Curvature-Adaptive Meta-Learning for Fast Adaptation to Manifold Data

Area of Science:

• Machine Learning
• Computational Geometry
• Optimization

Background:

• Existing meta-learning methods often assume data resides in Euclidean space, limiting their effectiveness for real-world applications with complex Riemannian manifold data.
• Fast adaptation to novel tasks is crucial, but current approaches struggle with the inherent geometric structures of non-Euclidean data.

Purpose of the Study:

• To propose a curvature-adaptive meta-learning method for fast adaptation to data residing on Riemannian manifolds.
• To develop a framework that effectively encodes complex manifold data into discriminative representations.
• To introduce adaptive schemes for constructing suitable product manifolds and optimizing learning rates and search directions.

Main Methods:

• Representing data in a product manifold of constant curvature spaces using a product manifold neural network.
• Implementing curvature generation for task-specific initialization and curvature updating for adaptive optimization.
• Evaluating the method on few-shot classification, few-shot regression, and reinforcement learning tasks.

Main Results:

• The proposed method achieves fast adaptation to manifold data by producing suitable curvature.
• The curvature generation and updating schemes enable the construction of appropriate product manifolds and adaptive optimization.
• Substantial improvements were observed compared to meta-learning methods that ignore data geometry.

Conclusions:

• The curvature-adaptive meta-learning approach effectively handles manifold data, offering significant performance gains.
• This method provides a more adaptive and efficient optimization paradigm for meta-learning on complex data geometries.
• The findings highlight the importance of incorporating geometric properties into meta-learning algorithms.