On the Hardness of the L1-L2 Regularization Problem
Abstract
The sparse linear reconstruction problem is a core problem in signal processing which aims to recover sparse solutions to linear systems. The original problem regularized by the total number of nonzero components (also known as L0 regularization) is well-known to be NP-hard. The relaxation of the L0 regularization by using the L1 norm offers a convex reformulation, but is only exact under certain conditions (e.g., restricted isometry property) which might be NP-hard to verify. To overcome the computational hardness of the L0 regularization problem while providing tighter results than the L1 relaxation, several alternate optimization problems have been proposed to find sparse solutions. One such problem is the L1-L2 minimization problem, which is to minimize the difference of the L1 and L2 norms subject to linear constraints. This paper proves that solving the L1-L2 minimization problem is NP-hard. Specifically, we prove that it is NP-hard to minimize the L1-L2 regularization function subject to linear constraints. Moreover, it is also NP-hard to solve the unconstrained formulation that minimizes the sum of a least squares term and the L1-L2 regularization function. Furthermore, restricting the feasible set to a smaller one by adding nonnegative constraints does not change the NP-hardness nature of the problems.
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