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The convergence guarantees of a non-convex approach for sparse recovery using regularized least squares

Existing literatures suggest that sparsity is more likely to be induced with non-convex penalties, but the corresponding algorithms usually suffer from multiple local minima. In this paper, we introduce a class of sparsity-inducing penalties and provide the convergence guarantees of a non-convex approach for sparse recovery using regularized least squares. Theoretical analysis demonstrates that under some certain conditions, if the non-convexity of the penalty is below a threshold (which is in inverse proportion to the distance between the initialization and the sparse signal), the sparse signal can be stably recovered. Numerical simulations are implemented to verify the theoretical results in this paper and to compare the performance of this approach with other references.