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$\mathbb{SL}(n)$ Representation Learning: An Intrinsic Mixed-Curvature Space with Higher Curvature Capacities and Deeper Order-Aware Composition — arXiv:2609.15083

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.15083
License (as archived): CC0-1.0
Archived: 2026-10-16T00:11:19.519Z
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How to cite

$\mathbb{SL}(n)$ Representation Learning: An Intrinsic Mixed-Curvature Space with Higher Curvature Capacities and Deeper Order-Aware Composition — arXiv:2609.15083 — arXiv (en), https://arxiv.org/abs/2609.15083 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_4ff021fb0eda (Metraipolis archive dc_4ff021fb0eda, sha256 061554fcc069f7effc3abd82e169b820b0cda7180aa360eba0c72b3f7ce95dad)

BibTeX

@misc{dc4ff021fb0eda,
  title = {$mathbbSL(n)$ Representation Learning: An Intrinsic Mixed-Curvature Space with Higher Curvature Capacities and Deeper Order-Aware Composition — arXiv:2609.15083},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.15083},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_4ff021fb0eda; sha256 061554fcc069f7effc3abd82e169b820b0cda7180aa360eba0c72b3f7ce95dad},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.15083. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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.

Tema degli esploratori: agenti e intelligenza artificiale (retro-catalogo arXiv: annunci del 2026-09-15 in cs.LG).
Categorie arXiv: Machine Learning (cs.LG).
Pagina del lavoro: https://arxiv.org/abs/2609.15083 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-15T23:54:14.737Z dalla pagina https://arxiv.org/catchup/cs.LG/2026-09-15?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).