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

TL;DR

SPD matrix learning provides a geometric framework for neuroimaging by representing covariance/connectivity and tensor-derived data on the SPD manifold $\mathcal{S}_{++}^n$, enabling principled analysis with Riemannian metrics. It surveys four principal SPD metrics ($\text{AIRM}$, $\text{LEM}$, $\text{LCM}$, $\text{BWM}$) and geometric statistics (Fréchet mean, PGA, geodesic regression), then unifies geometric shallow learning and geometric deep learning for SPD-valued data. The review covers a broad spectrum of applications—from classical BCI decoding to longitudinal fMRI analyses and AI-driven problems like dynamic connectivity and generative modeling—illustrating how SPD representations preserve intrinsic structure and support cross-modal fusion. It also discusses optimization and practical challenges (covariance estimation, computational cost, interpretability) and outlines future directions at the intersection of geometric statistics and AI, such as self-supervised learning and flow-based models on SPD spaces.

Abstract

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.

Figures (2)

• Figure 1: SPD Matrix Learning Paradigm. Different neuroimaging measurements collected from modalities such as M/EEG, fMRI, MRI, DTI, and ECoG are first transformed into SPD matrices, which are regarded as elements of an SPD manifold. The middle panel of the schematic illustrates the SPD manifold $\mathcal{S}_{++}^n$ together with its tangent space $\mathcal{T}_e\mathcal{S}_{++}^n$ at the Fréchet mean $e$ computed from the dataset. A tangent vector $v_e \in \mathcal{T}_e\mathcal{S}_{++}^n$ and a geodesic $\gamma(t)$ passing through $e$ are shown. The tangent space at any point consists of all symmetric matrices of dimension $n(n+1)/2$. The exponential map $\exp_e(v_e)$ projects a tangent vector $v_e$ onto the SPD manifold, while the logarithmic map performs the inverse operation. On the right, SPD matrix learning denotes the collection of learning methods operating on SPD manifolds from the perspective of geometric statistics. It represents an interdisciplinary area formed at the intersection of neuroimaging, computer science, and applied mathematics.
• Figure 2: Matrix Dimensionality, Accuracy, and Efficiency in EEG Classification: MDM vs Tensor-CSPNet: This figures shows the within-session classification accuracy and runtime per training epoch for MDM and two variants of Tensor-CSPNet evaluated on the BNCI2015001 EEG dataset. The dataset contains recordings from 12 subjects across a total of 28 sessions, and the reported performance is obtained using 10-fold cross-validation. In both panels, each dot represents the average performance on a single session, whereas the horizontal bar indicates the mean across all 28 sessions. Tensor-CSPNet is evaluated using two architectural configurations, each employing a single BiMap layer. Both configurations map the input EEG covariance matrices to SPD matrices of different output dimensions (e.g., 6 or 13), where the output dimension reflects the number of projected channels in the learned SPD representation. The dim=6 configuration effectively performs a dimensionality reduction by approximately half compared to the default configuration dim=13, resulting in reduced runtime and slightly lower average accuracy.

Theorems & Definitions (1)

• Definition : Riemannian Metric