Predictive linear and nonlinear models based on kernel machines or deep neural networks have been used to discover dependencies among time series. This paper proposes an efficient nonlinear modeling approach for multiple time series, with a complexity comparable to linear vector autoregressive (VAR) models while still incorporating nonlinear interactions among different time-series variables. The modeling assumption is that the set of time series is generated in two steps: first, a linear VAR process in a latent space, and second, a set of invertible and Lipschitz continuous nonlinear mappings that are applied per sensor, that is, a component-wise mapping from each latent variable to a variable in the measurement space. The VAR coefficient identification provides a topology representation of the dependencies among the aforementioned variables. The proposed approach models each component-wise nonlinearity using an invertible neural network and imposes sparsity on the VAR coefficients to reflect the parsimonious dependencies usually found in real applications. To efficiently solve the formulated optimization problems, a custom algorithm is devised combining proximal gradient descent, stochastic primal-dual updates, and projection to enforce the corresponding constraints. Experimental results on both synthetic and real data sets show that the proposed algorithm improves the identification of the support of the VAR coefficients in a parsimonious manner while also improving the time-series prediction, as compared to the current state-of-the-art methods.

翻译：基于核机器或深度神经网络的预测性线性和非线性模型已被用于发现时间序列间的依赖关系。本文提出一种高效的非线性多时间序列建模方法，其计算复杂度与线性向量自回归（VAR）模型相当，同时仍能捕捉不同时间序列变量间的非线性交互作用。该建模假设时间序列集合通过两步生成：首先在隐空间中运行线性VAR过程，其次对每个传感器施加一组可逆且Lipschitz连续的非线性映射，即从每个隐变量到测量空间变量的逐分量映射。VAR系数的辨识提供了上述变量间依赖关系的拓扑表征。所提方法采用可逆神经网络对逐分量非线性进行建模，并对VAR系数施加稀疏性约束，以反映实际应用中通常存在的简约依赖关系。为高效求解所构建的优化问题，我们设计了一种结合近端梯度下降、随机原始-对偶更新及投影约束的自定义算法。在合成数据集与真实数据集上的实验结果表明，与现有最先进方法相比，所提算法能以更精简的方式提升VAR系数支撑集的辨识能力，同时改进时间序列预测性能。