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[Submitted on 2 Oct 2026] Title:Expanding Attention: A Taylor-Series Route to Linear-Time Transformers with a Fixed Number of Recurrent States View a PDF of the paper titled Expanding Attention: A Taylor-Series Route to Linear-Time Transformers with a Fixed Number of Recurrent States, by Zicong Huang 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. Subjects: Machine Learning (cs.LG) Cite as: arXiv:2610.20637 [cs.LG] (or arXiv:2610.20637v1 [cs.LG] for this version) https://doi.org/10.48550/arXiv.2610.20637 arXiv-issued DOI via DataCite Submission history From: Zicong Huang [view email] [v1] Fri, 2 Oct 2026 03:39:07 UTC (259 KB) Full-text links: Access Paper: View a PDF of the paper titled Expanding Attention: A Taylor-Series Route to Linear-Time Transformers with a Fixed Number of Recurrent States, by Zicong Huang View PDF HTML (experimental) TeX Source view license Additional Features Audio Summary Current browse context: cs.LG new | recent | 2026-10 Change to browse by: cs References & Citations NASA ADS Google Scholar Semantic Scholar Loading... Data provided by: Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) IArxiv recommender toggle IArxiv Recommender (What is IArxiv?) Author Venue Institution Topic About arXivLabs 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. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)