arxiv: 2605.12085 · v1 · submitted 2026-05-12 · 🧮 math.NA · cs.NA

Recognition: 2 theorem links

· Lean Theorem

A Line--Search--Based Stochastic Gradient Method for 3D Computed Tomography

Elena Morotti, Federica Porta, Ilaria Trombini, Tatiana A. Bubba, Valeria Ruggiero

Pith reviewed 2026-05-13 04:09 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords 3D computed tomographystochastic gradient methodline searchforward-backward algorithminverse problemsvolumetric reconstructionmini-batch samplingCT reconstruction

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The pith

A line-search stochastic gradient method with full-projection mini-batches accelerates 3D CT reconstruction without training data.

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

The paper introduces FB-LISA, a forward-backward generalization of a line-search stochastic gradient algorithm, to solve the high-dimensional inverse problem of volumetric reconstruction from CT data. It forms stochastic mini-batches from complete 2D projections rather than smaller subsets, which keeps the physical geometry of the acquisition intact and produces faster progress during the first iterations. The approach repurposes optimization ideas common in deep learning to reduce the computational burden of forward and back-projections while avoiding any need for training data or learned priors.

Core claim

FB-LISA applies forward-backward splitting to a line-search-based stochastic gradient method so that mini-batches of full 2D projections can be used on the large-scale 3D CT operator; the resulting scheme yields substantial speed-ups in early iterations while preserving the physical structure of the acquisition process and requiring no training data or learned priors.

What carries the argument

FB-LISA, the forward-backward line-search stochastic gradient algorithm that samples mini-batches consisting of entire 2D projections to address the high-dimensional 3D CT inverse problem.

Load-bearing premise

The forward-backward generalization of the line-search stochastic gradient method keeps reliable convergence and reconstruction quality when applied to the high-dimensional 3D CT operator with full-projection mini-batches.

What would settle it

A test on a standard 3D CT benchmark dataset in which FB-LISA either diverges, fails to reach acceptable error in a reasonable number of iterations, or yields image quality metrics markedly worse than a deterministic line-search method would disprove the central claim.

Figures

Figures reproduced from arXiv: 2605.12085 by Elena Morotti, Federica Porta, Ilaria Trombini, Tatiana A. Bubba, Valeria Ruggiero.

**Figure 2.** Figure 2: Relative error vs execution time in seconds for Case 3. [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

read the original abstract

We introduce FB-LISA, a forward-backward (FB) generalization of a recently proposed line-search-based stochastic gradient algorithm to address the imaging problem of volumetric reconstruction in Computed Tomography, a substantially high demanding problem, which involves orders of magnitude of data, a high computational burden for forward and backprojection, and memory requirements that push current GPU architectures to their limits. Our formulation employs stochastic mini-batches composed of full 2D projections, preserving the physical structure of the acquisition process while enabling significant speed-ups during early iterations. The resulting method demonstrates how concepts traditionally associated with deep learning can be repurposed to accelerate large-scale inverse problems, without relying on training data or learned priors.

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.

Desk Editor's Note private letter to a colleague

FB-LISA is a direct forward-backward extension of a line-search stochastic gradient method to 3D CT using full-projection mini-batches, but the speed-up claims need concrete numbers and comparisons to land.

read the letter

The main point is that the authors generalize a recent line-search stochastic gradient algorithm into a forward-backward version (FB-LISA) for volumetric CT reconstruction. They form mini-batches from complete 2D projections rather than random subsets of rays, which keeps the acquisition geometry intact and targets faster early progress on these memory-heavy problems without any training data or learned priors.

Referee Report

0 major / 2 minor

Summary. The manuscript introduces FB-LISA, a forward-backward generalization of a line-search-based stochastic gradient algorithm for volumetric reconstruction in 3D Computed Tomography. The formulation employs stochastic mini-batches of full 2D projections to preserve the physical structure of the acquisition process while enabling speed-ups in early iterations, and it repurposes concepts from deep learning to accelerate large-scale inverse problems without training data or learned priors.

