The last decade has witnessed an experimental revolution in data science and machine learning, epitomised by deep learning methods. Indeed, many high-dimensional learning tasks previously thought to be beyond reach -- such as computer vision, playing Go, or protein folding -- are in fact feasible with appropriate computational scale. Remarkably, the essence of deep learning is built from two simple algorithmic principles: first, the notion of representation or feature learning, whereby adapted, often hierarchical, features capture the appropriate notion of regularity for each task, and second, learning by local gradient-descent type methods, typically implemented as backpropagation. While learning generic functions in high dimensions is a cursed estimation problem, most tasks of interest are not generic, and come with essential pre-defined regularities arising from the underlying low-dimensionality and structure of the physical world. This text is concerned with exposing these regularities through unified geometric principles that can be applied throughout a wide spectrum of applications. Such a 'geometric unification' endeavour, in the spirit of Felix Klein's Erlangen Program, serves a dual purpose: on one hand, it provides a common mathematical framework to study the most successful neural network architectures, such as CNNs, RNNs, GNNs, and Transformers. On the other hand, it gives a constructive procedure to incorporate prior physical knowledge into neural architectures and provide principled way to build future architectures yet to be invented.
核心贡献 · Key contributions
提出几何深度学习作为埃尔朗根纲领式的统一框架：从对称性与不变性原则推导神经网络架构。 Proposes Geometric Deep Learning as an Erlangen-style unification: deriving neural architectures from symmetry and invariance principles.
定义了五个几何域——网格、群、图、测地线与规范场——作为数据和信号背后的共同结构。 Defines five geometric domains—grids, groups, graphs, geodesics, gauges—as a common structure underlying data and signals.
表明 CNN、RNN、GNN 和 Transformer 都是相同几何先验——对称性、稳定性和尺度分离——的具体实例。 Shows CNNs, RNNs, GNNs, and Transformers are instances of the same geometric priors: symmetry, stability, and scale separation.
引入了一个结合对称性、形变稳定性和尺度分离的蓝图，用于学习稳定的高维表示。 Introduces a blueprint combining symmetry, deformation stability, and scale separation to learn stable high-dimensional representations.
提供了一种构造性方法，将物理先验知识融入神经网络，并为设计未来架构给出原则性路径。 Provides a constructive approach to incorporate physical prior knowledge and to design future neural architectures.
强调非线性等变映射与非线性激活组合，可在避免平凡平均的情况下构造富有表达力的不变表示。 Highlights that nonlinear equivariant maps compose with nonlinearities to build expressive invariant representations without trivial averaging.
局限 · Limitations
重点关注表示学习，未涵盖自监督学习、生成建模和强化学习等流程。 Focuses on representation learning, excluding self-supervised learning, generative modelling, and reinforcement learning pipelines.
并不旨在全面总结所有几何深度学习研究，仅选取若干典型架构来阐述原理。 Does not aim to fully summarise all geometric deep learning research; selected architectures illustrate principles.
精确对称性假设往往不现实；形变稳定性只是近似的，且在复杂域上难以量化。 Exact symmetry assumptions are often unrealistic; deformation stability is approximate and difficult to quantify on complex domains.
该几何蓝图可能无法完全涵盖所有现代设计，尤其是超越置换等变的基于注意力机制的 Transformer。 The geometric blueprint may not fully capture all modern designs, notably attention-based Transformers beyond permutation equivariance.
对等变架构在大规模问题中的计算成本或实际扩展性讨论较少。 Little discussion of computational cost or practical scaling of equivariant architectures across large-scale problems.