# Low Rank and Structured Modeling of High-dimensional Vector Autoregressions

## TL;DR

This paper introduces a novel regularized approach for estimating high-dimensional vector autoregressive models with low-rank and structured sparse components, improving network inference in complex time series data.

## Contribution

It develops a regularized framework combining nuclear norm and lasso penalties, along with a fast algorithm, to better capture low-rank and structured sparsity in VAR models.

## Key findings

- Improved estimation accuracy over standard sparse VAR methods.
- Theoretical bounds on estimation error rates.
- Effective performance demonstrated on synthetic and real datasets.

## Abstract

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) penalty. Further, and subsequently establish non-asymptotic 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.