Optimal Dictionary for Least Squares Representation

Dictionaries are collections of vectors used for representations of random vectors in Euclidean spaces. Recent research on optimal dictionaries is focused on constructing dictionaries that offer sparse representations, i.e., l0-optimal representations. Here we consider the problem of finding optimal dictionaries with which representations of samples of a random vector are optimal in an l2-sense: optimality of representation is defined as attaining the minimal average l2-norm of the coefficients used to represent the random vector. With the help of recent results on rank-1 decompositions of symmetric positive semidefinite matrices, we provide an explicit description of l2-optimal dictionaries as well as their algorithmic constructions in polynomial time.