Time series analysis is a widespread task in Natural Sciences, Social Sciences, and Engineering. A fundamental problem is finding an expressive yet efficient-to-compute representation of the input time series to use as a starting point to perform arbitrary downstream tasks. In this paper, we build upon recent works that use the Signature of a path as a feature map and investigate a computationally efficient technique to approximate these features based on linear random projections. We present several theoretical results to justify our approach and empirically validate that our random projections can effectively retrieve the underlying Signature of a path. We show the surprising performance of the proposed random features on several tasks, including (1) mapping the controls of stochastic differential equations to the corresponding solutions and (2) using the Randomized Signatures as time series representation for classification tasks. When compared to corresponding truncated Signature approaches, our Randomizes Signatures are more computationally efficient in high dimensions and often lead to better accuracy and faster training. Besides providing a new tool to extract Signatures and further validating the high level of expressiveness of such features, we believe our results provide interesting conceptual links between several existing research areas, suggesting new intriguing directions for future investigations.
翻译:时间序列分析是自然科学、社会科学和工程领域的一项广泛任务。一个根本问题是找到一个清晰但高效的输入时间序列的表达方式,作为执行任意下游任务的起点。在本文件中,我们以最近使用路径签名作为特征地图的工程为基础,并调查一种基于线性随机预测的计算高效技术,以近似这些特征。我们提出了一些理论结果,以证明我们的随机预测能够有效地检索路径的基本标志。我们展示了若干任务的拟议随机特性的惊人性能,其中包括:(1) 绘制对相应解决方案的随机差异方程式的控制图,(2) 使用随机签名作为分类任务的时间序列。与相应的松散签名方法相比,我们的随机签名在高维度方面计算效率更高,并往往导致更准确和更快的培训。除了提供一种新的工具来提取签名和进一步验证这些特征的高度清晰度外,我们认为我们的成果为几个现有研究领域提供了令人感兴趣的概念联系,为今后调查提供了新的方向。