# Minimax Estimation of Partially-Observed Vector AutoRegressions

arXiv: https://arxiv.org/abs/2106.09327 · HTML: https://mlsanity.com/paper/2106.09327/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 75 references that are in the map.
- First posted on arXiv 17 Jun 2021.

## Builds on (most cited first, 9 of 9)

- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [A Rewriting System for Convex Optimization Problems](https://mlsanity.com/paper/1709.04494/) (2017)
- [Parallelizing the dual revised simplex method](https://mlsanity.com/paper/1503.01889/) (2015)
- [JuMP: A Modeling Language for Mathematical Optimization](https://mlsanity.com/paper/1508.01982/) (2015)
- [Regularized estimation in sparse high-dimensional time series models](https://mlsanity.com/paper/1311.4175/) (2013)
- [Oracle Inequalities for High Dimensional Vector Autoregressions](https://mlsanity.com/paper/1311.0811/) (2013)
- [Estimating Structured Vector Autoregressive Model](https://mlsanity.com/paper/1602.06606/) (2016)
- [A Direct Estimation of High Dimensional Stationary Vector Autoregressions](https://mlsanity.com/paper/1307.0293/) (2013)
- [Missing Data in Sparse Transition Matrix Estimation for Sub-Gaussian Vector Autoregressive Processes](https://mlsanity.com/paper/1802.09511/) (2018)

## Abstract (arXiv)

High-dimensional time series are a core ingredient of the statistical modeling toolkit, for which numerous estimation methods are known.But when observations are scarce or corrupted, the learning task becomes much harder.The question is: how much harder? In this paper, we study the properties of a partially-observed Vector AutoRegressive process, which is a state-space model endowed with a stochastic observation mechanism.Our goal is to estimate its sparse transition matrix, but we only have access to a small and noisy subsample of the state components.Interestingly, the sampling process itself is random and can exhibit temporal correlations, a feature shared by many realistic data acquisition scenarios.We start by describing an estimator based on the Yule-Walker equation and the Dantzig selector, and we give an upper bound on its non-asymptotic error.Then, we provide a matching minimax lower bound, thus proving near-optimality of our estimator.The convergence rate we obtain sheds light on the role of several key parameters such as the sampling ratio, the amount of noise and the number of non-zero coefficients in the transition matrix.These theoretical findings are commented and illustrated by numerical experiments on simulated data.