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Riemannian Multinomial Logistics Regression for SPD Neural Networks
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Proposes Riemannian multinomial logistics regression (SPD MLR) classifiers for SPD neural networks under pullback Euclidean metrics (parameterized LEM and LCM), with an intrinsic interpretation of the LogEig classifier.
Problem
Existing SPD neural networks use Euclidean classifiers on tangent/approximated spaces, which distort the intrinsic geometry of SPD manifolds; a classifier that respects SPD geometry is needed.
Approach
Derives a closed-form SPD multinomial logistic regression under any pullback Euclidean metric by computing the geodesic margin distance to SPD hyperplanes, avoiding gyro structures; instantiates it for (α,β)-LEM and (θ)-LCM and shows the standard LEM variant is equivalent to LogEig MLR under LEM-based RSGD.
Key results
- On Radar with SPDNet, (1,1)-LEM SPD MLR reaches 95.64±0.83 vs 92.88±1.05 for LogEig MLR on the 2-layer architecture.
- On HDM05 with SPDNet, (0.5)-LCM SPD MLR reaches 65.66±0.73 vs 57.42±1.31 for LogEig MLR on [93,30].
- On Hinss2021, (1.5)-LCM SPD MLR improves inter-session balanced accuracy from 53.83 to 56.43 over the LogEig MLR baseline; LEM/LCM MLRs train up to ~9x faster than Gyro-AIM on HDM05 (117 classes).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the (1,1)-LEM classifier instead of LogEig MLR | SPDNet · Radar | accuracy: 92.88±1.05 -> 95.64±0.83; 93.47±0.45 -> 95.87±0.58 | Helped |
| Use the (1,1)-LEM variant instead of standard (1,0)-LEM (ablation) | SPDNet · Radar | accuracy: 93.55±1.21 -> 95.64±0.83; 94.60±0.70 -> 95.87±0.58 | Helped |
| Use the (0.5)-LCM classifier instead of LogEig MLR | SPDNet · HDM05 | accuracy: 57.42±1.31 -> 65.66±0.73; 60.69±0.66 -> 65.79±0.63; 60.76±0.80 -> 65.71±0.75 | Helped |
| Use (0.5)-LCM instead of standard (1)-LCM (ablation) | SPDNet · HDM05 | accuracy: 62.04±1.05 -> 65.66±0.73; 62.11±2.11 -> 65.79±0.63; 62.89±2.09 -> 65.71±0.75 | Helped |
| Use the (1.5)-LCM classifier instead of LogEig MLR | SPDDSMBN · Hinss2021 inter-session | balanced accuracy: 53.83±9.77 -> 56.43±8.79 | Helped |
| Use the (1.5)-LCM classifier instead of LogEig MLR | SPDDSMBN · Hinss2021 inter-subject | balanced accuracy: 49.68±7.88 -> 51.65±5.90 | Helped |
| Use the (1.5)-LCM variant instead of standard (1)-LCM (ablation) | SPDDSMBN · Hinss2021 inter-session | balanced accuracy: 55.71±8.57 -> 56.43±8.79 | Helped |
| Use the (1.5)-LCM variant instead of standard (1)-LCM (ablation) | SPDDSMBN · Hinss2021 inter-subject | balanced accuracy: 51.60±8.43 -> 51.65±5.90 | No clear effect |
| Use LEM-based SPD MLR instead of Gyro-AIM SPD MLR | SPDNet · HDM05 | training time (s/epoch): 31.64 -> 4.7 | Helped |
Surprising findings
- The standard LEM-based SPD MLR is saturated on HDM05 and performs similarly to LogEig MLR, which the authors attribute to the equivalence shown in Prop. 5.1.
- Gyro-AIM MLR, despite AIM's theoretical affine invariance, is often not better than the Euclidean-tangent LogEig baseline and is outperformed by deformed LEM/LCM MLRs.
Limitations
- Only PEM-based metrics (LEM, LCM and variants) are covered; AIM-based closed-form margin distance is not solved.
- Scaling to Euclidean backbones with large-dimensional SPD features is computationally burdensome for Riemannian optimization.
Experimental setup
Datasets: Radar HDM05 Hinss2021
Benchmarks: Radar HDM05 Hinss2021
Metrics: accuracy balanced accuracy training time per epoch
Baselines: LogEig MLR Gyro-AIM SPD MLR SPDNet SPDDSMBN
Builds on: SPDNet Hyperbolic neural networks Gyro SPD MLR TSMNet Pullback Euclidean metrics
Closest papers by meaning
- RMLR: Extending Multinomial Logistic Regression into General Geometries (NeurIPS 2024 · similarity 0.93)
- Siegel Neural Networks (NeurIPS 2025 · similarity 0.90)
- Matrix Manifold Neural Networks++ (ICLR 2024 · similarity 0.87)
- Neural networks on Symmetric Spaces of Noncompact Type (ICLR 2025 · similarity 0.86)
- A Lie Group Approach to Riemannian Batch Normalization (ICLR 2024 · similarity 0.86)
- Building Transformation Layers for Riemannian Neural Networks (NeurIPS 2026 · similarity 0.86)
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