Structured state-space models (SSMs) such as S4, stemming from the seminal
work of Gu et al., are gaining popularity as effective approaches for modeling
sequential data. Deep SSMs demonstrate outstanding performance across a diverse
set of domains, at a reduced training and inference cost compared to
attention-based transformers. Recent developments show that if the linear
recurrence powering SSMs allows for multiplicative interactions between inputs
and hidden states (e.g. GateLoop, Mamba, GLA), then the resulting architecture
can surpass in both in accuracy and efficiency attention-powered foundation
models trained on text, at scales of billion parameters. In this paper, we give
theoretical grounding to this recent finding using tools from Rough Path
Theory: we show that when random linear recurrences are equipped with simple
input-controlled transitions (selectivity mechanism), then the hidden state is
provably a low-dimensional projection of a powerful mathematical object called
the signature of the input -- capturing non-linear interactions between tokens
at distinct timescales. Our theory not only motivates the success of modern
selective state-space models such as Mamba but also provides a solid framework
to understand the expressive power of future SSM variants.

基于 Rough Path Theory，本研究证明了具备选择性机制的随机线性递归在输入控制条件下可产生低维投射的隐藏状态，并阐述了现代选择性状态空间模型的成功和未来 SSM 变体的表达能力。