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Wrapped Gaussian on the manifold of Symmetric Positive Definite Matrices

ICML 2025 · Probabilistic and scientific modeling › Molecules and proteins · approach: Sparse graphical and tensor models · 0 citations (OpenAlex)

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Defines a non-isotropic wrapped Gaussian on the SPD manifold via the exponential map, derives its density, CLT and parameter equivalence, fits it by MLE, and builds discriminant classifiers (Ho-WDA, He-WDA) that generalize MDM and tangent-space LDA/QDA.

Problem

Classical Euclidean Gaussians do not respect the intrinsic Riemannian geometry of symmetric positive definite (SPD) matrix data, and prior Riemannian Gaussians are either isotropic, centered on the tangent space, or hard to normalize and estimate.

Approach

Wrap a non-centered multivariate Gaussian from the tangent space at a point p onto the AIRM manifold of SPD matrices using only the exponential map, obtain its closed-form density via the Jacobian of Exp, and show that MDM and tangent-space LDA/QDA are maximum-likelihood classifiers under this model, leading to new Ho-WDA and He-WDA classifiers.

Key results

• Ho-WDA achieves the best accuracy on Pavia Univ. (87.36), Salinas (71.20), and BreizhCrops (54.66), outperforming MDM and TS-LDA on the large datasets.
• On BNCI2014004 and Zhou2016 (under 1,000 matrices), MDM or TS-LDA performs best (78.71 and 91.18 accuracy) while the WDA variants perform less well.
• Synthetic MLE experiments show estimation error decreases as the number of samples grows from 100 to 10,000, with dimension mainly affecting covariance estimation.

Measured effects reported in the paper

Surprising findings

• He-WDA does not consistently beat Ho-WDA, since per-class covariance estimates are noisy with few samples.

Limitations

• No closed-form MLE for the base point p; optimized with Riemannian conjugate gradient, cost grows as O(d^4) parameters for full covariance.
• Exponential and logarithmic maps are a computational bottleneck.
• Wrapped-Gaussian classifiers underperform MDM and TS-LDA on small BCI datasets.

Experimental setup

Datasets: BNCI2014004 Zhou2016 AirQuality Indiana Pines Pavia Univ. Salinas Textile BreizhCrops

Benchmarks: BNCI2014004 Zhou2016 AirQuality Indiana Pines Pavia Univ. Salinas Textile BreizhCrops

Metrics: accuracy AIRM distance estimation error l2 error on mu

Baselines: MDM TS-LDA TS-QDA

Builds on: Affine Invariant Riemannian Metric wrapped Gaussian (hyperbolic/symmetric space) Riemannian conjugate gradient Minimum Distance to Mean classifier

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