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MG-SpaIR is a training-data-free framework for restoring a clean image from a single observation corrupted by a mixture of blur, downsampling, noise, and missing pixels. Building on implicit neural representations (INRs), we introduce a multi-grade residual hierarchy that progressively refines the reconstruction from low to high spatial frequencies across grades, improving representational fidelity and mitigating spectral limitations. To stabilize reconstruction optimization and suppress INR-induced artifacts, we further propose an explicit sparse proximal regularization (e.g.,$$\ell _0$$ ${\ell }_0$ type) applied directly in the high-resolution image domain, which discourages spurious high-frequency patterns while preserving sharp structures. The resulting optimization is solved efficiently via a multi-grade proximal alternating scheme, and we establish convergence guarantees for the associated updates under standard regularity conditions. Experiments on mixed-degradation benchmarks demonstrate that MG-SpaIR consistently outperforms strong training-data-free baselines such as Deep Image Prior, providing a stable, interpretable, and data-efficient alternative to conventional learning-based restoration methods. 

Sparse signal recovery has been a cornerstone of advancements in data processing and imaging. Recently, the squared ratio of $$ℓ_1$$ to $$ℓ_2$$ norms, $$(ℓ_1/ℓ_2)^2,$$ has been introduced as a sparsity-prompting function, showing superior performance compared to traditional $$ℓ_1$$ minimization, particularly in challenging scenarios with high coherence and dynamic range. This paper explores the integration of the proximity operator of $$(ℓ_1/ℓ_2)^2$$ and $$ℓ_1/ℓ_2$$ into efficient optimization frameworks, including the Accelerated Proximal Gradient (APG) and Alternating Direction Method of Multipliers (ADMM). We rigorously analyze the convergence properties of these algorithms and demonstrate their effectiveness in compressed sensing and image restoration applications. Numerical experiments highlight the advantages of our proposed methods in terms of recovery accuracy and computational efficiency, particularly under noise and high-coherence conditions. 

Abstract This paper investigates the computation of proximity operators for scale and signed permutation invariant functions. A scale invariant function remains unchanged under uniform scaling, while a signed permutation invariant function retains its structure despite permutations and sign changes applied to its input variables. Noteworthy examples include the$$\ell _0$$ ${\ell }_0$ function, the ratio of$$\ell _1/\ell _2$$ ${\ell }_1/{\ell }_2$ , and its square, with their proximity operators being particularly crucial in sparse signal recovery. We delve into the properties of scale and signed permutation invariant functions, delineating the computation of their proximity operators into three sequential steps: the$${\varvec{w}}$$ $w$ -step,r-step, andd-step. These steps collectively form a procedure termed as WRD, with the$${\varvec{w}}$$ $w$ -step being of utmost importance and requiring careful treatment. Leveraging this procedure, we present a method for explicitly and efficiently computing the proximity operator of$$(\ell _1/\ell _2)^2$$ ${({\ell }_1/{\ell }_2)}^2$ and introduce an algorithm for the proximity operator of$$\ell _1/\ell _2$$ ${\ell }_1/{\ell }_2$ . Numerical experiments on sparse signal recovery corroborate the analysis and show that first-order methods equipped with these proximity operators outperform$$\ell _1$$ ${\ell }_1$ -based baselines in reconstruction accuracy. 

Abstract In the realm of image processing and analysis, image restoration stands out as a pivotal area, addressing the challenge of reconstructing degraded or distorted images. Typically categorized as an inverse problem, image restoration often leverages regularization techniques to enhance the quality of reconstructed images. This paper introduces an image restoration model that incorporates regularization through structured sparsity promoting functions (SPFs). The proposed model’s objective function structure is a key aspect, prompting the exploration of various formulations conducive to the application of existing algorithms. Among the algorithms considered are the inertial proximal algorithm for nonconvex optimization (iPiano), difference of convex algorithm (DCA), proximal linearized DCA, proximal DCA with extrapolation, double-proximal gradient algorithm, and the alternating direction method of multipliers (ADMM). To facilitate algorithmic application, the objective function is reformulated in multiple ways, ensuring compatibility with the diverse optimization approaches mentioned above. The comparative analysis of these algorithms serves as a benchmark for evaluating their performance concerning the proposed image restoration model. This study contributes to the understanding of the effectiveness of different optimization strategies in the context of structured sparsity promoting functions for image restoration. 

Abstract This paper introduces a nonconvex approach for sparse signal recovery, proposing a novel model termed the$$\tau _2$$ ${\tau }_2$ -model, which utilizes the squared$$\ell _1/\ell _2$$ ${\ell }_1/{\ell }_2$ norms for this purpose. Our model offers an advancement over the$$\ell _0$$ ${\ell }_0$ norm, which is often computationally intractable and less effective in practical scenarios. Grounded in the concept of effective sparsity, our approach robustly measures the number of significant coordinates in a signal, making it a powerful alternative for sparse signal estimation. The$$\tau _2$$ ${\tau }_2$ -model is particularly advantageous due to its computational efficiency and practical applicability. We detail two accompanying algorithms based on Dinkelbach’s procedure and a difference of convex functions strategy. The first algorithm views the model as a linear-constrained quadratic programming problem in noiseless scenarios and as a quadratic-constrained quadratic programming problem in noisy scenarios. The second algorithm, capable of handling both noiseless and noisy cases, is based on the alternating direction linearized proximal method of multipliers. We also explore the model’s properties, including the existence of solutions under certain conditions, and discuss the convergence properties of the algorithms. Numerical experiments with various sensing matrices validate the effectiveness of our proposed model.