L1-Norm Minimization Algorithms And Applications

The sparse representation theory has been a hot topic today and has great influences onsignal analysis and processing. The sparse representation based method has been widelyapplied to the field of image processing, pattern recognition, and automatic control. A keystep in this method is finding sparse solutions. In mathematics, the0-norm minimization isused to get a sparse solution. However, since the non-convex characteristic of the0-norm, itis difficult to be directly solved. In practical, an efficient and widely used method is replacingthe0-norm with1-norm minimization to get an approximate sparse solution, because thatthe1-norm minimization is convex, which has a unique global optimal solution.The core knowledge of this paper is1-norm minimization and its fast algorithms, and themain jobs include the following four aspects:Firstly, we compare three methods, including e₀、 e_p（e₀<e_p <e₁）and1-norm minimizationto seek sparse solutions, and introduce two types algorithms for1-norm minimization,including primal-dual interior point method and iterative shrinkage algorithm, which are usedin the sparse representation based method for EEG signal classification. The experimentalresults show that, both of these two algorithms have high classification accuracy when thenumber of training samples is relative large, while the iterative shrinkage algorithm has lowercomputational complexity than the other one;Secondly, we propose a mixed1-2minimization model which is an extension of1-norm minimization. At the same time, the iterative shrinkage based fast algorithm is alsopresented, and the computational complexity and a hardware implementation method of thisalgorithm are also studied in this paper.Thirdly, we use the proposed mixed1-2minimization model for real-time optical powermonitoring in the high-capacity WDM optical networks. In this method, the sparse andsmoothness characteristics of the optical power spectrum are used for modeling and solving.The simulation results confirm that accurate and fast optical power spectrum estimation canbe achieved with a low-cost tunable optical filter by using the proposed method.Lastly, in EEG signal classification, we use the proposed mixed1-2minimization modelfor dictionary learning to improve accuracy and reduce computational complexity. In thismethod, the sparse and class characteristics of the coefficients of training samples are used for modeling, and then a dictionary with small size and good discriminative ability can be solved,which can be used in the sparse representation classification method. The experimental resultsshow that the proposed dictionary learning method can greatly improve the accuracy of thesparse representation based method, and significantly reduce the computational complexity ofthe classification algorithm at the same time.