Complex-valued sparse reconstruction via arctangent regularization

• We present the arctangent regularization (ATANR), which handle the complex-valued sparse reconstruction directly.
• The ATANR has the ability of dealing with the problems of large size efficiently.
• The ATANR is less sensitive to the regularization parameter.
• The ATANR shows better performance than the conventional ℓ1 regularization methods.

Complex-valued sparse reconstruction is conventionally solved by transforming it into real-valued problems. However, this method might not work efficiently and correctly, especially when the size of the problem is large, or the mutual coherence is high. In this paper, we present a novel algorithm called the arctangent regularization (ATANR), which can handle the complex-valued problems of large size and high mutual coherence directly. The ATANR is implemented with the iterative least squares (IRLS) framework, and accelerated by the dimension reduction and active set selection steps. Further, we summarize and analyze the common properties of a penalty kernel which is suitable for sparse reconstruction. The analyses show that the key difference, between the arctangent kernel and the ℓ1 norm, is that the first order derivative of ATANR is close to zero for a nonzero variable. This will make ATANR less sensitive to the regularization parameter λ than ℓ1 regularization methods. Finally, lots of numerical experiments validate that ATANR usually has better performance than the conventional ℓ1 regularization methods, not only for the random signs ensemble, but also for the sensing matrix with high mutual coherence, such as the resolution enhancement case.