Tight Risk Bound for High Dimensional Time Series Completion

TL;DR

This paper develops a theoretical framework for low-rank matrix completion in high-dimensional, dependent time series data, showing that existing methods are effective and can even perform better under certain conditions.

Contribution

It introduces a general model for multivariate dependent time series and proves that least-square methods with rank penalties achieve near-optimal reconstruction error.

Findings

Reconstruction error matches that of independent data cases.

Additional properties like periodicity or smoothness can improve convergence rates.

Theoretical support for applying low-rank completion to dependent time series.

Abstract

Initially designed for independent datas, low-rank matrix completion was successfully applied in many domains to the reconstruction of partially observed high-dimensional time series. However, there is a lack of theory to support the application of these methods to dependent datas. In this paper, we propose a general model for multivariate, partially observed time series. We show that the least-square method with a rank penalty leads to reconstruction error of the same order as for independent datas. Moreover, when the time series has some additional properties such as periodicity or smoothness, the rate can actually be faster than in the independent case.