Classification of multivariate signals using the Hilbert transform and Riemannian geometry

2026/06/24 by Martin Gemborn Nilsson, Bo Bernhardsson
Mathematics · Physics and Astronomy · #Mathematical Analysis and Transform Methods #Morphological variations and asymmetry #Statistical Mechanics and Entropy

paper · doi:10.1016/j.sigpro.2026.110786

Abstract

We describe two Hilbert transform-based methods for augmenting covariance matrices from multivariate signals. The two methods are shown to be isometric under the Riemannian affine-invariant metric on the manifold of symmetric/Hermitian positive definite matrices. The augmented representations distinguish cases that standard covariances clearly cannot and, when paired with a Riemannian minimum distance to mean classifier, improve classification of both synthetic data and real electroencephalography (EEG) data without introducing any extra hyperparameters. This novel combination of methods also significantly improves tangent space classifier accuracy on the same dataset, outperforming state-of-the-art classifiers with similarly sized parameter grids. We also examine how the augmented covariances interact with the minimum distance to mean classifier and show how multivariate cross-covariance functions behave under the Hilbert transform.

Citations

• A well-conditioned estimator for large-dimensional covariance matrices

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