$\mathbb{SL}(n)$ Representation Learning: An Intrinsic Mixed-Curvature Space with Higher Curvature Capacities and Deeper Order-Aware Composition
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 cap...
| Authors | Xingrun Li, Yusuke Mukuta, Xin Yang et al. |
|---|---|
| Primary category | cs.LG |
| Published | 14 September 2026 |
| Last updated at source | 19 September 2026 |
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