论文标题
参数词典的翻译不变插值
Translation-invariant interpolation of parametric dictionaries
论文作者
论文摘要
在此通信中，我们解决了近似参数词典原子的问题，这通常是在“连续”词典中稀疏表示的上下文中遇到的。我们专注于翻译不变词典的情况，其中原子之间的内部产物仅取决于参数之间的差异。我们调查了以下一般性问题：字典$的近似近似值是否会插入原子的一部分，同时保留原始词典的翻译不变性？我们得出了表征存在这种“插值”和“翻译不变”的低级近似值的必要条件。此外，我们提供了这种词典存在时的封闭形式表达式。我们说明了我们在二维各向同性高斯词典中的结果的适用性。我们表明，在这种特定的设置中，所提出的近似框架的表现优于标准泰勒近似。
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.
