Minimization of Transformed L1 Penalty: Theory, Difference of Convex Function Algorithm, and Robust Application in Compressed Sensing

Shuai Zhang University of California, Irvine Jack Xin University of California, Irvine

Information Theory mathscidoc:1802.19004

Mathematical Programming, Series B, 2018
We study the minimization problem of a non-convex sparsity promoting penalty func- tion, the transformed l1 (TL1), and its application in compressed sensing (CS). The TL1 penalty inter- polates l0 and l1 norms through a nonnegative parameter a ∈ (0, +∞), similar to lp with p ∈ (0, 1], and is known to satisfy unbiasedness, sparsity and Lipschitz continuity properties. We first consider the constrained minimization problem, and discuss the exact recovery of l0 norm minimal solution based on the null space property (NSP). We then prove the stable recovery of l0 norm minimal solution if the sensing matrix A satisfies a restricted isometry property (RIP). We formulated a normalized problem to overcome the lack of scaling property of the TL1 penalty function. For a general sensing matrix A, we show that the support set of a local minimizer corresponds to linearly independent columns of A. Next, we present difference of convex algorithms for TL1 (DCATL1) in computing TL1-regularized con- strained and unconstrained problems in CS. The DCATL1 algorithm involves outer and inner loops of iterations, one time matrix inversion, repeated shrinkage operations and matrix-vector multiplications. The inner loop concerns an l1 minimization problem on which we employ the Alternating Direction Method of Multipliers (ADMM). For the unconstrained problem, we prove convergence of DCATL1 to a stationary point satisfying the first order optimality condition. In numerical experiments, we identify the optimal value a = 1, and compare DCATL1 with other CS algorithms on two classes of sensing ma- trices: Gaussian random matrices and over-sampled discrete cosine transform matrices (DCT). Among existing algorithms, the iterated reweighted least squares method based on l1/2 norm is the best in sparse recovery for Gaussian matrices, and the DCA algorithm based on l1 minus l2 penalty is the best for over-sampled DCT matrices. We find that for both classes of sensing matrices, the performance of DCATL1 algorithm (initiated with l1 minimization) always ranks near the top (if not the top), and is the most robust choice insensitive to the conditioning of the sensing matrix A. DCATL1 is also com- petitive in comparison with DCA on other non-convex penalty functions commonly used in statistics with two hyperparameters.
No keywords uploaded!
[ Download ] [ 2018-02-14 11:29:34 uploaded by jack ] [ 637 downloads ] [ 0 comments ]
@inproceedings{shuai2018minimization,
  title={Minimization of Transformed L1 Penalty: Theory, Difference of Convex Function Algorithm, and Robust Application in Compressed Sensing },
  author={Shuai Zhang, and Jack Xin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180214112934641655928},
  booktitle={Mathematical Programming, Series B},
  year={2018},
}
Shuai Zhang, and Jack Xin. Minimization of Transformed L1 Penalty: Theory, Difference of Convex Function Algorithm, and Robust Application in Compressed Sensing . 2018. In Mathematical Programming, Series B. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180214112934641655928.
Please log in for comment!
 
 
Contact us: office-iccm@tsinghua.edu.cn | Copyright Reserved