# Not All Dimensions Are Equal: Outlier-Aware Attention Linearization with Grouped Quadratic Kernels

arXiv: https://arxiv.org/abs/2610.13674 · HTML: https://mlsanity.com/paper/2610.13674/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 27 of its 29 references that are in the map.
- First posted on arXiv 9 Oct 2026.

## Builds on (most cited first, 27 of 27)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [Efficient Streaming Language Models with Attention Sinks](https://mlsanity.com/paper/2309.17453/) (2023)
- [Transformers are RNNs: Fast Autoregressive Transformers with Linear Attention](https://mlsanity.com/paper/2006.16236/) (2020)
- [Rethinking Attention with Performers](https://mlsanity.com/paper/2009.14794/) (2020)
- [LLM.int8(): 8-bit Matrix Multiplication for Transformers at Scale](https://mlsanity.com/paper/2208.07339/) (2022)
- [RWKV: Reinventing RNNs for the Transformer Era](https://mlsanity.com/paper/2305.13048/) (2023)
- [Retentive Network: A Successor to Transformer for Large Language Models](https://mlsanity.com/paper/2307.08621/) (2023)
- [Random Feature Attention](https://mlsanity.com/paper/2103.02143/) (2021)
- [Massive Activations in Large Language Models](https://mlsanity.com/paper/2402.17762/) (2024)
- [Simple linear attention language models balance the recall-throughput tradeoff](https://mlsanity.com/paper/2402.18668/) (2024)
- [The Hedgehog & the Porcupine: Expressive Linear Attentions with Softmax Mimicry](https://mlsanity.com/paper/2402.04347/) (2024)
- [BERT Busters: Outlier Dimensions that Disrupt Transformers](https://mlsanity.com/paper/2105.06990/) (2021)
- [Finetuning Pretrained Transformers into RNNs](https://mlsanity.com/paper/2103.13076/) (2021)
- [PolySketchFormer: Fast Transformers via Sketching Polynomial Kernels](https://mlsanity.com/paper/2310.01655/) (2023)
- [Systematic Outliers in Large Language Models](https://mlsanity.com/paper/2502.06415/) (2025)
- [Liger: Linearizing Large Language Models to Gated Recurrent Structures](https://mlsanity.com/paper/2503.01496/) (2025)
- [DiJiang: Efficient Large Language Models through Compact Kernelization](https://mlsanity.com/paper/2403.19928/) (2024)
- [On Learning the Transformer Kernel](https://mlsanity.com/paper/2110.08323/) (2021)
- [Linear Transformers with Learnable Kernel Functions are Better In-Context Models](https://mlsanity.com/paper/2402.10644/) (2024)
- [TaylorShift: Shifting the Complexity of Self-Attention from Squared to Linear (and Back) using Taylor-Softmax](https://mlsanity.com/paper/2403.02920/) (2024)
- [Exact Conversion of In-Context Learning to Model Weights in Linearized-Attention Transformers](https://mlsanity.com/paper/2406.02847/) (2024)
- [Transformers Are Born Biased: Structural Inductive Biases at Random Initialization and Their Practical Consequences](https://mlsanity.com/paper/2602.05927/) (2026)
- [Spectraformer: A Unified Random Feature Framework for Transformer](https://mlsanity.com/paper/2405.15310/) (2024)
- [The Structural Origin of Attention Sink: Variance Discrepancy, Super Neurons, and Dimension Disparity](https://mlsanity.com/paper/2605.06611/) (2026)

## Abstract (arXiv)

Linearizing softmax attention offers a route to efficient Transformer inference, but preserving pretrained attention remains challenging. We identify an important source of approximation error overlooked by standard linearization methods: a small number of outlier dimensions contribute disproportionately to query-key dot products and account for a large fraction of the approximation error. Motivated by this heterogeneity, we propose Outlier-Aware Attention Linearization (OAL), a dimension-aware method for constructing quadratic attention approximations. Starting from a second-order Taylor approximation, we introduce a Hadamard-product formulation that makes individual query-key contributions explicit. This formulation relaxes globally shared coefficients to support dimension-specific linear terms and cross-dimensional quadratic interactions while preserving separable evaluation. To retain this flexibility with fewer parameters, OAL groups dimensions with similar contribution scales to share coefficients, while allowing distinct treatment of the few outliers. The resulting positive quadratic kernel is calibrated against pretrained attention outputs and admits linear-time evaluation with respect to sequence length for fixed head dimensions. On Qwen2.5-0.5B, OAL reduces frozen-attention output relative MSE by approximately 8.9% compared with DiJiang. Replacing middle-layer attention modules and jointly fine-tuning the kernel coefficients with LoRA further lowers WikiText-103 perplexity from 31.71 to 27.50 compared with a learned global quadratic, while improving accuracy on PIQA, HellaSwag, and ARC-E. These results demonstrate the value of outlier-aware dimension grouping for both attention approximation and pretrained-model conversion. The code is available at https://github.com/wuqianyue09/OAL