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RMLR: Extending Multinomial Logistic Regression into General Geometries
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Proposes RMLR, a framework for Riemannian multinomial logistic regression on general manifolds requiring only the Riemannian logarithm, instantiated as five families of SPD MLRs under power-deformed metrics and a Lie MLR on SO(n).
Problem
Existing Riemannian MLR methods rely on specific geometric properties (generalized law of sines, gyro structures, flat pullback metrics), limiting their applicability to general manifolds for classifying manifold-valued features.
Approach
Reformulate Euclidean MLR via margin distance to hyperplanes using Riemannian trigonometry and geodesic distance, yielding a closed-form RMLR p(y=k|S) ∝ exp(<log_{P_k} S, A_k>_{P_k}) that needs only the Riemannian logarithm; instantiate on SPD under five power-deformed metrics and on SO(n) with bi-invariant metric.
Key results
- Lie MLR improves LieNet on HDM05 from 76.92% to 78.24% mean accuracy and reduces convergence failures from 8/20 to 2/20 folds.
- SPD MLR under 2θ-BWM improves SPDNet on HDM05 by up to 14.23% over LogEig MLR; SPD MLR under θ-LCM gives 4.46% gain for TSMNet inter-subject on Hinss2021.
- On RResNet, SPD MLR under θ-EM improves HDM05 accuracy from 58.17% to 71.89% (+13.72%).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| replace tangent-space LogEig MLR with intrinsic SPD MLR | SPDNet · Radar | accuracy: 92.88±1.05 -> 95.64±0.83 | Helped |
| use (θ,α,β)-EM (0.5,1.0,1/30) instead of LogEig MLR | SPDNet · HDM05 | accuracy: 57.42±1.31 -> 71.65±0.88 | Helped |
| use θ-LCM (1.5) instead of LogEig MLR | TSMNet · Hinss2021 inter-session | balanced accuracy: 53.83±9.77 -> 56.43±8.79 | Mixed |
| use θ-LCM (0.5) instead of LogEig MLR | TSMNet · Hinss2021 inter-subject | balanced accuracy: 49.68±7.88 -> 54.14±8.36 | Helped |
| use (θ,α,β)-EM instead of LogEig MLR | RResNet · HDM05 | accuracy: 58.17 ± 2.07 -> 71.89 ± 0.60 | Helped |
| use θ-LCM instead of LogEig MLR | RResNet · NTU60 | accuracy: 45.22 ± 1.23 -> 53.63 ± 0.95 | Helped |
| use intrinsic SPD MLR, best reported setting (α,β)-LEM | SPDGCN · Disease, Cora, and Pubmed | mean accuracy | Helped |
| replace tangent-space LogEig MLR with Lie MLR | LieNet · G3D and HDM05 | mean accuracy: 87.91±0.90 -> 89.13±1.7; 76.92±1.27 -> 78.24±1.03 | Helped |
| replace LogEig MLR with 2θ-BWM | direct classification · HDM05 | accuracy: 48.43 ± 1.25 -> 66.77 ± 1.34 | Helped |
| use a deformed metric with selected hyperparameters rather than the standard metric (ablation) | SPD MLRs · Radar and HDM05 | accuracy | Helped |
| use θ-LCM rather than LogEig MLR | direct classification · Radar | accuracy: 91.93 ± 1.30 -> 95.67 ± 0.61 | Helped |
| use (θ,α,β)-EM rather than LogEig MLR | direct classification · Hinss2021 inter-session | accuracy: 39.76 ± 7.60 -> 45.78 ± 8.51 | Helped |
Surprising findings
- Deformed SPD MLRs under BWM and EM, which are geodesically incomplete, yield the largest gains, though gyro-based frameworks cannot derive them.
Limitations
- Each class requires a Riemannian parameter P_k and Euclidean parameter A_k, so the classification layer over-parameterizes as the number of classes grows.
- Numerical instability of parallel transport under BWM required a Lie group alternative and a custom Lyapunov backpropagation.
Experimental setup
Datasets: Radar HDM05 Hinss2021 NTU60 Disease Cora Pubmed G3D
Benchmarks: Radar HDM05 Hinss2021 NTU60 Disease Cora Pubmed G3D
Metrics: accuracy balanced accuracy mean±std over folds
Baselines: LogEig MLR gyro SPD MLR flat SPD MLR SPSD gyro MLR
Builds on: Euclidean multinomial logistic regression Hyperbolic Neural Networks gyro SPD MLR flat SPD MLR SPDNet
Closest papers by meaning
- Riemannian Multinomial Logistics Regression for SPD Neural Networks (CVPR 2024 · similarity 0.93)
- Matrix Manifold Neural Networks++ (ICLR 2024 · similarity 0.89)
- Building Transformation Layers for Riemannian Neural Networks (NeurIPS 2026 · similarity 0.88)
- Understanding Matrix Function Normalizations in Covariance Pooling through the Lens of Riemannian Geometry (ICLR 2025 · similarity 0.87)
- Riemannian Geometric-based Meta Learning (AAAI 2025 · similarity 0.87)
- Learning to Normalize on the SPD Manifold under Bures-Wasserstein Geometry (CVPR 2025 · similarity 0.87)
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