参数字典的平移不变插值
Translation-invariant interpolation of parametric dictionaries
摘要：
在这次交流中，我们解决的问题近似的原子的参数字典，通常遇到的情况下，稀疏表示在“连续”字典。我们专注于的情况下，不变量字典，原子之间的内积只取决于参数之间的差异。我们调查以下一般性问题：是否有一些低秩近似的字典$插入一个子集的原子，同时保持原字典的平移不变的性质？我们得到了存在这种“插值”和“不变量”的低秩近似的充分必要条件。此外，我们提供了这样的字典时，它存在的封闭形式的表达式。我们说明了我们的结果在二维各向同性高斯字典的情况下的适用性。我们表明，在这个特定的设置，建议的近似框架优于标准泰勒近似。
英文摘要：
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.
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