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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.20896 |
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| _version_ | 1866910062663958528 |
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| author | Liu, Zhaoyi Zhang, Haichuan Li, Ang |
| author_facet | Liu, Zhaoyi Zhang, Haichuan Li, Ang |
| contents | Hyper-Connections (HC) generalize residual connections into multiple streams, employing residual matrices for cross-stream feature mixing to enrich model expressivity. However, unconstrained mixing disrupts the identity mapping property intrinsic to the residual connection, causing unstable training. To address this, Manifold-Constrained Hyper-Connections (mHC) and its variant restrict these matrices to the Birkhoff polytope (doubly stochastic matrices) via Sinkhorn iterations or permutation-based parameterizations. We reveal three limitations of this polytope constraint: (1) identity degeneration, where learned matrices collapse around the identity and diminish cross-stream interactions, (2) an expressivity bottleneck, as the non-negativity constraint prevents subtractive feature disentanglement, and (3) parameterization inefficiencies, manifesting as unstable Sinkhorn iterations or the factorial-scaling overhead of permutation-based parameterizations. To overcome these flaws, we propose Spectral-Sphere-Constrained Hyper-Connections (sHC). By geometrically shifting the feasible set from a rigid polytope to a spectral norm sphere, sHC allows negative entries, unlocking subtractive interactions for selective feature diversification. This shift eliminates unstable Sinkhorn projections and factorial parameterization, enabling expressive, non-degenerate residual matrices while preserving training stability. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_20896 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Beyond the Birkhoff Polytope: Spectral-Sphere-Constrained Hyper-Connections Liu, Zhaoyi Zhang, Haichuan Li, Ang Machine Learning Artificial Intelligence Hyper-Connections (HC) generalize residual connections into multiple streams, employing residual matrices for cross-stream feature mixing to enrich model expressivity. However, unconstrained mixing disrupts the identity mapping property intrinsic to the residual connection, causing unstable training. To address this, Manifold-Constrained Hyper-Connections (mHC) and its variant restrict these matrices to the Birkhoff polytope (doubly stochastic matrices) via Sinkhorn iterations or permutation-based parameterizations. We reveal three limitations of this polytope constraint: (1) identity degeneration, where learned matrices collapse around the identity and diminish cross-stream interactions, (2) an expressivity bottleneck, as the non-negativity constraint prevents subtractive feature disentanglement, and (3) parameterization inefficiencies, manifesting as unstable Sinkhorn iterations or the factorial-scaling overhead of permutation-based parameterizations. To overcome these flaws, we propose Spectral-Sphere-Constrained Hyper-Connections (sHC). By geometrically shifting the feasible set from a rigid polytope to a spectral norm sphere, sHC allows negative entries, unlocking subtractive interactions for selective feature diversification. This shift eliminates unstable Sinkhorn projections and factorial parameterization, enabling expressive, non-degenerate residual matrices while preserving training stability. |
| title | Beyond the Birkhoff Polytope: Spectral-Sphere-Constrained Hyper-Connections |
| topic | Machine Learning Artificial Intelligence |
| url | https://arxiv.org/abs/2603.20896 |