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Minimization of Transformed L₁ Penalty: Theory, Difference of Convex Function Algorithm, and Robust Application in Compressed Sensing
Shuai Zhang, Jack Xin
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We study the minimization problem of a non-convex sparsity promoting penalty function, the transformed l₁ (TL1), and its application in compressed sensing (CS). The TL1 penalty interpolates 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). Next, we present difference of convex algorithms for TL1 (DCATL1) in computing TL1-regularized constrained and unconstrained problems in CS. 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 matrices: Gaussian random matrices and over-sampled discrete cosine transform matrices (DCT). 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 competitive in comparison with DCA on other non-convex penalty functions commonly used in statistics with two hyperparameters.
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