Translation-invariant interpolation of parametric dictionaries

TL;DR

This paper develops a method for low-rank, translation-invariant interpolation of parametric dictionaries, providing conditions for existence and explicit formulas, with applications to Gaussian dictionaries showing improved approximation over Taylor methods.

Contribution

It introduces a novel framework for translation-invariant low-rank interpolation of parametric dictionaries, with explicit solutions and applicability to Gaussian dictionaries.

Findings

Derived necessary and sufficient conditions for translation-invariant low-rank approximation.

Provided closed-form expressions for the interpolating dictionaries when they exist.

Demonstrated improved approximation over Taylor methods in Gaussian dictionary case.

Abstract

In this communication, we address the problem of approximating the atoms of a parametric dictionary, 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 atoms only depends on the difference between parameters. We investigate the following general question: is there some low-rank approximation of the dictionary $ which interpolates a subset of atoms while preserving the translation-invariant nature of the original dictionary? 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.