KromHC: Manifold-Constrained Hyper-Connections with Kronecker-Product Residual Matrices

The success of Hyper-Connections (HC) in neural networks (NN) has also highlighted issues related to training instability and restricted scalability. The Manifold-Constrained Hyper-Connections (mHC) mitigate these challenges by projecting the residual connection space onto a Birkhoff polytope, however, it faces two issues: 1) its iterative Sinkhorn-Knopp (SK) algorithm does not always yield exactly doubly stochastic residual matrices; 2) mHC incurs a prohibitive $\mathcal{O}(n^3C)$ parameter complexity with $n$ as the width of the residual stream and $C$ as the feature dimension. The recently proposed mHC-lite reparametrizes the residual matrix via the Birkhoff-von-Neumann theorem to guarantee double stochasticity, but also faces a factorial explosion in its parameter complexity, $\mathcal{O} \left( nC \cdot n! \right)$. To address both challenges, we propose KromHC, which uses the $\underline{\text{Kro}}$necker products of smaller doubly stochastic matrices to parametrize the residual matrix in $\underline{\text{mHC}}$. By enforcing manifold constraints across the factor residual matrices along each mode of the tensorized residual stream, KromHC guarantees exact double stochasticity of the residual matrices while reducing parameter complexity to only $\mathcal{O}(n^2C)$. Experiments show that KromHC matches or even outperforms other state-of-the-art (SOTA) mHC variants, while requiring significantly fewer trainable parameters. The code is at https://github.com/wz1119/KromHC.