The Birkhoff Geometry of Manifold-Constrained Hyper-Connections: Two Channels, Vertex Viscosity, and Sinkhorn as a Retraction
Hyper-connections widen the residual stream of a Transformer to $n$ parallel streams. Their manifold-constrained version (mHC) mixes the streams at each layer with a doubly stochastic matrix, which it computes by Sinkhorn normalization of exponentiated logits. We give a geometric theory of this design on the Birkhoff polytope. First, a doubly stochastic mixer splits the stream into a mean channel, on which mHC is exactly a residual network, and a difference channel, which each layer contracts by its second singular value $σ_2 \le 1 - n \min_{ij} H_{ij}$. Thus the extra width is a fading memory with a horizon of $1/(1-σ_2)$ layers, and among nonnegative mixers only the permutations do not ...
| Authors | Xiaoyu Li, Zhizhou Sha, Chiwun Yang |
|---|---|
| Primary category | cs.LG |
| Published | 5 October 2026 |
| Last updated at source | 5 October 2026 |
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