SL n space improves graph and order data representation accuracy
$\mathbb{SL}(n)$ Representation Learning: An Intrinsic Mixed-Curvature Space with Higher Curvature Capacities and Deeper Order-Aware Composition
Machine Learning
Summary
Some data has complex shapes and relationships that are hard to capture using simple geometry. The authors introduce a new mathematical space, called SL(n), which naturally combines different types of geometric curvature in one setting. This space also keeps track of the order in data, which helps understand sequences and compositions better. When tested on various graph and ordered data tasks, SL(n) performed better than existing methods, reducing errors and capturing deeper patterns.
What this means in practice
- •For machine learning engineers: Represent complex graph data with improved accuracy using SL(n) geometry in model embeddings.
- •For natural language processing teams: Enhance sequence modeling by capturing higher order dependencies through SL(n)'s order-sensitive representations.
Authors
Xingrun Li, Yusuke Mukuta, Xin Yang, Yinyu Ye, Tatsuya Harada
Abstract
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.