A complete characterization of optimal dictionaries for least squares representation
  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.
