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$\mathbb{SL}(n)$ Representation Learning: An Intrinsic Mixed-Curvature Space with Higher Curvature Capacities and Deeper Order-Aware Composition

arXiv · AI, language, vision and robotics · article · Sep 14, 2026 · UTC

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 posit

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