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Optimally sparse representation in general (nonorthogonal) dictionaries via \(\ell^1\) minimization. (English) Zbl 1064.94011

Summary: Given a dictionary D\(= \{\underline{d}_k\}\) of vectors \(\underline{d}_k\), we seek to represent a signal \(\underline{S}\) as a linear combination \(\underline{S}= \sum_k \gamma(k) \underline{d}_k\), with scalar coefficients \(\gamma(k)\). 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 \(\underline{S}\) 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 \(\gamma\). 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:

94A29 Source coding
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
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References:

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