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

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.

Machine Learning (cs.LG)

arXiv:2610.20637 [cs.LG]

(or arXiv:2610.20637v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2610.20637 Focus to learn more arXiv-issued DOI via DataCite

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