Nonconvex Latent Optimally Partitioned Block-Sparse Recovery via Log-Sum and Minimax Concave Penalties

We propose two nonconvex regularization methods, LogLOP-l2/l1 and AdaLOP-l2/l1, for recovering block-sparse signals with unknown block partitions. These methods address the underestimation bias of existing convex approaches by extending log-sum penalty and the Minimax Concave Penalty (MCP) to the block-sparse domain via novel variational formulations. Unlike Generalized Moreau Enhancement (GME) and Bayesian methods dependent on the squared-error data fidelity term, our proposed methods are compatible with a broad range of data fidelity terms. We develop efficient Alternating Direction Method of Multipliers (ADMM)-based algorithms for these formulations that exhibit stable empirical convergence. Numerical experiments on synthetic data, angular power spectrum estimation, and denoising of nanopore currents demonstrate that our methods outperform state-of-the-art baselines in estimation accuracy.

A Convex-Nonconvex Framework for Enhancing Minimization Induced Penalties

This paper presents a novel framework for nonconvex enhancement of minimization induced (MI) penalties while preserving the overall convexity of associated regularization models. MI penalties enable the adaptation to certain signal structures via minimization, but often underestimate significant components owing to convexity. To overcome this shortcoming, we design a generalized Moreau enhanced minimization induced (GME-MI) penalty by subtracting from the MI penalty its generalized Moreau envelope. While the proposed GME-MI penalty is nonconvex in general, we derive an overall convexity condition for the GME-MI regularized least-squares model. Moreover, we present a proximal splitting algorithm with guaranteed convergence to a globally optimal solution of the GME-MI model under the overall convexity condition. Numerical examples illustrate the effectiveness of the proposed framework.

Convex Estimation of Sparse-Smooth Power Spectral Densities from Mixtures of Realizations with Application to Weather Radar

In this paper, we propose a convex optimization-based estimation of sparse and smooth power spectral densities (PSDs) of complex-valued random processes from mixtures of realizations. While the PSDs are related to the magnitude of the frequency components of the realizations, it has been a major challenge to exploit the smoothness of the PSDs because penalizing the difference of the magnitude of the frequency components results in a nonconvex optimization problem that is difficult to solve. To address this challenge, we design the proposed model that jointly estimates the complex-valued frequency components and the nonnegative PSDs, which are respectively regularized to be sparse and sparse-smooth. By penalizing the difference of the nonnegative variable that estimates the PSDs, the proposed model can enhance the smoothness of the PSDs via convex optimization. Numerical experiments on the phased array weather radar, an advanced weather radar system, demonstrate that the proposed model achieves superior estimation accuracy to the existing sparse estimation models combined with or without post-smoothing.