When does OMP achieve exact recovery with continuous dictionaries?

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, setting means that dictionary is made up an infinite uncountable number atoms. In this work, we rely Hilbert structure observation space to express our as property kernel defined by inner product between two Using extension Tropp's Exact Recovery Condition, identify key assumptions allowing analyze OMP setting. Under these assumptions, unambiguously identifies exactly k steps atom parameters from any observed linear combination These play role so-called support representation traditional recovery. paper, and set satisfy conditions are said be admissible. one-dimensional setting, exhibit family kernels relying completely monotone functions which admissibility holds parameters. For higher dimensional parameter spaces, analysis turns out more subtle. An additional assumption, axis admissibility, imposed ensure form delayed (in at most kD steps, where D dimension space). Furthermore, derived under algebraic condition involving finite subset atoms (built recovered). We show latter technical simplify case Laplacian kernels, us derive simple k-step exact recovery, carry coherence-based terms minimum separation assumption recovered.