# TEMPER: Tensorized Efficient Manifold-constrained Parameterization for Expressive Residual Routing

arXiv: https://arxiv.org/abs/2608.07851 · HTML: https://mlsanity.com/paper/2608.07851/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 20 references that are in the map.
- First posted on arXiv 8 Aug 2026.

## Builds on (most cited first, 9 of 9)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Tensorizing Neural Networks](https://mlsanity.com/paper/1509.06569/) (2015)
- [TensorGPT: Efficient Compression of Large Language Models based on Tensor-Train Decomposition](https://mlsanity.com/paper/2307.00526/) (2023)
- [mHC-lite: You Don't Need 20 Sinkhorn-Knopp Iterations](https://mlsanity.com/paper/2601.05732/) (2026)
- [TensorLLM: Tensorising Multi-Head Attention for Enhanced Reasoning and Compression in LLMs](https://mlsanity.com/paper/2501.15674/) (2025)
- [KromHC: Manifold-Constrained Hyper-Connections with Kronecker-Product Residual Matrices](https://mlsanity.com/paper/2601.21579/) (2026)
- [Domain-Aware Tensor Network Structure Search](https://mlsanity.com/paper/2505.23537/) (2025)
- [Tucker Attention: A generalization of approximate attention mechanisms](https://mlsanity.com/paper/2603.30033/) (2026)

## Built on by (most cited first, 1 of 1)

- [The Birkhoff Geometry of Manifold-Constrained Hyper-Connections: Two Channels, Vertex Viscosity, and Sinkhorn as a Retraction](https://mlsanity.com/paper/2610.06653/) (2026)

## Abstract (arXiv)

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 underlineTensorized underlineEfficient underlineManifold-constrained underlineParameterization for underlineExpressive Residual underlineRouting (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.