Learning: Wavelet-Dictionaries and Continuous Dictionaries

There are several approximative algorithms for learning a dictionary from given signals, some also in combination with additional properties, but no one of them combines the properties of shift- and scale-invariance with a fast algorithm for coding. We introduce an algorithm, learning a dictionary being composed of a number of wavelet bases, by minimizing an error measure with side conditions induced by the lifting scheme. Later on we apply this algorithm to problems in the fields of mechanical engineering and musics.Secondly questions concerning a continuous generalization of the dictionary learning problem are not treated till now. Concerning this we define a corresponding error functional, depending on two variables, equivalent to the discrete case. We investigate the existence of a minimizer of this non-convex functional. Furthermore we point out a practical way to obtain at least a local minimum using a generalization of the conditional gradient algorithm.