Significance. If the claimed convergence behavior and reconstruction quality hold, the work would be significant for practical acceleration of high-dimensional CT problems on limited hardware. It offers a structure-preserving stochastic approach that bridges optimization ideas from machine learning with classical inverse problems, potentially reducing computational and memory demands without requiring external training data.

minor comments (2)

[Abstract] The abstract asserts speed-ups and structure preservation but would benefit from a brief quantitative statement (e.g., iteration count or wall-clock reduction) to ground the central claim, even if full results appear later in the manuscript.
[Methods] Notation for the stochastic mini-batch selection (full 2D projections) should be introduced with a clear equation or diagram in the methods section to make the preservation of acquisition physics explicit.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work on FB-LISA and for recommending minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper introduces FB-LISA as a forward-backward generalization of a recently proposed line-search stochastic gradient method for 3D CT reconstruction, using stochastic mini-batches of full 2D projections to preserve acquisition physics and accelerate early iterations without training data. No equations, fitted parameters, or explicit predictions are shown in the abstract or description that reduce by construction to inputs. The formulation is presented as a direct methodological extension that repurposes deep learning concepts for inverse problems, with the central claim remaining independent and self-contained rather than tautological or forced by self-citation chains. No self-definitional steps, fitted-input predictions, or load-bearing uniqueness theorems appear.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract supplies no explicit free parameters, axioms, or invented entities; the method is described only at the level of algorithmic structure.

pith-pipeline@v0.9.0 · 5425 in / 1039 out tokens · 75183 ms · 2026-05-13T04:09:53.835828+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel (J(x) = ½(x + x⁻¹) − 1 is the unique calibrated reciprocal cost) unclear
FB–LISA ... proximal operator ... line-search inequality ... f_Nk(¯x(k)) ≤ ... + (1/(2α_k))∥¯x(k)−x(k)∥² ... Theorem 1 (almost-sure stationarity of limit points)
IndisputableMonolith/Foundation/DimensionForcing.lean reality_from_one_distinction (8-tick period and D=3 forced) unclear
mini-batch size increase ... N_t = min(n_max, max(⌈C/(ε̂_k + ⌈n/N_{t-1}⌉)⌉, N_0)) ... 8-tick or φ-ladder absent

Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages

[1]

A. H. Andersen and A. C. Kak. Simultaneous algebraic reconstruction technique (sart): a superior implementation of the art algorithm.Ultrasonic imaging, 6(1):81–94, 1984

work page 1984
[2]

Barzilai and J

J. Barzilai and J. M. Borwein. Two-point step size gradient methods.IMA Journal of Numerical Analysis, 8:141–148, 1988

work page 1988
[3]

Beck.First-order methods in optimization

A. Beck.First-order methods in optimization. SIAM, 2017

work page 2017
[4]

Bonettini, I

S. Bonettini, I. Loris, F. Porta, and M. Prato. Variable metric inexact line-search-based methods for nonsmooth optimization.SIAM Journal on Optimization, 26(2):891–921, 2016

work page 2016
[5]

Bottou, F

L. Bottou, F. E. Curtis, and J. Nocedal. Optimization methods for large-scale machine learning.SIAM Review, 60(2):223–311, 2018

work page 2018
[6]

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein. Distributed optimization and statistical learning via the alternating direction method of multipliers.Foundations and Trends in Machine learning, 3(1):1–22, 2011

work page 2011
[7]

Cavicchioli, J

R. Cavicchioli, J. C. Hu, E. Loli Piccolomini, E. Morotti, and L. Zanni. Gpu accel- eration of a model-based iterative method for digital breast tomosynthesis.Scientific reports, 10(1):43, 2020

work page 2020
[8]

Chambolle, C

A. Chambolle, C. Delplancke, M. J. Ehrhardt, C.-B. Sch¨ onlieb, and J. Tang. Stochastic primal–dual hybrid gradient algorithm with adaptive step sizes.Journal of Mathemat- ical Imaging and Vision, 66:294–313, 2024. 14

work page 2024
[9]

Chambolle, M

A. Chambolle, M. J. Ehrhardt, P. Richt´ arik, and C.-B. Sch¨ onlieb. Stochastic primal- dual hybrid gradient algorithm with arbitrary sampling and imaging applications. SIAM Journal on Optimization, 28:2783–2808, 2018

work page 2018
[10]

Chambolle and T

A. Chambolle and T. Pock. A first-order primal-dual algorithm for convex problems with applications to imaging.Journal of Mathematical Imaging and Vision, 40(1):120– 145, 2011

work page 2011
[11]

P. L. Combettes and V. R. Wajs. Signal recovery by proximal forward-backward split- ting.Multiscale Modeling & Simulation, 4(4):1168–1200, 2005

work page 2005
[12]

L. Condat. A primal–dual splitting method for convex optimization involving lips- chitzian, proximable and linear composite terms.Journal of optimization theory and applications, 158(2):460–479, 2013

work page 2013
[13]

