A complete characterization of optimal dictionaries for least squares representation

Mohammed Rayyan Sheriff, Debasish Chatterjee

Published 2017-10-18Version 1

Dictionaries are collections of vectors used for representations of elements in Euclidean spaces. While recent research on optimal dictionaries is focussed on providing sparse (i.e., $\ell_0$-optimal,) representations, here we consider the problem of finding optimal dictionaries such that representations of samples of a random vector are optimal in an $\ell_2$-sense. For us, optimality of representation is equivalent to minimization of the average $\ell_2$-norm of the coefficients used to represent the random vector, with the lengths of the dictionary vectors being specified a priori. With the help of recent results on rank-$1$ decompositions of symmetric positive semidefinite matrices and the theory of majorization, we provide a complete characterization of $\ell_2$-optimal dictionaries. Our results are accompanied by polynomial time algorithms that construct $\ell_2$-optimal dictionaries from given data.