Expanding Attention: A Taylor-Series Route to Linear-Time Transformers with a Fixed Number of Recurrent States

Computer Science > Machine Learning

[Submitted on 2 Oct 2026]

Title:Expanding Attention: A Taylor-Series Route to Linear-Time Transformers with a Fixed Number of Recurrent States

View PDF HTML (experimental)Abstract:We expand the exponential kernel of attention into a truncated Taylor series of $n$ terms and factorize each term into a tensor-power inner product, which collapses the causal history into exactly $n$ recurrent states, each updated by one rank-one outer product per token and independent of context length $N$. The recursion is lossless with respect to the truncated kernel ($2.2 \times 10^{-16}$ in float64), and the states are additive across chunks ($5.7 \times 10^{-14}$), so prefill admits a parallel scan. Measured exponents are $t \propto N^{2.28}$ for softmax against $N^{0.91}$ for states, with a crossover at $N \approx 728$ and $20.8\times$ at $N = 8192$; per-token decoding latency is constant at $7.3\,\mu$s against $N^{1.04}$ for softmax, giving $76.0\times$ at $N = 65536$. Accuracy improves exponentially in $n$ while cost grows combinatorially, since the $m$-th state is $\binom{d+m-1}{m}$ wide---a bound confirmed by SVD---so the optimum is second order, $n = 3$, at $6.5 \times 10^{-4}$ error and $9{,}945$ elements per head. Query and key normalization is a prerequisite: without it the argument exceeds the convergence radius and adding terms stops helping.

Additional Features

References & Citations

Loading...

Bibliographic and Citation Tools

Bibliographic Explorer (What is the Explorer?)

Connected Papers (What is Connected Papers?)

Litmaps (What is Litmaps?)

scite Smart Citations (What are Smart Citations?)

Code, Data and Media Associated with this Article

alphaXiv (What is alphaXiv?)

CatalyzeX Code Finder for Papers (What is CatalyzeX?)

DagsHub (What is DagsHub?)

Gotit.pub (What is GotitPub?)

Hugging Face (What is Huggingface?)

ScienceCast (What is ScienceCast?)

Demos

Recommenders and Search Tools

Influence Flower (What are Influence Flowers?)

CORE Recommender (What is CORE?)

IArxiv Recommender

(What is IArxiv?)

arXivLabs: experimental projects with community collaborators

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.