# Time-varying Autoregression with Low Rank Tensors

## TL;DR

This paper introduces a scalable, low-rank tensor approach for learning interpretable, time-varying autoregressive models from multivariate time series, capable of handling high-dimensional data and incorporating priors.

## Contribution

The paper proposes a novel windowed, tensor-based method for modeling time-varying autoregressive systems, with proven convergence and demonstrated effectiveness on synthetic and real data.

## Key findings

- Successfully identifies true rank in noisy synthetic data.
- Outperforms existing methods in scalability and accuracy.
- Effectively models diverse real-world datasets.

## Abstract

We present a windowed technique to learn parsimonious time-varying autoregressive models from multivariate timeseries. This unsupervised method uncovers interpretable spatiotemporal structure in data via non-smooth and non-convex optimization. In each time window, we assume the data follow a linear model parameterized by a system matrix, and we model this stack of potentially different system matrices as a low rank tensor. Because of its structure, the model is scalable to high-dimensional data and can easily incorporate priors such as smoothness over time. We find the components of the tensor using alternating minimization and prove that any stationary point of this algorithm is a local minimum. We demonstrate on a synthetic example that our method identifies the true rank of a switching linear system in the presence of noise. We illustrate our model's utility and superior scalability over extant methods when applied to several synthetic and real-world example: two types of time-varying linear systems, worm behavior, sea surface temperature, and monkey brain datasets.