This review discusses how neuroimaging data can be represented as symmetric positive-definite (SPD) matrices and analyzed using the Riemannian geometry of the SPD manifold. It surveys the evolution from modality-specific SPD representations to geometric shallow and deep learning methods, emphasizing how these approaches maintain structural constraints while integrating modern AI techniques. The paper frames SPD matrix learning as a bridge between classical geometric statistics and contemporary machine learning in neuroimaging and brain‑computer interface research.
By Ce Ju, Reinmar Kobler, Antoine Collas, Motoaki Kawanabe, Cuntai Guan, Bertrand Thirion
arXiv:2604.22494v2 Announce Type: replace-cross
Abstract: We introduce two federated learning frameworks for the classical SPDnet model operating on symmetric positive definite (SPD) matrices with St...
By Thibault Pautrel, Florent Bouchard, Ammar Mian, Guillaume Ginolhac
arXiv:2607. 19305v1 Announce Type: cross Abstract: Deep neural networks on manifold-valued representations have attracted growing interest, but many basic components remain tied to specific manifolds, rely on Euclidean approximations, or require costly and numerically fragile geometric operations.
By Chen Ziheng
arXiv:2604. 20308v2 Announce Type: replace Abstract: Graph neural networks face two fundamental challenges rooted in the linear structure of Euclidean vector spaces: (1) Current architectures represent geometry through vectors (directions, gradients), yet many tasks require matrix-valued representations that capture relationships between directions-such as how atomic orientations covary in a molecule.
By Yuhan Peng, Junwen Dong, Yuzhi Zeng, Hao Li, Ce Ju, Huitao Feng, Diaaeldin Taha, Anna Wienhard, Kelin Xia
arXiv:2607. 19305v2 Announce Type: replace-cross Abstract: Deep neural networks on manifold-valued representations have attracted growing interest, but many basic components remain tied to specific manifolds, rely on Euclidean approximations, or require costly and numerically fragile geometric operations.
By Chen Ziheng
The paper introduces a Nested Inductive Bias framework that uses a two‑stage diffeomorphic composition to pull back non‑Euclidean target geometries onto symmetric positive definite (SPD) manifolds. This approach allows the construction of curvature‑aligned Riemannian classifiers that respect both matrix constraints and the intrinsic relational geometry of data. Empirical results on kinematic, signal processing, and synthetic benchmarks show that class separability degrades when metric curvature does not match the data distribution, and the authors also propose the Rational Conformal Metric (RCM) for robust vectorized architectures.
By Tushar Das
arXiv:2609.30487v1 Announce Type: cross
Abstract: We here develop a functional neural network, termed MatFAE, for learning trajectories on the Riemannian manifold of symmetric positive definite (SPD)...
By Samuel V. Singh, Mimi Zhang
arXiv:2606. 25456v1 Announce Type: new Abstract: Brain-Computer Interface (BCI) based on electroencephalography (EEG) enables direct interaction between the brain and external environments and has significant applications in assistive technologies, medical rehabilitation, and entertainment.
By Shaocheng Jin, Tao Zhou, Rui Wang, Ziheng Chen, Xiaoqing Luo, Xiaojun Wu, Josef Kittler
arXiv:2607. 09710v1 Announce Type: new Abstract: Tabular classification is often governed by local, condition-triggered rules rather than smooth global patterns.
By Tian Li, Lucy Robinson, Varun Ojha, Huizhi Liang
arXiv:2609.21039v1 Announce Type: new
Abstract: A pervasive structural pattern in modern deep learning is the linear factorization block: a submodule of the form $W = BA$ in which two parameter matri...
By Emanuele Zangrando, Marco Sutti, Francesco Tudisco
arXiv:2609.35436v2 Announce Type: replace
Abstract: Recently, deep neural networks on manifold-valued representations have garnered significant attention across various machine learning applications....
By Ziheng Chen
This work studies neural architectures for classifying symmetric positive-definite matrices, focusing on congruence-like layers, in which the input matrix is multiplied on the left and right by a (possibly rectangular) weight matrix $W$ and its transpose. Such layers lie at the core of the celebrated SPDNet and have also been employed independently for dimensionality reduction on positive-definite data.