Higher-order Linear Attention
Authors: name: Yifan Zhang, name: Zhen Qin, name: Quanquan Gu
Organization: math-ai

The quadratic cost of scaled dot-product attention is a central obstacle to
scaling autoregressive language models to long contexts. Linear-time attention
and State Space Models (SSMs) provide scalable alternatives but are typically
restricted to first-order or kernel-based approximations, which can limit
expressivity. We introduce Higher-order Linear Attention (HLA), a causal,
streaming mechanism that realizes higher interactions via compact prefix
sufficient statistics. In the second-order case, HLA maintains a constant-size
state and computes per-token outputs in linear time without materializing any
n times n matrices. We give closed-form streaming identities, a strictly
causal masked variant using two additional summaries, and a chunk-parallel
training scheme based on associative scans that reproduces the activations of a
serial recurrence exactly. We further outline extensions to third and higher
orders. Collectively, these results position HLA as a principled, scalable
building block that combines attention-like, data-dependent mixing with the
efficiency of modern recurrent architectures. Project Page:
https://github.com/yifanzhang-pro/HLA.