# ButterflyMoE: Compression-Scalable Ternary Experts via Structured Butterfly Orbits

arXiv: https://arxiv.org/abs/2601.13563 · HTML: https://mlsanity.com/paper/2601.13563/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in MoE large language models, one of 12 directions in Mixture-of-experts.
- Builds on 17 of its 17 references that are in the map.
- First posted on arXiv 20 Jan 2026.

## Placed in

- Mixture-of-experts / [MoE large language models](https://mlsanity.com/moe/moe-large-language-models/)

## Builds on (most cited first, 17 of 17)

- [Switch Transformers: Scaling to Trillion Parameter Models with Simple and Efficient Sparsity](https://mlsanity.com/paper/2101.03961/) (2021)
- [Estimating or Propagating Gradients Through Stochastic Neurons for Conditional Computation](https://mlsanity.com/paper/1308.3432/) (2013)
- [SmoothQuant: Accurate and Efficient Post-Training Quantization for Large Language Models](https://mlsanity.com/paper/2211.10438/) (2022)
- [QuaRot: Outlier-Free 4-Bit Inference in Rotated LLMs](https://mlsanity.com/paper/2404.00456/) (2024)
- [Loss Surfaces, Mode Connectivity, and Fast Ensembling of DNNs](https://mlsanity.com/paper/1802.10026/) (2018)
- [Git Re-Basin: Merging Models modulo Permutation Symmetries](https://mlsanity.com/paper/2209.04836/) (2022)
- [SpinQuant: LLM quantization with learned rotations](https://mlsanity.com/paper/2405.16406/) (2024)
- [The Era of 1-bit LLMs: All Large Language Models are in 1.58 Bits](https://mlsanity.com/paper/2402.17764/) (2024)
- [On the Representation Collapse of Sparse Mixture of Experts](https://mlsanity.com/paper/2204.09179/) (2022)
- [Monarch: Expressive Structured Matrices for Efficient and Accurate Training](https://mlsanity.com/paper/2204.00595/) (2022)
- [Learning Fast Algorithms for Linear Transforms Using Butterfly Factorizations](https://mlsanity.com/paper/1903.05895/) (2019)
- [MoE-I2: Compressing Mixture of Experts Models through Inter-Expert Pruning and Intra-Expert Low-Rank Decomposition](https://mlsanity.com/paper/2411.01016/) (2024)
- [QMoE: Practical Sub-1-Bit Compression of Trillion-Parameter Models](https://mlsanity.com/paper/2310.16795/) (2023)
- [Mixture Compressor for Mixture-of-Experts LLMs Gains More](https://mlsanity.com/paper/2410.06270/) (2024)
- [Mixture of Quantized Experts (MoQE): Complementary Effect of Low-bit Quantization and Robustness](https://mlsanity.com/paper/2310.02410/) (2023)
- [MoBE: Mixture-of-Basis-Experts for Compressing MoE-based LLMs](https://mlsanity.com/paper/2508.05257/) (2025)
- [PuzzleMoE: Efficient Compression of Large Mixture-of-Experts Models via Sparse Expert Merging and Bit-packed inference](https://mlsanity.com/paper/2511.04805/) (2025)

## Abstract (arXiv)

In current Mixture of Experts (MoE) architectures, linear memory scaling is present, the memory grows as the number of experts increases. N independent expert weight matrices require O(N · d2) memory which exceeds the memory budget of edge devices. Current compression methods like quantization, pruning, and low-rank factorization reduce constant factors, but the scaling bottleneck is still unresolved. We introduce ButterflyMoE, a method which treats experts not as independent matrices but as geometric reorientations of a shared quantized substrate. Diversity amongst the experts arises from viewing different angles of the shared capacity and not from redundant storage. Learned rotations are applied to a shared ternary prototype. With this, each expert yields O(d2 + N · d log d) memory-reducing per-expert cost from O(d2) to O(d log d). The key insight is that training these rotations with quantization reduces activation outliers and stabilizes extreme low-bit training where other static methods collapse. Across language modeling benchmarks, ButterflyMoE achieves 80× memory reduction at 8 experts with a highly favorable memory-accuracy tradeoff.At this 80x compression ButterflyMoE outperforms an equal memory dense baseline, showing that orbital parameterization extracts fundamentally more utility per byte. When scaled up to 256 experts, ButterflyMoE asymptotically compresses the memory by 150 ×. ButterflyMoE reduces the constant factor of linear scaling with compression ratio growing with the expert count.