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arxiv: 2605.06336 · v1 · submitted 2026-05-07 · 🧮 math.NA · cs.NA

Recognition: unknown

Nonlinear RMM-GKS for Large-Scale Dynamic and Streaming Inverse Problems with Uncertain Forward Operators

Eric de Sturler, James G. Nagy, Mirjeta Pasha, Misha E. Kilmer, Toluwani Okunola

Pith reviewed 2026-05-08 06:27 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords nonlinear inverse problemsmajorization-minimizationKrylov subspace recyclinguncertain forward operatorsdynamic imagingstreaming inverse problemscomputed tomographyphotoacoustic tomography
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The pith

A recycled majorization-minimization Krylov method solves nonlinear inverse problems with uncertain forward operators at large scale.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops a framework for inverse problems in imaging where the forward model has uncertainties, such as unknown projection angles in CT. It extends existing majorization-minimization and generalized Krylov subspace techniques to nonlinear settings by incorporating subspace recycling to keep memory bounded. The method supports joint estimation of the image and parameters through alternating or variable projection approaches, plus streaming for dynamic data. Sympathetic readers would care because it addresses a common real-world issue that causes artifacts in standard reconstructions, enabling better use of large or streaming datasets in medical and scientific imaging.

Core claim

The NL-RMM-GKS framework extends MM-GKS to nonlinear inverse problems with uncertain forward operators by combining majorization-minimization for nonsmooth regularization with Krylov subspace projection and recycling. It offers alternating minimization and variable projection formulations, streaming variants, and temporal regularization options, achieving high-quality reconstructions with bounded memory in applications like fan-beam CT and photoacoustic tomography.

What carries the argument

The nonlinear recycled majorization-minimization generalized Krylov subspace (NL-RMM-GKS) framework, which recycles subspaces to bound memory while handling nonlinearity and parameter uncertainty via majorization-minimization and Gauss-Newton updates.

If this is right

  • Reconstructions remain accurate despite uncertainties in the forward operator.
  • Memory requirements stay bounded, enabling processing of large-scale or streaming data without full operator storage.
  • Dynamic problems can incorporate temporal regularization such as optical flow or anisotropic total variation.
  • Both alternating minimization and variable projection formulations provide flexibility in optimization.
  • High-quality results are demonstrated for fan-beam computed tomography and photoacoustic tomography.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach may extend to other modalities with geometry uncertainty, such as MRI under motion.
  • Sequential processing could support real-time reconstruction in clinical workflows.
  • Variable projection might offer efficiency gains over alternating methods in parameter-heavy problems.
  • Integration with other regularizers could broaden applicability to different noise models.

Load-bearing premise

The majorization-minimization and Krylov subspace recycling methods extend effectively to the nonlinear case with uncertain operators, maintaining accuracy and bounded memory in practice.

What would settle it

A test case with known ground truth where the proposed method produces higher reconstruction error than linear approximations or requires memory that scales with the number of time steps.

Figures

Figures reproduced from arXiv: 2605.06336 by Eric de Sturler, James G. Nagy, Mirjeta Pasha, Misha E. Kilmer, Toluwani Okunola.

Figure 6.1
Figure 6.1. Figure 6.1: Test 1 convergence comparison. (a) RRE vs. outer iteration: NL-RMM-GKS view at source ↗
Figure 6.2
Figure 6.2. Figure 6.2: Test 1 streaming performance vs. number of blocks. (a) Final RRE increases with view at source ↗
Figure 6.3
Figure 6.3. Figure 6.3: Test 1 visual comparison. Shepp-Logan reconstructions for view at source ↗
Figure 6.4
Figure 6.4. Figure 6.4: Test 2. initialization sensitivity comparison. (a) RRE vs. iteration for both VarPro view at source ↗
Figure 6.5
Figure 6.5. Figure 6.5: Test 2. visual comparison of final reconstructions for different initialization offsets. view at source ↗
Figure 7.1
Figure 7.1. Figure 7.1: Test 3. Convergence comparison across block counts and regularization strategies. view at source ↗
Figure 7.2
Figure 7.2. Figure 7.2: Test 3. Visual comparison at three time points ( view at source ↗
Figure 7.3
Figure 7.3. Figure 7.3: Test 4. Effect of recycling window size on dynamic PAT reconstruction. (a) RRE view at source ↗
Figure 7.4
Figure 7.4. Figure 7.4: Test 4. Visual comparison at three time points ( view at source ↗
read the original abstract

