The research team, led by Mohammad Akram of the Islamic University of Madinah together with Umar Ishtiaq, Khaleel Ahmad, and Ioan-Lucian Popa, proves that convergence survives under a far more permissive condition called enriched nonexpansivity. A map in this class is allowed to be expansive, to stretch distances, as long as it becomes nonexpansive once it is averaged with the identity operator. Formally, an operator T is b-enriched when a specific inequality holds that bounds how much T can expand points, with the amount of permissible expansion absorbed by averaging. The enriched class strictly contains the nonexpansive one, so every previously certifiable denoiser still qualifies, but an entire new territory of expressive networks opens up alongside it. Crucially, the enrichment constant b is not merely an abstraction; it becomes a computable number that a program can extract from a trained network.

The heart of the paper is a reduction, and it has a satisfying geometric flavor. The authors show that when the composite Plug-and-Play operator is enriched, an inertial Mann or Ishikawa iteration, classical fixed-point schemes that move only part of the way toward the operator’s output and add a momentum-like correction, becomes term by term identical to an ordinary iteration on a hidden nonexpansive operator called the generator. The inertial correction turns out to be exactly the quantity that cancels the overshoot introduced by the expansion. In other words, the rigidity is not imposed on the network at all; it is moved off the network and onto the iteration, where averaging repairs the loss of structure automatically. The fixed points of the expansive composite and its tame generator are the same set, so the convergent iteration lands on a genuine Plug-and-Play solution.

Turning this into usable software required a certificate, a way of checking for any given trained denoiser whether the enrichment hypothesis actually holds. The team’s main characterization theorem states that a smooth operator is b-enriched precisely when its displacement, the difference between the identity and the operator, is infinitesimally cocoercive with a modulus tied to b. The smallest admissible enrichment constant is one less than the supremum of a Rayleigh-type quotient of that displacement, a ratio that measures how far the displacement moves a direction relative to how much of that motion is aligned with the direction itself. This quotient is exact rather than merely sufficient, it is attained on non-scalar operators, it remains bounded near the identity where coarser bounds degenerate, and it can be evaluated through Jacobian-vector products that automatic differentiation supplies at the cost of roughly one network evaluation each.

The numbers that emerge are striking. On a linear surrogate problem in 120 dimensions, constructed so that every constant is available in closed form, the exact certificate widened the admissible step-size window by about 31 percent compared with the standard field-of-values bound, and the computed values matched the ground truth. The surrogate itself made the point vividly: an expansive operator with a spectral norm above one, admitted by no nonexpansive condition, was certified with a finite enrichment constant, and averaging it yielded a generator whose norm measured just below one. On real image restoration tasks with trained denoisers, the team measured the certified constant for each composite operator individually, using Lanczos iterations and bisection built on the same differentiable primitives.

The headline experimental result concerns Gaussian deblurring. A denoiser trained with a relaxed Lipschitz target produced a composite operator with a Lipschitz constant of 2.08, a value that no nonexpansive, firmly nonexpansive, or contractive-residual condition in the existing literature would accept. Yet its symmetric-part maximum stayed below one, and the exact characterization certified it with an enrichment constant of just 0.60, more than fifteen times tighter than the field-of-values relaxation for the same operator. The gap is a direct measurement of non-normality: the deep network expands strongly in directions poorly aligned with the residual, something the pair of conventional spectral quantities cannot see. Most remarkably, this expansive operator restored images within four hundredths of a decibel of a certified nonexpansive composite of identical architecture, meaning the larger, more expressive class was reached at no measured cost in quality.

Certification alone would be a curiosity without control over the iteration itself, so the authors built a safeguarded step-size controller on top of the theory. Any adaptive policy that stays inside the certified window preserves the convergence guarantee, whatever the policy has learned, so speed can be trained while safety is decided by the certificate. Within that architecture they derived a closed-form greedy relaxation, a Barzilai-Borwein-type step for the fixed-point residual, which roughly halved the iteration count of the constant-step scheme in both the surrogate and the image experiments. An amortized policy trained by unrolled differentiation through the solver matched the greedy rule without ever violating its guarantee, and the team verified empirically that stepping beyond the certified window by a factor of two caused divergence, confirming that the certificates mark a real stability boundary rather than an overly cautious one.

The framework also proved robust in ways that matter for practice. Convergence rates degrade gracefully, with the enrichment constant entering the residual bound as a simple known factor while the order of convergence is untouched. The theory tolerates denoiser evaluation errors of summable norm, covering finite-precision arithmetic and early-stopped inner solvers, which is exactly the situation encountered in real pipelines. A splitting-agnostic extension carries the entire construction beyond forward-backward splitting to Plug-and-Play ADMM and other schemes, since all that is ever required is a finite enrichment constant for the fixed-point operator of the scheme, checkable through the same eigenvalue computations.

The authors are careful about scope. The certificate is first order and measured on a finite sample, so the reported guarantees hold on the region actually certified rather than the whole image domain without further work. On the linear surrogate, an unrelaxed Picard iteration that carries no guarantee happened to be the fastest method, a reminder that acceleration claims here are comparisons among safeguarded schemes. The restoration study used small residual networks on modest image crops, so the reconstruction numbers are not a bid for state-of-the-art performance. What the results establish is arguably more consequential: the enlarged operator class is genuinely occupied by networks the existing guarantees reject, and using it costs nothing measurable in quality.

The deeper significance of the work may lie beyond imaging. Wherever a learned operator is iterated to a fixed point, in deep equilibrium models, learned optimizers, or unrolled reconstruction networks, the same tension between expressivity and guaranteed convergence reappears. The transfer-return mechanism offers a uniform recipe: certify the composite as enriched, transfer to the nonexpansive generator, iterate with the inertial correction, and transfer back. Whether that recipe extends to those settings remains an open conjecture, but the authors have made it checkable, reducing the entire question to eigenvalue computations that automatic differentiation already supports. In a field where convergence proofs have traditionally been bought by constraining networks into submission, the idea that the mathematics can instead bend around the network is a quiet revolution with loud implications.

Subject of Research: Convergence theory and certification for enriched-nonexpansive Plug-and-Play image restoration with learned denoisers

Article Title: Enriched-nonexpansive Plug-and-Play image restoration

Article References: Akram, M., Ishtiaq, U., Ahmad, K., & Popa, I.-L. (2026). Enriched-nonexpansive Plug-and-Play image restoration. Frontiers in Applied Mathematics and Statistics, 12, Article 1945678. https://doi.org/10.3389/fams.2026.1945678

Image Credits: AI Generated

DOI: 10.3389/fams.2026.1945678

Keywords: Plug-and-Play restoration, enriched nonexpansive mappings, fixed-point iteration, inertial Mann iteration, denoisers, image deblurring, super-resolution, convergence guarantees, cocoercivity, automatic differentiation, spectral certification, ADMM