https://hal.inria.fr/inria-00576643Gribonval, RémiRémiGribonvalDepartment of Mathematics [Columbia] - University of South Carolina [Columbia]Nielsen, MortenMortenNielsenDepartment of Mathematics [Columbia] - University of South Carolina [Columbia]Approximate Weak Greedy AlgorithmsHAL CCSD2001[INFO.INFO-TS] Computer Science [cs]/Signal and Image Processing[SPI.SIGNAL] Engineering Sciences [physics]/Signal and Image processingGribonval, Rémi2011-03-15 09:32:332021-10-20 00:17:252011-03-16 09:48:59enJournal articleshttps://hal.inria.fr/inria-00576643/document10.1023/A:1012255021470application/pdf1We 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 as detailed discussion of how a "real-world" implementation of the weak greedy algorithm, when one has to take into account floating point arithmetic and other types of finite precision errors, can be modeled by the new algorithm.