This paper proposes an optimization framework that can efficiently deal with convex data-fidelity constraints onto which the metric projections are difficult to compute. Although such an involved data-fidelity constraint is expected to play an important role in signal recovery under non-Gaussian noise contamination, the said difficulty precludes existing algorithms from solving convex optimization problems with the constraint. To resolve this dilemma, we introduce a fixed point set characterization of involved data-fidelity constraints based on a certain computable quasi-nonexpansive mapping. This characterization enables us to mobilize the hybrid steepest descent method to solve convex optimization problems with such a constraint. The proposed framework can handle a variety of involved data-fidelity constraints in a unified manner, without geometric approximation to them. In addition, it requires no computationally expensive procedure such as operator inversion and inner loop. As applications of the proposed framework, we provide image restoration under several types of non-Gaussian noise contamination with illustrative examples.

The convex envelopes of the direct discrete measures, for the sparsity of vectors or for the low-rankness of matrices, have been utilized extensively as practical penalties in order to compute a globally optimal solution of the corresponding regularized least-squares models. Motivated mainly by the ideas in Zhang (2010 Ann. Stat . 38 894–942; Selesnick 2017 IEEE Trans. Signal Process . 65 4481–94; Yin et al 2019 IEEE Trans. Signal Process . 67 2595–607) to exploit nonconvex penalties in the regularized least-squares models without losing their overall convexities, this paper presents the linearly involved generalized Moreau enhanced (LiGME) model as a unified extension of such utilizations of nonconvex penalties. The proposed model can admit multiple nonconvex penalties without losing its overall convexity and thus is applicable to much broader scenarios in the sparsity-rank-aware signal processing. Under the general overall-convexity condition of the LiGME model, we also present a novel proximal splitting type algorithm of guaranteed convergence to a globally optimal solution. Numerical experiments in typical examples of the sparsity-rank-aware signal processing demonstrate the effectiveness of the LiGME models and the proposed proximal splitting algorithm.

This paper addresses the selection of a desirable solution among all the solutions of a convex optimization problem (referred to as the first-stage problem) mainly for inverse problems in signal processing. This is realized in the framework of hierarchical convex optimization, i.e., minimizing another convex function over the solution set of the first-stage problem. Hierarchical convex optimization is an ideal strategy when the first-stage problem has infinitely many solutions because of the non-strict convexity of its objective function, which could arise in various scenarios, e.g., convex feasibility problems. To this end, first, the fixed point set characterization behind a primal-dual splitting type method is incorporated into the framework of hierarchical convex optimization, which enables the framework to cover a broad class of first-stage problem formulations. Then, a pair of efficient algorithmic solutions to the hierarchical convex optimization problem, as certain realizations of the hybrid steepest descent method, are provided with guaranteed convergence. We also present a specialized form of the proposed framework to focus on a typical scenario of inverse problems, and show its application to signal interpolation.

This paper proposes a novel linear estimator named stochastic MV-PURE estimator, developed for the stochastic linear model, and designed to provide improved performance over the linear minimum mean square error (MMSE) Wiener estimator in cases prevailing in practical, real-world settings, where at least some of the second-order statistics of the random vectors under consideration are only imperfectly known. The proposed estimator shares its main mathematical idea and terminology with the recently introduced minimum-variance pseudo-unbiased reduced-rank estimator (MV-PURE), developed for the linear regression model. The proposed stochastic MV-PURE estimator minimizes the mean square error (MSE) of its estimates subject to rank constraint and inducing minimum distortion to the target random vector. Therefore, the stochastic MV-PURE combines the techniques of the reduced rank Wiener filter (named in this paper RR-MMSE) and the distortionless-constrained estimator (named in this paper C-MMSE), in order to achieve greater robustness against noise or model errors than RR-MMSE and C-MMSE. Furthermore, to ensure that the stochastic MV-PURE estimator combines the reduced-rank and minimum-distortion approaches in the MSE-optimal way, we propose a rank selection criterion which minimizes the MSE of the estimates obtained by the stochastic MV-PURE. As a numerical example, we employ the stochastic MV-PURE, RR-MMSE, C-MMSE, and MMSE estimators as linear receivers in a MIMO wireless communication system. This example is chosen as a typical signal processing scenario, where the statistical information on the data, on which the estimates are built, is only imperfectly known. We verify that the stochastic MV-PURE achieves the lowest MSE and symbol error rate (SER) in such settings by employing the proposed rank selection criterion.

Hierarchical convex optimization concerns two-stage optimization problems: the first stage problem is a convex optimization; the second stage problem is the minimization of a convex function over the solution set of the first stage problem. For the hierarchical convex optimization, the hybrid steepest descent method (HSDM) can be applied, where the solution set of the first stage problem must be expressed as the fixed point set of a certain nonexpansive operator. In this paper, we propose a nonexpansive operator that yields a computationally efficient update when it is plugged into the HSDM. The proposed operator is inspired by the update of the linearized augmented Lagrangian method. It is applicable to characterize the solution set of recent sophisticated convex optimization problems found in the context of inverse problems, where the sum of multiple proximable convex functions involving linear operators must be minimized to incorporate preferable properties into the minimizers. For such a problem formulation, there has not yet been reported any nonexpansive operator that yields an update free from the inversions of linear operators in cases where it is utilized in the HSDM. Unlike previously known nonexpansive operators, the proposed operator yields an inversion-free update in such cases. As an application of the proposed operator plugged into the HSDM, we also present, in the context of the so-called superiorization, an algorithmic solution to a convex optimization problem over the generalized convex feasible set where the intersection of the hard constraints is not necessarily simple.