Interpolating and translation-invariant approximations of parametric dictionaries

In this paper, we address the problem of approximating the atoms of a parametric dictionary A = {α(θ) : θ ∈ Θ}, commonly encountered in the context of sparse representations in “continuous” dictionaries. We focus on the case of translation-invariant dictionaries, where the inner product between α(θ) and α(θ') only depends on the difference θ - θ'. We investigate the following general question: is there some low-rank approximation of A which interpolates a subset of atoms {α(θ
_j
)}
_j=1
^J
while preserving the translation-invariant nature of the original dictionary? In this paper, we derive necessary and sufficient conditions characterizing the existence of such an “interpolating” and “translation-invariant” low-rank approximation. Moreover, we provide closed-form expressions of such a dictionary when it exists. We illustrate the applicability of our results in the case of a two-dimensional isotropic Gaussian dictionary. We show that, in this particular setup, the proposed approximation framework outperforms standard Taylor approximation.