Optimally sparse representation in general (nonorthogonal) dictionaries via
minimization. (English) Zbl 1064.94011
Summary: Given a dictionary D of vectors , we seek to represent a signal as a linear combination , with scalar coefficients . In particular, we aim for the sparsest representation possible. In general, this requires a combinatorial optimization process. Previous work considered the special case where D is an overcomplete system consisting of exactly two orthobases and has shown that, under a condition of mutual incoherence of the two bases, and assuming that has a sufficiently sparse representation, this representation is unique and can be found by solving a convex optimization problem: specifically, minimizing the 1 norm of the coefficients . In this article, we obtain parallel results in a more general setting, where the dictionary D can arise from two or several bases, frames, or even less structured systems. We sketch three applications: separating linear features from planar ones in 3D data, noncooperative multiuser encoding, and identification of over-complete independent component models.
|94A12||Signal theory (characterization, reconstruction, filtering, etc.)|