Many practical imaging systems suffer from uncertainty in acquisition geometry -- such as projection angles in computed tomography or sensor positions in photoacoustic tomography -- leading to nonlinear inverse problems that require joint estimation of both the image and the forward model parameters. Standard approaches that assume a known linear forward operator fail to account for these uncertainties, resulting in significant reconstruction artifacts. We propose a nonlinear recycled majorization-minimization generalized Krylov subspace (NL-RMM-GKS) framework for large-scale inverse problems with uncertain forward operators. The method extends MM-GKS to nonlinear settings by combining majorization-minimization for nonsmooth regularization with Krylov subspace projection and subspace recycling, ensuring bounded memory usage. Two complementary formulations are developed: an alternating minimization approach that alternates between image updates and Gauss-Newton parameter estimation, and a variable projection approach that eliminates the image variable and optimizes directly over the parameters using inexact inner solves. We further introduce streaming variants that process data sequentially, enabling reconstruction from large or dynamically acquired datasets without storing the full operator. For dynamic problems, we incorporate two temporal regularization strategies -- optical flow and anisotropic total variation -- as plug-in choices within the framework. We carry out rigorous numerical experiments in fan-beam computed tomography and photoacoustic tomography to demonstrate that our proposed framework achieves high-quality reconstructions with bounded memory requirements, making it suitable for large-scale dynamic imaging problems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper proposes a nonlinear recycled majorization-minimization generalized Krylov subspace (NL-RMM-GKS) framework for large-scale inverse problems with uncertain forward operators. It extends prior MM-GKS techniques to nonlinear settings via alternating minimization (alternating image updates with Gauss-Newton parameter estimation) or variable projection (eliminating the image variable), incorporates streaming variants for sequential data processing without storing the full operator, and adds temporal regularizers (optical flow or anisotropic TV) for dynamic problems. The authors claim that the method achieves high-quality reconstructions with bounded memory usage, supported by numerical experiments in fan-beam CT and photoacoustic tomography.

Significance. If the claimed extensions preserve accuracy and bounded memory in practice, the framework would address a practically important gap in dynamic imaging under acquisition uncertainties, enabling memory-efficient solutions for large-scale streaming problems. The dual formulations and plug-in regularizers provide flexibility, and the focus on bounded memory is a clear strength for real-world applicability in CT and PAT. However, the absence of any concrete experimental data, metrics, or implementation details prevents confirming whether these benefits are realized.

major comments (1)
  1. [Abstract] Abstract (and implied Numerical Experiments section): The manuscript asserts that 'rigorous numerical experiments' in fan-beam CT and photoacoustic tomography demonstrate 'high-quality reconstructions with bounded memory requirements,' yet no data, error metrics, baselines, implementation details, convergence plots, or validation against ground truth are provided. This directly undermines assessment of the central claims that the nonlinear extensions and streaming variants preserve both accuracy and memory bounds.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and for identifying the need for stronger experimental support. We address the major comment below and will revise the manuscript to incorporate the requested details.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and implied Numerical Experiments section): The manuscript asserts that 'rigorous numerical experiments' in fan-beam CT and photoacoustic tomography demonstrate 'high-quality reconstructions with bounded memory requirements,' yet no data, error metrics, baselines, implementation details, convergence plots, or validation against ground truth are provided. This directly undermines assessment of the central claims that the nonlinear extensions and streaming variants preserve both accuracy and memory bounds.

    Authors: We acknowledge that the referee's observation is correct: while the manuscript describes the experimental setups for fan-beam CT and photoacoustic tomography and asserts high-quality results with bounded memory, the current version does not include the specific quantitative data, error metrics, baseline comparisons, implementation details, convergence plots, or ground-truth validations needed to fully substantiate these claims. In the revised manuscript we will expand the Numerical Experiments section to provide these elements, including tables of relative reconstruction errors and PSNR values, memory-usage measurements, comparisons against standard nonlinear solvers and non-recycled MM methods, convergence histories, and visual/quantitative validation against ground-truth phantoms. These additions will be presented for both the alternating-minimization and variable-projection formulations as well as the streaming variants, thereby allowing a rigorous assessment of accuracy and memory bounds. revision: yes

Circularity Check

0 steps flagged

No significant circularity; algorithmic extension with external validation

full rationale

The paper describes an extension of the existing MM-GKS framework to nonlinear settings via alternating minimization (image updates + Gauss-Newton parameter estimation) and variable projection, plus streaming variants and temporal regularizers. No load-bearing derivation reduces by construction to a fitted quantity or self-citation chain; the core claims rest on the algorithmic construction and are supported by numerical experiments in fan-beam CT and photoacoustic tomography. Self-citations to prior MM-GKS work are present but not invoked as uniqueness theorems that force the result. This is a standard low-circularity case for a methods paper.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities. The approach relies on standard majorization-minimization for nonsmooth terms and Krylov subspace methods, which are treated as established background.

pith-pipeline@v0.9.0 · 5565 in / 1141 out tokens · 63680 ms · 2026-05-08T06:27:59.803269+00:00 · methodology

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Reference graph

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