发表机构
The University of Tokyo; Stanford University（东京大学; 斯坦福大学）
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 $\mathbb{SL}(n)$ 空间作为混合曲率表示几何，通过简单约束实现最大曲率容量和深层序感知组合，在图基准上显著优于现有流形方法。
AI 中文摘要
混合曲率表示学习旨在捕捉单一曲率机制无法充分建模的丰富几何结构。现有方法主要依赖乘积流形，这需要手动指定不同曲率空间如何组合，并将其曲率贡献在各因子间分离。我们引入了 $\mathbb{SL}(n)$ 空间，这是一种由简单的 $\det(A)=1$ 约束和左不变 Schatten-$p$ Finsler 结构定义的表示几何。尽管构造极简，$\mathbb{SL}(n)$ 围绕一个公共旗杆表现出逐点负、零和正旗曲率，而其混合曲率和曲率耦合容量相对于内在几何上界渐近达到最大。在几何之外，其非交换群结构提供了固有的序敏感性，其非幂零李代数允许在任意深度存在非零嵌套李括号，从而实现深层序感知组合。实验上，$\mathbb{SL}(n)$ 在不同规模的图基准上持续优于广泛的表示流形基线。它在 KEGG 上将平均失真相对于最强基线降低了 $44.3\\%$，在 HumanCyc 上降低了 $40.5\\%$，并在 OGBL-PPA 上将 Hits@20 提高了 $42.8\\%$。在 Flickr30k-Order 上的实验进一步支持其从有序组合中捕捉高阶依赖的能力。这些结果共同表明，一个看似简单的结构约束如何在统一的表示空间内产生出乎意料的丰富几何、容量和组合能力。
英文摘要
Mixed-curvature representation learning seeks to capture rich geometric structures that cannot be adequately modeled by a single curvature regime. Existing approaches largely rely on product manifolds, which require manually specifying how different curvature spaces are combined and separate their curvature contributions across factors. We introduce the $\mathbb{SL}(n)$ space, a representation geometry defined by the simple $\det(A)=1$ constraint and a left invariant Schatten-$p$ Finsler structure. Despite this minimal construction, $\mathbb{SL}(n)$ exhibits pointwise negative, zero, and positive flag curvature around a common flagpole, while its mixed-curvature and curvature-coupling capacities are asymptotically maximal relative to the intrinsic geometric upper bound. Beyond geometry, its noncommutative group structure provides inherent order sensitivity, and its non-nilpotent Lie algebra admits nonzero nested Lie brackets at arbitrary depth, enabling deep order-aware composition. Empirically, $\mathbb{SL}(n)$ consistently outperforms a broad range of representation manifold baselines across graph benchmarks at different scales. It reduces average distortion over the strongest baselines by $44.3\%$ on KEGG and $40.5\%$ on HumanCyc, and improves Hits@20 by $42.8\%$ on OGBL-PPA. Experiments on Flickr30k-Order further support its ability to capture higher order dependencies from ordered composition. Together, these results show how a seemingly simple structural constraint can yield unexpectedly rich geometry, capacity, and composition within a unified representation space.
Comments35 pages, 8 figures
