Matrix Approximation with Side Information: When Column Sampling is Enough

TL;DR

This paper introduces a new matrix approximation method that leverages limited column sampling and structural side information, with applications in quantum chemistry where full matrix computation is costly.

Contribution

It presents a novel algorithm that estimates matrix subspaces using partial data and side information, along with a spectral error bound that accounts for side information inaccuracies.

Findings

The algorithm accurately estimates matrix subspaces with limited column data.

The spectral error bound scales with the signal-to-noise ratio.

Simulations demonstrate the effectiveness of side information in matrix approximation.

Abstract

A novel matrix approximation problem is considered herein: observations based on a few fully sampled columns and quasi-polynomial structural side information are exploited. The framework is motivated by quantum chemistry problems wherein full matrix computation is expensive, and partial computations only lead to column information. The proposed algorithm successfully estimates the column and row-space of a true matrix given a priori structural knowledge of the true matrix. A theoretical spectral error bound is provided, which captures the possible inaccuracies of the side information. The error bound proves it scales in its signal-to-noise (SNR) ratio. The proposed algorithm is validated via simulations which enable the characterization of the amount of information provided by the quasi-polynomial side information.