# Missing Data in Sparse Transition Matrix Estimation for Sub-Gaussian Vector Autoregressive Processes

## TL;DR

This paper develops new theoretical tools for estimating sparse transition matrices in high-dimensional vector autoregressive models with missing data, addressing challenges posed by dependent covariates and partial observations.

## Contribution

It introduces a novel concentration inequality and a new interaction measure to handle missing data in dependent, high-dimensional time series models.

## Key findings

- Established a concentration result for innovation noise with convex concentration property
- Derived bounds for transition matrix estimation under missing data
- Provided theoretical guarantees for sparse VAR models with partial observations

## Abstract

High-dimensional time series data exist in numerous areas such as finance, genomics, healthcare, and neuroscience. An unavoidable aspect of all such datasets is missing data, and dealing with this issue has been an important focus in statistics, control, and machine learning. In this work, we consider a high-dimensional estimation problem where a dynamical system, governed by a stable vector autoregressive model, is randomly and only partially observed at each time point. Our task amounts to estimating the transition matrix, which is assumed to be sparse. In such a scenario, where covariates are highly interdependent and partially missing, new theoretical challenges arise. While transition matrix estimation in vector autoregressive models has been studied previously, the missing data scenario requires separate efforts. Moreover, while transition matrix estimation can be studied from a high-dimensional sparse linear regression perspective, the covariates are highly dependent and existing results on regularized estimation with missing data from i.i.d.~covariates are not applicable. At the heart of our analysis lies 1) a novel concentration result when the innovation noise satisfies the convex concentration property, as well as 2) a new quantity for characterizing the interactions of the time-varying observation process with the underlying dynamical system.