Discrete empirical interpolation in the tensor t-product framework

TL;DR

This paper introduces t-Q-DEIM, a tensor-based extension of DEIM that preserves data structure, improves approximation accuracy, and reduces computational costs in tensor-valued data applications.

Contribution

The paper develops a tensor t-product based DEIM extension (t-Q-DEIM) that avoids data reshaping, providing better structural preservation and efficiency.

Findings

t-Q-DEIM outperforms standard DEIM in approximation accuracy

The method significantly reduces computational costs

Numerical experiments on five datasets validate the approach

Abstract

The discrete empirical interpolation method (DEIM) is a well-established approach, widely used for state reconstruction using sparse sensor/measurement data, nonlinear model reduction, and interpretable feature selection. We introduce the tensor t-product Q-DEIM (t-Q-DEIM), an extension of the DEIM framework for dealing with tensor-valued data. The proposed approach seeks to overcome one of the key drawbacks of DEIM, viz., the need for matricizing the data, which can distort any structural and/or geometric information. Our method leverages the recently developed tensor t-product algebra to avoid reshaping the data. In analogy with the standard DEIM, we formulate and solve a tensor-valued least-squares problem, whose solution is achieved through an interpolatory projection. We develop a rigorous, computable upper bound for the error resulting from the t-Q-DEIM approximation. Using five different tensor-valued datasets, we numerically illustrate the better approximation properties of t-Q-DEIM and the significant computational cost reduction it offers.