This paper presents new theoretical results on sparse recovery guarantees for a greedy algorithm, Orthogonal Matching Pursuit (OMP), in the context of continuous parametric dictionaries. Here, the continuous setting means that the dictionary is made up of an infinite uncountable number of atoms. In this work, we rely on the Hilbert structure of the observation space to express our recovery results as a property of the kernel defined by the inner product between two atoms. Using a continuous extension of Tropp's Exact Recovery Condition, we identify key assumptions allowing to analyze OMP in the continuous setting. Under these assumptions, OMP unambiguously identifies in exactly k steps the atom parameters from any observed linear combination of k atoms. These parameters play the role of the so-called support of a sparse representation in traditional sparse recovery. In our paper, any kernel and set of parameters that satisfy these conditions are said to be admissible.

In the one-dimensional setting, we exhibit a family of kernels relying on completely monotone functions for which admissibility holds for any set of atom parameters. For higher dimensional parameter spaces, the analysis turns out to be more subtle. An additional assumption, so-called axis admissibility, is imposed to ensure a form of delayed recovery (in at most $k^D$ steps, where D is the dimension of the parameter space). Furthermore, guarantees for recovery in exactly k steps are derived under an additional algebraic condition involving a finite subset of atoms (built as an extension of the set of atoms to be recovered). We show that the latter technical conditions simplify in the case of Laplacian kernels, allowing us to derive simple conditions for k-step exact recovery, and to carry out a coherence-based analysis in terms of a minimum separation assumption between the atoms to be recovered.

本文介绍了在连续参数字典的背景下，针对贪心算法（正交匹配追踪，OMP）的稀疏恢复保证的新理论结果。在此，连续设置意味着字典由无限数量的原子组成。在这项工作中，我们依靠观察空间的希尔伯特结构将恢复结果表示为由两个原子之间的内积定义的核的性质。使用Tropp的“精确恢复条件”的连续扩展，我们确定了允许在连续设置下分析OMP的关键假设。在这些假设下，OMP从k的任何观察到的线性组合中准确地以k步准确地识别原子参数。原子。这些参数在传统的稀疏恢复中起所谓的稀疏表示支持的作用。在我们的论文中，任何满足这些条件的内核和参数集都被认为是可以接受的。

在一维环境中，我们展示了一系列依赖于完全单调函数的内核，这些函数对于任何原子参数集均具有可接纳性。对于更高维的参数空间，分析变得更加微妙。一个额外的假设，即所谓的轴受理，被施加以确保延迟恢复的形式（在至多${ķ}^d$步骤，其中D是参数空间的尺寸）。此外，在涉及有限原子子集（作为要恢复的原子集的扩展而构建）的附加代数条件下，得出了精确地以k步进行恢复的保证。我们表明，在拉普拉斯内核的情况下，后一种技术条件得以简化，这使我们能够导出用于k步精确恢复的简单条件，并根据要恢复的原子之间的最小分离假设进行基于相干性的分析。