A complete characterization of optimal dictionaries for least squares representation

Vectors in Euclidean spaces are typically represented using a standard collection of vectors called a Dictionary. Given a data, it is of interest to find a good dictionary that offers desirable representation of the data. In this article, we consider the problem of finding optimal dictionaries with which representations of data points are optimal in an l(2)-sense: optimality of representation is defined as attaining the minimal average squared l(2)-norm of the coefficients used for representation. With the help of recent results on Majorization, rank-1 decompositions of symmetric positive semidefinite matrices, we provide an explicit description of l(2)-optimal dictionaries as well as their algorithmic constructions in polynomial time. (C) 2020 Elsevier Inc. All rights reserved.