Optimal dictionary for least squares representation
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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., ℓ_0-optimal representations. Here we consider the problem of finding optimal dictionaries with which representations of samples of a random vector are 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 random vector. 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.
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