🤖 AI Summary
This work addresses the limited expressivity of conventional parameter optimization paradigms by proposing direct optimization of metric tensor fields on fixed-topology manifolds, enabling data-driven dynamic evolution of model geometry. Methodologically, leveraging discrete differential geometry, the manifold is represented as a triangular mesh, with the metric parameterized via edge lengths and optimized efficiently using automatic differentiation. Theoretically, the framework establishes a profound analogy between metric optimization and the Einstein–Hilbert action in general relativity. Crucially, geometric complexity adapts automatically while preserving topology, thereby enhancing model expressivity and generalization and effectively mitigating overfitting. The resulting framework provides a novel paradigm for scientific modeling, robust representation learning, and geometric deep learning—unifying geometric reasoning with differentiable optimization in a principled, topology-preserving manner.
📝 Abstract
This paper proposes a novel paradigm for machine learning that moves beyond traditional parameter optimization. Unlike conventional approaches that search for optimal parameters within a fixed geometric space, our core idea is to treat the model itself as a malleable geometric entity. Specifically, we optimize the metric tensor field on a manifold with a predefined topology, thereby dynamically shaping the geometric structure of the model space. To achieve this, we construct a variational framework whose loss function carefully balances data fidelity against the intrinsic geometric complexity of the manifold. The former ensures the model effectively explains observed data, while the latter acts as a regularizer, penalizing overly curved or irregular geometries to encourage simpler models and prevent overfitting. To address the computational challenges of this infinite-dimensional optimization problem, we introduce a practical method based on discrete differential geometry: the continuous manifold is discretized into a triangular mesh, and the metric tensor is parameterized by edge lengths, enabling efficient optimization using automatic differentiation tools. Theoretical analysis reveals a profound analogy between our framework and the Einstein-Hilbert action in general relativity, providing an elegant physical interpretation for the concept of "data-driven geometry". We further argue that even with fixed topology, metric optimization offers significantly greater expressive power than models with fixed geometry. This work lays a solid foundation for constructing fully dynamic "meta-learners" capable of autonomously evolving their geometry and topology, and it points to broad application prospects in areas such as scientific model discovery and robust representation learning.