Low Rank and Structured Modeling of High-Dimensional Vector Autoregressions

Network modeling of high-dimensional time series data is a key learning task due to its widespread use in a number of application areas, including macroeconomics, finance, and neuroscience. While the problem of sparse modeling based on vector autoregressive models (VAR) has been investigated in depth in the literature, more complex network structures that involve low rank and group sparse components have received considerably less attention, despite their presence in data. Failure to account for low-rank structures results in spurious connectivity among the observed time series, which may lead practitioners to draw incorrect conclusions about pertinent scientific or policy questions. In order to accurately estimate a network of Granger causal interactions after accounting for latent effects, we introduce a novel approach for estimating low-rank and structured sparse high-dimensional VAR models. We introduce a regularized framework involving a combination of nuclear norm and lasso (or group lasso) penalties. Subsequently, we establish nonasymptotic probabilistic upper bounds on the estimation error rates of the low-rank and the structured sparse components. We also introduce a fast estimation algorithm and finally demonstrate the performance of the proposed modeling framework over standard sparse VAR estimates through numerical experiments on synthetic and real data.