# Escaping the Capacity Ceiling: Routing on the Stiefel Manifold for Bilinear SPD Layers

arXiv: https://arxiv.org/abs/2605.31043 · HTML: https://mlsanity.com/paper/2605.31043/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 35 references that are in the map.
- First posted on arXiv 29 May 2026.

## Builds on (most cited first, 13 of 13)

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Outrageously Large Neural Networks: The Sparsely-Gated Mixture-of-Experts Layer](https://mlsanity.com/paper/1701.06538/) (2017)
- [Switch Transformers: Scaling to Trillion Parameter Models with Simple and Efficient Sparsity](https://mlsanity.com/paper/2101.03961/) (2021)
- [Learning to Prompt for Continual Learning](https://mlsanity.com/paper/2112.08654/) (2021)
- [A Riemannian Network for SPD Matrix Learning](https://mlsanity.com/paper/1608.04233/) (2016)
- [Riemannian batch normalization for SPD neural networks](https://mlsanity.com/paper/1909.02414/) (2019)
- [SPD domain-specific batch normalization to crack interpretable unsupervised domain adaptation in EEG](https://mlsanity.com/paper/2206.01323/) (2022)
- [The largest EEG-based BCI reproducibility study for open science: the MOABB benchmark](https://mlsanity.com/paper/2404.15319/) (2024)
- [Deep Riemannian Networks for End-to-End EEG Decoding](https://mlsanity.com/paper/2212.10426/) (2022)
- [Geometric Neural Network based on Phase Space for BCI-EEG decoding](https://mlsanity.com/paper/2403.05645/) (2024)
- [Riemannian Self-Attention Mechanism for SPD Networks](https://mlsanity.com/paper/2311.16738/) (2023)
- [SPD Learn: A Geometric Deep Learning Python Library for Neural Decoding Through Trivialization](https://mlsanity.com/paper/2602.22895/) (2026)

## Abstract (arXiv)

Deep networks on the symmetric positive-definite (SPD) manifold promise expressive representations by encoding data geometry as an inductive bias, but stacking BiMap layers with the standard ReEig nonlinearity often adds no capacity: on real, preconditioned EEG data, ReEig rarely activates, so the stack behaves as a single layer at any depth. In the worst case, when domains share no discriminative directions, we prove a single filter has a capacity ceiling, so it cannot fully align every domain at once. To overcome that, we propose SCAP (Stiefel Cross-Attention Pool), a layer implementing a family of Stiefel filters by combining a pool of K experts into a sample-specific bilinear map via cross-attention. We show that it matches a per-domain filter bank to first order with fewer experts than domains when domain-optimal filters span few directions near a shared tangent-space basepoint; in the worst case, its alignment empirically stays nearly flat as domains grow, escaping the fixed-filter ceiling. Naively trained, however, this routing can collapse to a fixed filter; we diagnose why and adapt three mechanisms to mitigate it. SCAP significantly improves balanced accuracy over fixed-filter SPDNet on all five cross-domain EEG motor-imagery datasets, and matches or exceeds three domain-adaptive baselines on four out of five.