## 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:

 [1] SIAM REV 43 pp 129– (2001) · Zbl 0979.94010 [2] PROCEEDINGS OF THE IEEE INTERNATIONAL SYMPOSIUM ON CIRCUITS AND SYSTEMS 4 pp 106– (1999) [3] Debrunner, IEEE transactions on image processing : a publication of the IEEE Signal Processing Society 6 (9) pp 1316– (1997) [4] MATH COMP 65 pp 1513– (1996) · Zbl 0853.42018 [5] IEEE TRANS SIGNAL PROCESSING 11 pp 670– (2002) [6] IEEE TRANS SIGNAL PROCESSING 41 pp 3397– (1993) · Zbl 0842.94004 [7] IEEE TRANS INF THEORY 47 pp 2845– (2001) · Zbl 1019.94503 [8] IEEE TRANS INF THEORY 48 pp 2558– (2002) · Zbl 1062.15001 [9] CRYPTOGRAPHY LECTURE NOTES IN COMPUTER SCIENCE 149 pp 71– (1983)
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