the Jacobian of the off-log map and its inverse, analyze the expressive capacity of structured Gaussian models, and establish conditions for geometric identifiability. Experiments on synthetic data and five public benchmarks across multiple domains demonstrate the effectiveness of CorTG.