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Optimally sparse representation in general (nonorthogonal) dictionaries via 1 minimization. (English) Zbl 1064.94011
Summary: Given a dictionary D={d ̲ k } of vectors d ̲ k , we seek to represent a signal S ̲ as a linear combination S ̲= k γ(k)d ̲ k , with scalar coefficients γ(k). 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 S ̲ 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 γ. 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.
MSC:
94A29Source coding
94A12Signal theory (characterization, reconstruction, filtering, etc.)