用黎曼几何优化元学习，提升小样本任务适应能力
Riemannian Geometric-based Meta Learning
- 在施蒂费尔流形上优化参数，强制正交约束增强表达力
- 在Omniglot等4个数据集上超越传统MAML，少样本性能更优
- 适合研究元学习与几何优化的学者，尤其关注参数空间结构
元学习旨在让模型用极少数据快速适应新任务。传统方法如MAML在欧几里得空间优化参数，难以捕捉复杂学习动态，尤其在少样本场景下表现受限。为此，我们提出Stiefel-MAML，将黎曼几何引入优化过程，在施蒂费尔流形（Stiefel manifold）上进行参数更新，该空间天然满足正交性约束。通过黎曼梯度计算与收缩操作，提升了参数表达能力并实现更高效优化。我们还设计了一种定义在施蒂费尔流形上的新型核函数损失，进一步增强模型对参数空间的探索能力。在Omniglot、Mini-ImageNet、FC-100和CUB等基准数据集上的实验表明，Stiefel-MAML在各类少样本任务中均持续优于传统MAML，验证了黎曼几何在元学习中的潜力，为后续基于不同几何结构优化的研究奠定基础。
原文摘要 · Abstract (English)
Meta-learning, or "learning to learn," aims to enable models to quickly adapt to new tasks with minimal data. While traditional methods like Model-Agnostic Meta-Learning (MAML) optimize parameters in Euclidean space, they often struggle to capture complex learning dynamics, particularly in few-shot learning scenarios. To address this limitation, we propose Stiefel-MAML, which integrates Riemannian geometry by optimizing within the Stiefel manifold, a space that naturally enforces orthogonality constraints. By leveraging the geometric structure of the Stiefel manifold, we improve parameter expressiveness and enable more efficient optimization through Riemannian gradient calculations and retraction operations. We also introduce a novel kernel-based loss function defined on the Stiefel manifold, further enhancing the model's ability to explore the parameter space. Experimental results on benchmark datasets--including Omniglot, Mini-ImageNet, FC-100, and CUB--demonstrate that Stiefel-MAML consistently outperforms traditional MAML, achieving superior performance across various few-shot learning tasks. Our findings highlight the potential of Riemannian geometry to enhance meta-learning, paving the way for future research on optimizing over different geometric structures.
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