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Sparse Representation (Cont.)

已有 5419 次阅读 2012-3-23 23:28 |个人分类:Signal processing|系统分类:科研笔记| OMP, coding, uniqueness, Satability

1. Uniqueness

Theorem 2.1. For an arbitrary pair of orthogonal basesΨ,Φwith mutual-coherenceμ(A), and for an arbitrary non-zero vector bIRn with representationsαandβcorrespondingly,the following inequality holds true:

Uncertainty Principle 1 :  

Theorem 2.2. Any two distinct solutions x1xof the linear system [Ψ;Φ]b

cannot both be very sparse, governed by the following uncertainty principle:

Uncertainty Principle 2 :     

We refer to this result as an uncertainty of redundant solutions, as we discuss here solutions to the underdetermined system.

(1) via SparkF

 

(2) via Mutual-Coherence

 


 

2. Equivalence

If

 

is empty, we can get the 1 norm get the same result as 0 norm, and it’s the sparsest solution.

3. Stability

 

Prove that:

 

4. OMP(Orthogonal-Matching-Pursuit)

 

Reference:
[1]  A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM Review, vol. 51, no. 1, pp. 34–81, 2009.  
[2]  Y.C. Pati, R. Rezaiifar, and P.S. Krishnaprasad, "Orthogonal matching pursuit: Recursive function approximation with applications to wavelet decomposition," in Proceedings of the 27th Asilomar Conference on Signals, Systems and Computers, Vol. 1, 1993, pp. 40–44.



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