发表机构
National Institute of Technology Jamshedpur（杰姆谢德布尔国家理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出嵌套归纳偏置框架，将非欧几何拉回至SPD流形，构建曲率对齐的黎曼分类器，实验表明度量曲率与数据分布对齐可提升类可分性，还提出RCM增强几何鲁棒性。
AI 中文摘要
在几何深度学习中，归纳偏置有两个主要功能：施加流形约束和嵌入关系先验。目前，SPD流形上的表示学习常依赖拉回欧氏度量（如Log-Euclidean度量）来满足前者，这类度量计算高效且可避免域边界违规，但会诱导出平坦几何，可能无法捕捉数据集的内在关系先验。虽然庞加莱度量等常被用于诱导与域对齐的关系先验，但将其从标准向量表示推广到SPD流形仍是挑战。为弥合这一差距，本文提出嵌套归纳偏置框架，利用两阶段微分同胚组合将非欧目标几何正式拉回至SPD流形，该框架可构建曲率对齐的黎曼分类器，同时兼顾矩阵约束与数据的潜在关系几何。在运动学、信号处理基准及合成实验的实证评估表明，除非度量曲率与内在数据分布对齐，否则深度流形网络的类可分性会退化。此外，针对标准向量化架构，本文提出有理共形度量（RCM），旨在通过约束表示空间建立针对异常值的最优几何鲁棒性。
英文摘要
In Geometric Deep Learning, inductive biases serve two primary functions: enforcing manifold constraints and embedding relational priors. Currently, representation learning on SPD manifolds frequently relies on pullback Euclidean metrics, such as the Log-Euclidean Metric, to satisfy the former. While computationally efficient in avoiding domain boundary violations, these metrics induce a flat geometry that may fail to capture the intrinsic relational priors of datasets. While metrics such as the Poincaré metric are widely utilized to induce domain-aligned relational priors, generalizing them from standard vector representations to the SPD manifold has remained a challenge. To bridge this gap, we introduce a Nested Inductive Bias framework that utilizes a two-stage diffeomorphic composition to formally pull back non-Euclidean target geometries onto the SPD manifold. This framework enables the construction of curvature-aligned Riemannian classifiers that simultaneously respect matrix constraints and the latent relational geometry of the data. Empirical evaluations on kinematic and signal processing benchmarks, together with synthetic experiments, demonstrate that deep manifold networks experience degradation in class separability unless the metric curvature aligns with the intrinsic data distribution. Furthermore, for standard vectorized architectures, we propose the Rational Conformal Metric (RCM), designed to establish state-of-the-art geometric robustness against outliers by bounding the representation space.
