# RunningTensor: Generalizing Linear Attention to Higher-Order Recurrent States

arXiv: https://arxiv.org/abs/2609.12814 · HTML: https://mlsanity.com/paper/2609.12814/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 44 references that are in the map.
- First posted on arXiv 11 Sep 2026.

## Builds on (most cited first, 30 of 35)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [Qwen2 Technical Report](https://mlsanity.com/paper/2407.10671/) (2024)
- [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)
- [The FineWeb Datasets: Decanting the Web for the Finest Text Data at Scale](https://mlsanity.com/paper/2406.17557/) (2024)
- [RWKV: Reinventing RNNs for the Transformer Era](https://mlsanity.com/paper/2305.13048/) (2023)
- [Simplified State Space Layers for Sequence Modeling](https://mlsanity.com/paper/2208.04933/) (2022)
- [Transformers learn in-context by gradient descent](https://mlsanity.com/paper/2212.07677/) (2022)
- [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)
- [Hungry Hungry Hippos: Towards Language Modeling with State Space Models](https://mlsanity.com/paper/2212.14052/) (2022)
- [Resurrecting Recurrent Neural Networks for Long Sequences](https://mlsanity.com/paper/2303.06349/) (2023)
- [Parallelizing Linear Transformers with the Delta Rule over Sequence Length](https://mlsanity.com/paper/2406.06484/) (2024)
- [Learning to (Learn at Test Time): RNNs with Expressive Hidden States](https://mlsanity.com/paper/2407.04620/) (2024)
- [Titans: Learning to Memorize at Test Time](https://mlsanity.com/paper/2501.00663/) (2024)
- [LSTM: A Search Space Odyssey](https://mlsanity.com/paper/1503.04069/) (2015)
- [Zoology: Measuring and Improving Recall in Efficient Language Models](https://mlsanity.com/paper/2312.04927/) (2023)
- [Transformer Quality in Linear Time](https://mlsanity.com/paper/2202.10447/) (2022)
- [An Empirical Study of Mamba-based Language Models](https://mlsanity.com/paper/2406.07887/) (2024)
- [Samba: Simple Hybrid State Space Models for Efficient Unlimited Context Language Modeling](https://mlsanity.com/paper/2406.07522/) (2024)
- [Parallelizing Linear Recurrent Neural Nets Over Sequence Length](https://mlsanity.com/paper/1709.04057/) (2017)

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

- [CyFA: Linear Sequence Modeling with Relative-Time-Partitioned Memory](https://mlsanity.com/paper/2609.36259/) (2026)

## Abstract (arXiv)

Linear attention and state-space models provide linear-time sequence modeling, but their recurrent memory remains a second-order tensor (a matrix), limiting the order of interactions that can be represented in the state. We introduce the RunningTensor, which generalizes this memory to an order-o tensor, updated by a rank-1 outer product and read by contracting against o-1 vector queries. Order 2 recovers linear attention; we study order 3 as a proof of concept, retaining both recurrent and parallel forms while remaining linear in sequence length T and improving working memory capacity from O(W2) to O(Wo). On synthetic multi-query associative recall, RunningTensor outperforms linear-attention and SSM baselines. After pretraining, it also improves performance on language-understanding and non-synthetic retrieval tasks, suggesting that higher-order recurrent state can provide useful additional memory capacity beyond matrix-valued state.