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Riemannian Geometric-based Meta Learning
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Proposes Stiefel-MAML, which runs MAML's inner and outer loops with Riemannian gradient descent and QR retraction on the Stiefel manifold, plus a kernel-based loss, and reports gains over MAML and first-order MAML on few-shot classification benchmarks.
Problem
MAML optimizes meta-parameters in Euclidean space, which may struggle to capture complex learning dynamics in few-shot learning and does not naturally enforce orthogonality constraints.
Approach
Constrain MAML parameters to the Stiefel manifold of orthonormal matrices, using Riemannian gradients and QR-based retraction in both inner and outer loops, together with a kernel-based loss defined via geodesic distance on the manifold.
Key results
- Stiefel-MAML averages +0.48% over MAML on Omniglot, +1.73% on Mini-ImageNet, +5.05% on FC-100, and +1.91% on CUB across 1- and 5-shot, 3/5/10-way settings.
- On ResNet-50 Omniglot, S-MAML is higher than MAML in most 1-shot settings but lower in the 5-shot 3-way setting (86.30 vs 89.26).
- Per-iteration time rises from about 1.1-1.6 s (MAML) to about 1.5-2.0 s (S-MAML) on Omniglot 1-shot.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace Euclidean MAML updates with Stiefel-manifold optimization | MAML · Omniglot | accuracy: +0.48% average improvement | Mixed |
| Replace Euclidean MAML updates with Stiefel-manifold optimization | MAML · Mini-ImageNet | accuracy: +1.73% average improvement | Helped |
| Replace Euclidean MAML updates with Stiefel-manifold optimization | MAML · FC-100 | accuracy: +5.05% | Helped |
| Replace Euclidean MAML updates with Stiefel-manifold optimization | MAML · CUB | accuracy: +1.91% average improvement | Helped |
| Replace Euclidean MAML updates with Stiefel-manifold optimization | MAML · 1-shot few-shot classification | accuracy: +1.88% | Helped |
| Replace Euclidean MAML updates with Stiefel-manifold optimization | MAML · 5-shot few-shot classification | accuracy: +2.81% | Helped |
| Replace Euclidean MAML updates with Stiefel-manifold optimization | MAML · Mini-ImageNet → CUB and CUB → Mini-ImageNet, 5-way | accuracy: 47.81 -> 47.97; 63.35 -> 65.61; 28.81 -> 30.94; 39.50 -> 40.13 | Helped |
| Replace Euclidean MAML updates with Stiefel-manifold optimization | MAML with ResNet-50 · Omniglot, 5-shot, 3-way | accuracy: 89.26 -> 86.30 | Hurt |
| Replace Euclidean MAML updates with Stiefel-manifold optimization | MAML with ResNet-50 · Omniglot, ResNet-50, other reported way/shot settings | accuracy | Mixed |
| Use Stiefel-manifold optimization instead of Euclidean optimization | MAML · Omniglot, 1-shot and 5-shot timing | seconds per iteration | Hurt |
Surprising findings
- On ResNet-50, S-MAML underperforms MAML in the 5-shot 3-way setting (86.30 vs 89.26).
Limitations
- Kernel-based loss is stated as a theorem without a rigorous proof or derivation shown in the text.
- Reported 5-way MAML results on Omniglot and Mini-ImageNet are copied from Finn et al. rather than rerun, and 10-way Mini-ImageNet is missing.
- Results are mixed on ResNet-50, with S-MAML losing in some settings.
Experimental setup
Datasets: Omniglot Mini-ImageNet FC-100 CUB
Benchmarks: Omniglot Mini-ImageNet FC-100 CUB
Metrics: accuracy 95% confidence interval Cohen's d sec/iter gradient norm
Baselines: MAML FO-MAML
Builds on: MAML first-order MAML Riemannian optimization
Closest papers by meaning
- Riemannian SAM: Sharpness-Aware Minimization on Riemannian Manifolds (NeurIPS 2023 · similarity 0.88)
- StelLA: Subspace Learning in Low-rank Adaptation using Stiefel Manifold (NeurIPS 2025 · similarity 0.88)
- Learning-Rate-Free Stochastic Optimization over Riemannian Manifolds (ICML 2024 · similarity 0.87)
- RMLR: Extending Multinomial Logistic Regression into General Geometries (NeurIPS 2024 · similarity 0.87)
- Building Transformation Layers for Riemannian Neural Networks (NeurIPS 2026 · similarity 0.86)
- Learning Distributions on Manifolds with Free-Form Flows (NeurIPS 2024 · similarity 0.86)
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