A complete characterization of optimal dictionaries for least squares representation

10/18/2017
by   Mohammed Rayyan Sheriff, et al.
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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., ℓ_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 ℓ_2-sense. For us, optimality of representation is equivalent to minimization of the average ℓ_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 ℓ_2-optimal dictionaries. Our results are accompanied by polynomial time algorithms that construct ℓ_2-optimal dictionaries from given data.

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