Interpolating and translation-invariant approximations of parametric dictionaries

摘要

In this paper, we address the problem of approximating the atoms of a parametric dictionary ${mathcal{A}} = left{ {aleft(hetaight):heta in Theta } ight}$, 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 a(θ) and a(θ′) only depends on the difference θ – θ′. We investigate the following general question: is there some low-rank approximation of ${mathcal{A}}$ which interpolates a subset of atoms $left{ {aleft({{heta _j}}ight)} ight}_{j = 1}^J$ 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.