# Variational Linear Attention: Stable Associative Memory for Long-Context Transformers

arXiv: https://arxiv.org/abs/2605.11196 · HTML: https://mlsanity.com/paper/2605.11196/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 11 references that are in the map.
- First posted on arXiv 11 May 2026.

## Builds on (most cited first, 7 of 7)

- [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)
- [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)
- [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)
- [Zoology: Measuring and Improving Recall in Efficient Language Models](https://mlsanity.com/paper/2312.04927/) (2023)

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

- [MMLA: Memory-Mediated Learning Architecture for Predictive Dual-State Adaptation](https://mlsanity.com/paper/2606.28876/) (2026)
- [The Orthogonalized Read Is a Removable Training Scaffold for Recurrent Memory](https://mlsanity.com/paper/2607.19390/) (2026)

## Abstract (arXiv)

Linear attention reduces the quadratic cost of softmax attention to O(T), but its memory state grows as O(T) in Frobenius norm, causing progressive interference between stored associations. We introduce Variational Linear Attention (VLA), which reframes the memory update as an online regularised least-squares problem with an adaptive penalty matrix maintained via the Sherman-Morrison rank-1 formula. We prove that normalising the write direction to unit length gives the recurrence Jacobian spectral norm exactly 1 for all sequence lengths and head dimensions (Proposition 2), and that the state norm is self-limiting under bounded inputs (Proposition 1). Empirically, VLA reduces |St|F by 109× relative to standard linear attention at T=1,000, achieves near-perfect exact-match accuracy on multi-query associative recall within the effective per-head memory regime (npairs < dh), maintaining substantially higher retrieval performance than DeltaNet and standard linear attention under increasing memory load, and maintains 62% accuracy at the per-head capacity boundary. A Triton-fused kernel achieves 14× speedup over sequential Python and O(T) scaling, crossing below softmax attention latency at approximately 43,000 tokens.