TBP-mHC: full expressivity for manifold-constrained hyper connections through transportation polytopes

Manifold-constrained hyper-connections (mHCs) have recently been proposed as a principled extension of hyper-connections, where the residual mixing matrices are constrained to be doubly stochastic via projection onto the Birkhoff polytope. In practical mHC implementations, this constraint is enforced by Sinkhorn-Knopp iterations, and the backward pass relies on unrolling the iterative solver. This design introduces substantial computation and memory overhead, and may also yield inaccurate projections when the algorithm converges slowly on challenging inputs, undermining the intended norm-control and stability guarantees of mHCs. In this work, we focus on the practically important 4x4 Birkhoff projection setting and develop an end-to-end acceleration framework. By leveraging the dual formulation, we reduce the problem to a three-dimensional unconstrained convex problem and solve it with Newton's method, achieving fast convergence and high accuracy. For the backward pass, we replace the unrolled differentiation with implicit differentiation, yielding exact gradients without storing intermediate states. To exploit massive parallelism, we design a warp-level CUDA kernel that uses only register-level primitives, avoiding global and shared memory I/O. Extensive experiments against representative open-source baselines demonstrate that the proposed solver yields substantially more reliable doubly stochastic projections -- especially when the input magnitude is large -- and achieves significant end-to-end speedups (including the backward pass), reaching over 20x acceleration at large batch sizes while maintaining orders of magnitude smaller marginal errors.

Residual connections rely on a static residual pathway, and are essential for training deep neural networks. Hyper-connections (HC) increase the expressivity of residual routing by incorporating multiple residual streams and learning dynamic information flow, while manifold-constrained (mHC) variants stabilize training through doubly stochastic residual mixing. However, a generator-level bottleneck remains in existing methods: they use dense, unstructured generators for pre-branch aggregation, residual mixing, and post-branch redistribution, which results in parameter count growing rapidly with the number of streams. To address this issue, we propose \underline{\textbf{T}}ensorized \underline{\textbf{E}}fficient \underline{\textbf{M}}anifold-constrained \underline{\textbf{P}}arameterization for \underline{\textbf{E}}xpressive Residual \underline{\textbf{R}}outing (\textbf{TEMPER}), which represents these generators as multi-way tensors over the input-stream, feature, and output-stream modes, and parameterizes them using tensor networks. Such a structured low-rank formulation is shown to preserve token-dependent manifold-constrained routing interface while substantially reducing parameter growth. It also promotes interpretability and intuition, as: i) tensor ranks control the dimensionality of the learned routing subspace, with full ranks recovering dense routing; while ii) the generator approximation errors bound differences in routing logits and, consequently, in the routed-block outputs. Comprehensive experiments show that TEMPER matches or outperforms existing methods across language modeling and commonsense reasoning tasks, while requiring substantially fewer additional parameters. At eight residual streams, TEMPER achieves the best CORE score while using about $84\%$ fewer additional parameters than mHC, thus showing a stronger performance-parameter efficiency trade-off.

Manifold-Constrained Hyper-Connections (mHC) introduce a stability-motivated variant of multi stream residual mixing by constraining residual stream mixing matrices to the manifold of doubly stochastic matrices via Sinkhorn-Knopp projection. In his work, we study whether mHC-style constrained multi-stream residual topology transfers effectively to state space model (SSM) language modeling. We implement a static mHC mechanism around an SSM block by expanding the residual stream into multiple parallel streams, aggregating streams into a single SSM input through simplex-constrained pre-mixing, scattering the SSM output back to streams through simplex-constrained post-mixing, and applying Sinkhorn-projected residual stream mixing at each layer. We further introduce stream-specialized adapters that add lightweight stream-specific capacity through a shared bottleneck with per-stream scaling, applied both before stream aggregation and after the SSM output prior to scattering. We evaluate baseline single-stream SSM, static mHC SSM, and mHC SSM with adapters on WikiText-2 using identical training settings and report checkpoint-based validation loss, perplexity, throughput, and peak GPU memory. Under the reported fair checkpoint evaluation, static mHC improves validation loss from 6.3507 to 6.2448 and reduces perplexity from 572.91 to 515.35, while mHC with adapters further improves validation loss to 6.1353 and perplexity to 461.88. These gains are accompanied by modest throughput reductions from 1025.52 to 964.81 and 938.90 tokens per second, and increased peak memory from 2365 MB to 2568 MB and 3092 MB. The results suggest that mHC-inspired constrained multi-stream residual mixing can yield measurable quality improvements in SSM language models and that stream-specialized adapter capacity can further enhance performance with predictable efficiency tradeoffs.