# Higher-order Linear Attention

arXiv: https://arxiv.org/abs/2510.27258 · HTML: https://mlsanity.com/paper/2510.27258/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 33 references that are in the map.
- First posted on arXiv 31 Oct 2025.

## Builds on (most cited first, 28 of 28)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [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)
- [Linformer: Self-Attention with Linear Complexity](https://mlsanity.com/paper/2006.04768/) (2020)
- [Transformers are SSMs: Generalized Models and Efficient Algorithms Through Structured State Space Duality](https://mlsanity.com/paper/2405.21060/) (2024)
- [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)
- [Gated Delta Networks: Improving Mamba2 with Delta Rule](https://mlsanity.com/paper/2412.06464/) (2024)
- [Gated Linear Attention Transformers with Hardware-Efficient Training](https://mlsanity.com/paper/2312.06635/) (2023)
- [Linear Transformers Are Secretly Fast Weight Programmers](https://mlsanity.com/paper/2102.11174/) (2021)
- [Hopfield Networks is All You Need](https://mlsanity.com/paper/2008.02217/) (2020)
- [Parallelizing Linear Transformers with the Delta Rule over Sequence Length](https://mlsanity.com/paper/2406.06484/) (2024)
- [Efficient Attention: Attention with Linear Complexities](https://mlsanity.com/paper/1812.01243/) (2018)
- [Learning to (Learn at Test Time): RNNs with Expressive Hidden States](https://mlsanity.com/paper/2407.04620/) (2024)
- [Random Feature Attention](https://mlsanity.com/paper/2103.02143/) (2021)
- [Titans: Learning to Memorize at Test Time](https://mlsanity.com/paper/2501.00663/) (2024)
- [Using Fast Weights to Attend to the Recent Past](https://mlsanity.com/paper/1610.06258/) (2016)
- [Eagle and Finch: RWKV with Matrix-Valued States and Dynamic Recurrence](https://mlsanity.com/paper/2404.05892/) (2024)
- [RWKV-7 "Goose" with Expressive Dynamic State Evolution](https://mlsanity.com/paper/2503.14456/) (2025)
- [Going Beyond Linear Transformers with Recurrent Fast Weight Programmers](https://mlsanity.com/paper/2106.06295/) (2021)
- [ATLAS: Learning to Optimally Memorize the Context at Test Time](https://mlsanity.com/paper/2505.23735/) (2025)
- [It's All Connected: A Journey Through Test-Time Memorization, Attentional Bias, Retention, and Online Optimization](https://mlsanity.com/paper/2504.13173/) (2025)
- [Lightning Attention-2: A Free Lunch for Handling Unlimited Sequence Lengths in Large Language Models](https://mlsanity.com/paper/2401.04658/) (2024)
- [MesaNet: Sequence Modeling by Locally Optimal Test-Time Training](https://mlsanity.com/paper/2506.05233/) (2025)
- [TransNormerLLM: A Faster and Better Large Language Model with Improved TransNormer](https://mlsanity.com/paper/2307.14995/) (2023)
- [Understanding Transformer from the Perspective of Associative Memory](https://mlsanity.com/paper/2505.19488/) (2025)

## Built on by (most cited first, 2 of 2)

- [Nested Learning: The Illusion of Deep Learning Architectures](https://mlsanity.com/paper/2512.24695/) (2025)
- [Raven: High-Recall Sequence Modeling with Sparse Memory Routing](https://mlsanity.com/paper/2607.25357/) (2026)

## Abstract (arXiv)

The quadratic cost of scaled dot-product attention is a central obstacle to scaling autoregressive language models to long contexts. Linear-time attention and State Space Models (SSMs) provide scalable alternatives but are typically restricted to first-order or kernel-based approximations, which can limit expressivity. We introduce Higher-order Linear Attention (HLA), a causal, streaming mechanism that realizes higher interactions via compact prefix sufficient statistics. In the second-order case, HLA maintains a constant-size state and computes per-token outputs in linear time without materializing any n × n matrices. We give closed-form streaming identities, a strictly causal masked variant using two additional summaries, and a chunk-parallel training scheme based on associative scans that reproduces the activations of a serial recurrence exactly. We further outline extensions to third and higher orders. Collectively, these results position HLA as a principled, scalable building block that combines attention-like, data-dependent mixing with the efficiency of modern recurrent architectures.