Optimal Dictionary for Least Squares Representation

Dictionaries are collections of vectors used for the representation of a class of vectors in Euclidean spaces. Recent research on optimal dictionaries is focused on constructing dictionaries that offer sparse representations, i.e., l(0)-optimal representations. Here we consider the problem of finding optimal dictionaries with which representations of a given class of vectors is optimal in an l(2)-sense: optimality of representation is defined as attaining the minimal average l(2)-norm of the coefficients used to represent the vectors in the given class. With the help of recent results on 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.