Approximate Weak Greedy Algorithms

We present a generalization of V. Temlyakov’s weak greedy algorithm, and give a sufficient condition for norm convergence of the algorithm for an arbitrary dictionary in a Hilbert space. We provide two counter-examples to show that the condition cannot be relaxed for general dictionaries. For a class of dictionaries with more structure, we give a more relaxed necessary and sufficient condition for convergence of the algorithm. We also provide a detailed discussion of how a “real-world” implementation of the weak greedy algorithm, where one has to take into account floating point arithmetic and other types of finite precision errors, can be modeled by the new algorithm.