Signatures are iterated path integrals of continuous and discrete-time processes, and their universal nonlinearity linearizes the problem of feature selection in time series data analysis. This paper studies the consistency of signature using Lasso regression, both theoretically and numerically. We establish conditions under which the Lasso regression is consistent both asymptotically and in finite sample. Furthermore, we show that the Lasso regression is more consistent with the It\^o signature for time series and processes that are closer to the Brownian motion and with weaker inter-dimensional correlations, while it is more consistent with the Stratonovich signature for mean-reverting time series and processes. We demonstrate that signature can be applied to learn nonlinear functions and option prices with high accuracy, and the performance depends on properties of the underlying process and the choice of the signature.

翻译：签名是连续和离散时间过程的迭代路径积分，其普适非线性特性将时间序列数据分析中的特征选择问题线性化。本文从理论和数值两方面研究了使用Lasso回归的签名一致性。我们建立了Lasso回归在渐近和有限样本下均保持一致性的条件。进一步研究表明：对于更接近布朗运动且维度间相关性较弱的时间序列和过程，Lasso回归与Itô签名的一致性更高；而对于均值回归型时间序列和过程，Lasso回归与Stratonovich签名的一致性更高。我们证明签名可用于高精度学习非线性函数和期权价格，其性能取决于基础过程的特性及签名的选择。