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Matrix Manifold Neural Networks++
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Extends matrix-manifold neural networks with gyrovector-style FC and convolutional layers for SPD manifolds, MLR on SPSD manifolds, and Grassmann GCNs with a differentiable projector-perspective logarithmic map.
Problem
Building general deep network components (FC, convolution, MLR, GCN) on matrix manifolds such as SPD and Grassmann manifolds, where existing SPD/Grassmann networks lack principled generalizations of standard DNN building blocks and the Grassmann logarithmic map in projector form is not differentiabl
Approach
Uses gyrovector-space theory to define SPD hypergyroplanes and FC/convolution layers via orthonormal bases for AI, LE and LC metrics; extends MLR to SPSD manifolds as structure space Gr-tilde x SPD; and expresses the projector-perspective Grassmann log map via an ONB-perspective log map computed with differentiable SVD.
Key results
- GyroSpsd++ reaches 97.90% on FPHA and 96.64% on NTU60, outperforming SPD baselines such as SPSD-AI (95.72%, 93.92%) and MLR-AI.
- Gr-GCN++ achieves 81.6% on Cora versus 68.1% for the ONB-perspective variant Gr-GCN-ONB, and 80.3% vs 76.2% on Pubmed.
- GyroSpd++ achieves 79.78% on HDM05, versus 70.26% for TGN and 54.18% for HypGRU.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace the SPD MLR layer with MLR in structure spaces (ablation) | GyroSpd++ · HDM05 | accuracy: 79.78 $\pm$ 1.42 -> 78.52 $\pm$ 1.34 | Hurt |
| Replace the SPD MLR layer with MLR in structure spaces (ablation) | GyroSpd++ · FPHA and NTU60 | accuracy: 96.84 $\pm$ 0.27 -> 97.90 $\pm$ 0.24; 95.28 $\pm$ 1.37 -> 96.64 $\pm$ 1.35 | Helped |
| Use projector perspective instead of ONB perspective (ablation) | Gr-GCN++ · Airport, Pubmed, and Cora | accuracy: 81.9 $\pm$ 1.2 -> 82.8 $\pm$ 0.7; 76.2 $\pm$ 1.5 -> 80.3 $\pm$ 0.5; 68.1 $\pm$ 1.0 -> 81.6 $\pm$ 0.4 | Helped |
| Set k=3 in the alternative Affine-Invariant metric (ablation) | GyroSpd++ · HDM05 and FPHA | accuracy: 79.78 $\pm$ 1.42 -> 79.12 $\pm$ 1.37; 96.84 $\pm$ 0.27 -> 96.16 $\pm$ 0.25 | Hurt |
| Set k=3 in the alternative Affine-Invariant metric (ablation) | GyroSpd++ · NTU60 | accuracy: 95.28 $\pm$ 1.37 -> 96.32 $\pm$ 1.33 | Helped |
| Set k=10 in the alternative Affine-Invariant metric (ablation) | GyroSpd++ · HDM05, FPHA, and NTU60 | accuracy: 79.78 $\pm$ 1.42 -> 78.25 $\pm$ 1.39; 96.84 $\pm$ 0.27 -> 95.91 $\pm$ 0.29; 95.28 $\pm$ 1.37 -> 94.44 $\pm$ 1.34 | Hurt |
| Set convolution output dimension m=10 instead of m=21 (ablation) | GyroSpd++ · FPHA | accuracy: 94.53 $\pm$ 0.31 -> 96.84 $\pm$ 0.27 | Hurt |
| Increase convolution output dimension from m=21 to m=30 (ablation) | GyroSpd++ · FPHA | accuracy: 96.84 $\pm$ 0.27 -> 96.80 $\pm$ 0.26 | No clear effect |
| Use Log-Euclidean metrics in both layers instead of AI-LE (ablation) | GyroSpd++ · FPHA | accuracy: 96.84 $\pm$ 0.27 -> 94.72 $\pm$ 0.25 | Hurt |
| Use Affine-Invariant metrics in both layers instead of AI-LE (ablation) | GyroSpd++ · FPHA | accuracy: 96.84 $\pm$ 0.27 -> 94.35 $\pm$ 0.29 | Hurt |
| Use Log-Euclidean metric for convolution and Affine-Invariant metric for MLR instead of AI-LE (ablation) | GyroSpd++ · FPHA | accuracy: 96.84 $\pm$ 0.27 -> 95.21 $\pm$ 0.26 | Hurt |
| Use Log-Cholesky metric for MLR instead of Log-Euclidean (ablation) | GyroSpd++ · FPHA | accuracy: 96.84 $\pm$ 0.27 -> 89.16 $\pm$ 0.26 | Hurt |
Surprising findings
- Using Log-Cholesky geometry for the MLR layer gives the worst accuracy on FPHA (89.16%) among the tested metric designs.
Limitations
- Gr-GCN++ training and testing times are roughly double those of the ONB variant.
- Computation time of SPD networks is high relative to baselines (e.g., 129 s per epoch for GyroSpd++ on FPHA).
Experimental setup
Datasets: HDM05 FPHA NTU RGB+D 60 Airport Pubmed Cora
Benchmarks: HDM05 FPHA NTU60 Airport Pubmed Cora
Metrics: mean accuracy standard deviation computation time per epoch model size (MB)
Baselines: SPDNet SPDNetBN GyroAI-HAUNet SPSD-AI MLR-AI HypGRU MS-G3D ST-TR TGN Gr-GCN-ONB
Builds on: SPDNet SPDNetBN GyroAI-HAUNet MLR-AI GCN
Closest papers by meaning
- Building Transformation Layers for Riemannian Neural Networks (NeurIPS 2026 · similarity 0.89)
- RMLR: Extending Multinomial Logistic Regression into General Geometries (NeurIPS 2024 · similarity 0.89)
- Understanding Matrix Function Normalizations in Covariance Pooling through the Lens of Riemannian Geometry (ICLR 2025 · similarity 0.87)
- Riemannian Multinomial Logistics Regression for SPD Neural Networks (CVPR 2024 · similarity 0.87)
- Neural networks on Symmetric Spaces of Noncompact Type (ICLR 2025 · similarity 0.87)
- Learning to Normalize on the SPD Manifold under Bures-Wasserstein Geometry (CVPR 2025 · similarity 0.86)
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