PADDLE: Proximal Algorithm for Dual Dictionaries LEarning

The representation of a signal as the superposition of elementary signals, or atoms, is the pillar of a number of research fields and analysis techniques. The best-known example of such methods is the Fourier transform, where the atoms form an orthonormal basis and every signal has a unique representation. Although an orthonormal basis would seem the most natural choice for decomposing a signal, overcomplete dictionaries (or frames) are nowadays commonplace and their use is both theoretically justified and supported by experimentally successful applications [1]. Tight frames are a class of overcomplete dictionaries with the particular property of ensuring that the optimal representation can still be recovered, as with orthonormal bases, by means of inner products of the signal and the dictionary. The goal of this paper is to introduce an algorithm - that we called PADDLE - capable of learning from data a dictionary endowed with properties similar to that of tight frames.