Mathematical Programming for Multidimensional Data Reconstruction and Its Progress

Abstract

Research on reconstructing original data from data that can only be partially observed due to noise or missing data has been ongoing for many years. Such problems are generally referred to as matrix estimation problems. The problem can be formulated when the data to be estimated can be defined as matrix variables in the problem of reconstructing partially observed data. When the properties of the target matrix are unknown, a common approach is a method called matrix rank minimization, which is known for its high estimation accuracy in various fields such as audio, image, and wireless communication. However, this method assumes that the data belong to a linear subspace, and if this assumption is not satisfied, the estimation accuracy significantly deteriorates. Therefore, in recent years, this assumption has been extended to manifolds, and various methods based on this assumption have been proposed. This paper reviews the progress of these methods and describes the latest techniques in this field.