Dossal, S

C. Dossal, S. Hurault, and N. Papadakis. Optimization with first order algorithms. arXiv preprint arXiv:2410.19506, 2024

work page arXiv 2024
[14]

M. J. Ehrhardt, ˇZ. Kereta, J. Liang, and J. Tang. A guide to stochastic optimisation for large-scale inverse problems.Inverse Problems, 41(5), 2025

work page 2025
[15]

M. J. Ehrhardt, P. J. Markiewicz, and C.-B. Sch¨ onlieb. Faster pet reconstruction with non-smooth priors by randomization and preconditioning.Physics in Medicine & Biology, 64:225019, 2019

work page 2019
[16]

Erdogan and J

H. Erdogan and J. A. Fessler. Ordered subsets algorithms for transmission tomography. Phys. Med. Biol., 44(11):2835, 1999

work page 1999
[17]

L. A. Feldkamp, L. C. Davis, and J. W. Kress. Practical cone-beam algorithm.Journal of the Optical Society of America A, 1(6):612–619, 1984

work page 1984
[18]

Franchini, F

G. Franchini, F. Porta, V. Ruggiero, and I. Trombini. A line search based proximal stochastic gradient algorithm with dynamical variance reduction.Journal of Scientific Computing, 94:23, 2023

work page 2023
[19]

Franchini, F

G. Franchini, F. Porta, V. Ruggiero, I. Trombini, and L. Zanni. A stochastic gradient method with variance control and variable learning rate for deep learning.Journal of Computational and Applied Mathematics, 451:116083, 2024

work page 2024
[20]

Frassoldati, G

G. Frassoldati, G. Zanghirati, and L. Zanni. New adaptive stepsize selections in gra- dient methods.J. Ind. Manag. Optim., 4(2):299–312, 2008

work page 2008
[21]

G. T. Herman and L. B. Meyer. Algebraic reconstruction techniques can be made computationally efficient (positron emission tomography application).IEEE Trans. Med. Imag., 12(3):600–609, 1993

work page 1993
[22]

A. C. Kak and M. Slaney.Principles of computerized tomographic imaging. SIAM, 2001. 15

work page 2001
[23]

D. Kim, S. Ramani, and J. A. Fessler. Combining ordered subsets and momentum for accelerated x-ray ct image reconstruction.IEEE Trans. Med. Imag., 34(1):167–178, 2015

work page 2015
[24]

Lazzaretti, Z

M. Lazzaretti, Z. Kereta, C. Estatico, and L. Calatroni. Stochastic gradient descent for linear inverse problems in variable exponent lebesgue spaces. InInternational Con- ference on Scale Space and Variational Methods in Computer Vision, pages 457–470. Springer, 2023

work page 2023
[25]

A. Meaney. Cone-beam computed tomography dataset of a walnut (1.1.0) [data set]. Zenodo, 2022

work page 2022
[26]

Papoutsellis, M

E. Papoutsellis, M. A. G. Duff, J. S. Jørgensen, S. Porter, C. Delplancke, G. Fardell, E. Pasca, and K. Thielemans. A modular approach to stochastic optimisation for inverse problems using the core imaging library.arXiv preprint arXiv:2603.21230, 2026

work page arXiv 2026
[27]

Polyak.Introduction to Optimization

B. Polyak.Introduction to Optimization. Optimization Software, New York, 1987

work page 1987
[28]

Reisenhofer, S

R. Reisenhofer, S. Bosse, G. Kutyniok, and T. Wiegand. A Haar Wavelet-Based Percep- tual Similarity Index for Image Quality Assessment.Signal Process. Image, 61:33–43, 2018

work page 2018
[29]

J. Tang, K. Egiazarian, and M. Davies. The limitation and practical acceleration of stochastic gradient algorithms in inverse problems. In IEEE, editor,ICASSP 2019- 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 7680–7684, 2019

work page 2019
[30]

J. Tang, K. Egiazarian, M. Golbabaee, and M. Davies. The practicality of stochas- tic optimization in imaging inverse problems.IEEE Transactions on Computational Imaging, 6:1471–1485, 2020

work page 2020
[31]

Van Aarle, W

W. Van Aarle, W. J. Palenstijn, J. Cant, E. Janssens, F. Bleichrodt, A. Dabravolski, J. De Beenhouwer, K. J. Batenburg, and J. Sijbers. Fast and flexible X-ray tomography using the ASTRA toolbox.Optics express, 24(22):25129–25147, 2016

work page 2016
[32]

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli. Image quality assessment: from error visibility to structural similarity.IEEE transactions on image processing, 13(4):600–612, 2004. 16

work page 2004