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

TL;DR

KromHC introduces a scalable, manifold-constrained hyper-connection method using Kronecker products to ensure exact doubly stochastic residual matrices, improving training stability and efficiency in neural networks.

Contribution

It proposes KromHC, a novel approach that employs Kronecker products of smaller matrices to efficiently parametrize residual matrices with exact stochasticity, reducing complexity and enhancing performance.

Findings

KromHC matches or outperforms state-of-the-art methods.

It significantly reduces parameter complexity to O(n^2 C).

Experiments validate improved stability and scalability.

Abstract

The success of Hyper-Connections (HC) in neural networks (NN) has also highlighted issues related to its 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 exact 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 \textbf{KromHC}, which uses the \underline{Kro}necker products of smaller doubly stochastic matrices to parametrize the residual matrix in \underline{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 $\mathcal{O}(n^2C)$. Comprehensive experiments demonstrate that KromHC matches or even outperforms state-of-the-art (SOTA) mHC variants, while requiring significantly fewer trainable parameters. The code is available at \texttt{https://github.com/wz1119/KromHC}.