[2504.18882] SPD Matrix Learning for Neuroimaging Analysis: Perspectives, Methods, and Challenges

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a summary:Neuroimaging provides essential tools for characterizing brain activity by measuring the strength of connectivity between remote areas, using different modalities that capture different aspects of connectivity. However, decoding meaningful neural signatures must deal with method-specific challenges, including measurement noise, spatial and temporal distortions, heterogeneous acquisition protocols, and limited sample sizes. A unified perspective emerges when these data are expressed through symmetric positive-valued (SPD) representations: across neuroimaging modalities, SPD-valued representations naturally lead to SPD matrices that capture dependencies between sensors or brain regions. Endowing the SPD space with Riemannian scales provides it with a non-Euclidean geometric structure, enabling principled statistical modeling and machine learning on the resulting manifold.

This review integrates machine learning methodologies operating on the SPD manifold within a unified framework called SPD matrix learning. SPD matrix learning brings conceptual clarity across multiple modalities, establishes continuity with decades of geometric statistics in neuroimaging, and positions SPD modeling as a methodological bridge between classical analysis and emerging AI-driven models. We have shown that (1) modeling on an SPD manifold is mathematically natural and numerically stable, preserving symmetry and positive definiteness while avoiding degeneracies inherent in Euclidean embeddings; (2) SPD matrix learning extends a wide range of geometric statistical tools used across neuroimaging; (3) SPD matrix learning incorporates new generation AI techniques, giving rise to a new class of neuroimaging problems that were previously out of reach. Taken together, SPD matrix learning provides a principled and forward-looking framework for the next generation of neuroimaging analyses.