The signature is a canonical representation of a multidimensional path over an interval. However, it treats all historical information uniformly, offering no intrinsic mechanism for contextualising the relevance of the past. To address this, we introduce the Exponentially Weighted Signature (EWS), generalising the Exponentially Fading Memory (EFM) signature from diagonal to general bounded linear operators. These operators enable cross-channel coupling at the level of temporal weighting together with richer memory dynamics including oscillatory, growth, and regime-dependent behaviour, while preserving the algebraic strengths of the classical signature. We show that the EWS is the unique solution to a linear controlled differential equation on the tensor algebra, and that it generalises both state-space models and the Laplace and Fourier transforms of the path. The group-like structure of the EWS enables efficient computation and makes the framework amenable to gradient-based learning, with the full semigroup action parametrised by and learned through its generator. We use this framework to empirically demonstrate the expressivity gap between the EWS and both the signature and EFM on two SDE-based regression tasks.

翻译：签名是对区间上多维路径的规范表示，然而它对所有历史信息一视同仁，缺乏内在地将过去相关性置于语境中的机制。针对这一问题，我们引入指数加权签名（EWS），将指数衰减记忆（EFM）签名从对角算子推广至一般有界线性算子。这些算子能够在时间加权层面实现跨通道耦合，并带来更丰富的记忆动力学行为（包括振荡、增长和状态依赖行为），同时保留经典签名的代数优势。我们证明EWS是张量代数上线性受控微分方程的唯一解，且它同时推广了状态空间模型以及路径的拉普拉斯变换和傅里叶变换。EWS的群结构特性使其能够高效计算，并适用于基于梯度的学习框架，其中完整的半群作用通过其生成元进行参数化和学习。我们基于该框架，在两个随机微分方程回归任务上通过实验证明了EWS相较于签名和EFM的表达能力差距。