Paper detail

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

Neuroimaging provides essential tools for characterizing brain activity by quantifying connectivity strength between remote regions, using different modalities that capture different aspects of connectivity. Yet, decoding meaningful neural signatures must contend with modality-specific challenges, including measurement noise, spatial and temporal distortions, heterogeneous acquisition protocols, and limited sample sizes. A unifying perspective emerges when these data are expressed through symmetric positive definite (SPD)-valued representations: across neuroimaging modalities, SPD-valued representations naturally give rise to SPD matrices that capture dependencies between sensors or brain regions. Endowing the SPD space with Riemannian metrics equips it with a non-Euclidean geometric structure, enabling principled statistical modeling and machine learning on the resulting manifold. This review consolidates machine learning methodologies that operate on the SPD manifold under a unified framework termed 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 paradigms. We show that (i) modeling on the SPD manifold is mathematically natural and numerically stable, preserving symmetry and positive definiteness while avoiding degeneracies inherent to Euclidean embeddings; (ii) SPD matrix learning extends a broad family of established geometric statistical tools used across neuroimaging; and (iii) SPD matrix learning integrates new-generation AI technologies, driving a new class of neuroimaging problems that were previously out of reach. Taken together, SPD matrix learning offers a principled and forward-looking framework for next-generation neuroimaging analytics.