提出加权路径签名，让模型更关注近期信息。
The Exponentially Weighted Signature
- 用可学习的加权机制替代传统均匀记忆，支持跨通道耦合
- 在两个随机微分方程回归任务中，性能超越经典签名和指数衰减签名
- 保留代数结构，适合梯度优化，适用于金融、信号等时序建模
路径签名是区间上多维路径的规范表示，但其对历史信息一视同仁，缺乏对过去相关性的内在建模。为此，我们提出指数加权签名（EWS），将指数衰减记忆（EFM）从对角形式推广至一般有界线性算子。该算子支持时间加权中的跨通道耦合，并具备振荡、增长及制度依赖等丰富记忆动态，同时保持经典签名的代数优势。我们证明EWS是张量代数上线性控制微分方程的唯一解，并广义化了状态空间模型以及路径的拉普拉斯与傅里叶变换。EWS的群作用结构支持高效计算，且整个半群动作可通过生成器参数化并学习。我们在两个基于SDE的回归任务中实证展示了EWS在表达能力上显著优于经典签名与EFM。
原文摘要 · Abstract (English)
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.
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