Dimensionality Reduction of SPD Data Based on Riemannian Manifold Tangent Spaces and Isometry.

Area of Science:

• Machine Learning
• Differential Geometry
• Data Science

Background:

• Symmetric Positive Definite (SPD) data commonly reside on nonlinear Riemannian manifolds, posing challenges for traditional Euclidean dimensionality reduction (DR) methods.
• High dimensionality of SPD data necessitates effective DR techniques, with bilinear transformations playing a crucial role.

Purpose of the Study:

• To propose a novel dimensionality reduction (DR) method for Symmetric Positive Definite (SPD) data that leverages Riemannian manifold tangent spaces and global isometry.
• To address the limitations of applying Euclidean DR methods directly to SPD matrices on nonlinear manifolds.

Main Methods:

• Developed the Riemannian Manifold Tangent Spaces and Global Isometry for SPD Data DR (RMTSISOM-SPDDR) method.
• Transferred bilinear transformations from SPD manifolds to their tangent spaces, which are Euclidean-isomorphic Hilbert spaces.
• Utilized log transformation and the affine invariant Riemannian metric (AIRM) to map SPD data to tangent spaces, preserving geodesic distances as Euclidean distances.

Main Results:

• The proposed RMTSISOM-SPDDR method effectively reduces the dimensionality of SPD data.
• Experimental results on five datasets demonstrate the superiority of RMTSISOM-SPDDR compared to five advanced SPD data DR algorithms.
• The method preserves the intrinsic properties of SPD data during the dimensionality reduction process.

Conclusions:

• RMTSISOM-SPDDR offers a robust and effective solution for dimensionality reduction of Symmetric Positive Definite (SPD) data.
• The integration of Riemannian geometry principles enhances the performance of DR techniques for SPD data.
• This approach provides a valuable tool for machine learning applications involving complex SPD data structures.