Joint Learning of Topology and Invertible Nonlinearities from Multiple Time Series

Discovery of causal dependencies among time series has been tackled in the past either by using linear models, or using kernel- or deep learning-based nonlinear models, the latter ones entailing great complexity. This paper proposes a nonlinear modelling technique for multiple time series that has a complexity similar to that of linear vector autoregressive (VAR), but it can account for nonlinear interactions for each sensor variable. The modelling assumption is that the time series are generated in two steps: i) a VAR process in a latent space, and ii) a set of invertible nonlinear mappings applied component-wise, mapping each sensor variable into a latent space. Successful identification of the support of the VAR coefficients reveals the topology of the interconnected system. The proposed method enforces sparsity on the VAR coefficients and models the component-wise nonlinearities using invertible neural networks. To solve the estimation problem, a technique combining proximal gradient descent (PGD) and projected gradient descent is designed. Experiments conducted on real and synthetic data sets show that the proposed algorithm provides an improved identification of the support of the VAR coefficients, while improving also the prediction capabilities.