What question did this study set out to answer?

The aim is to adapt classical learning algorithms to work with non-Euclidean data structures using Riemannian geometry.

February 12, 2026Open Access

From Correlations to Manifolds: A Geometric Approach to Classifications

Key Points

The aim is to adapt classical learning algorithms to work with non-Euclidean data structures using Riemannian geometry.
Utilized the UCI Wine dataset for real data analysis.
Obtained correlation matrices from raw feature vectors with class-consistent windows.
Defined points on the SPD manifold using these matrices.
Projected the matrices onto the tangent space around the Fréchet mean.
Input the resulting data into a Euclidean-form SVM classifier.
Achieved integration of manifold-based representation with classical machine learning methods.
Enhanced classification performance was noted compared to traditional approaches.

Abstract

This study addresses a machine learning application based on Riemannian geometry. The main objective is to demonstrate how classical learning algorithms, particularly Support Vector Machine (SVM), can be adapted for non-Euclidean data structures. The UCI Wine dataset is used as a real dataset. Instead of directly applying the algorithm to raw feature vectors, correlation matrices are obtained from these vectors through class-consistent windows, defining points on the SPD manifold. These matrices are then projected onto the tangent space around the Fréchet mean using the Riemannian logarithm map and provided as input to a Euclidean-form SVM classifier. Thus, the integration of manifold-based representation with classical methods is achieved, and its contribution to classification performance is examined.

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