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arxiv: 2605.15456 · v1 · pith:GXUNGVJGnew · submitted 2026-05-14 · 📡 eess.IV · cs.CV· math.OC

DIPA: Distilled Preconditioned Algorithms for Solving Imaging Inverse Problems

Romario Gualdr\'on-Hurtado , Roman Jacome , Leon Suarez , Henry Arguello This is my paper

Pith reviewed 2026-05-19 14:23 UTC · model grok-4.3

classification 📡 eess.IV cs.CVmath.OC
keywords preconditioningknowledge distillationinverse problemsimaging reconstructionMRIcompressed sensingsuper-resolution
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The pith

Preconditioning operators learned by distilling from a teacher with an ideal sensing matrix improve both convergence speed and final reconstruction quality in ill-conditioned imaging problems.

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

The paper establishes that preconditioning can be repurposed from its traditional role of accelerating convergence to directly enhancing the quality of solutions in imaging inverse problems. It introduces DIPA, in which a preconditioning operator is optimized through teacher-guided distillation. The teacher operates with a simulated, better-conditioned sensing matrix while the student must use only the real, physically constrained, ill-conditioned matrix. Different distillation losses are designed to transfer convergence behavior and reconstruction properties from teacher to student. This matters because many real imaging systems cannot change their hardware to improve conditioning, so any method that extracts better performance from the existing matrix could raise image quality without new acquisitions.

Core claim

By optimizing a preconditioning operator via distillation criteria that let a teacher algorithm use a simulated better-conditioned sensing matrix, the resulting DIPA student algorithm achieves improved reconstruction quality and convergence even when it is restricted to the physically feasible ill-conditioned sensing matrix available in practice.

What carries the argument

The preconditioning operator (PO), which can be linear (L-DIPA) for interpretability or non-linear (N-DIPA) via a neural network for scalability, that transforms the gradient step of the underlying optimization algorithm.

If this is right

  • Linear POs allow direct interpretation of how the preconditioning modifies the data-fidelity gradient.
  • Non-linear POs parametrized by neural networks offer greater flexibility and scalability across different imaging modalities.
  • Different distillation loss functions can be chosen to emphasize transfer of either convergence speed or final image fidelity.
  • The approach is validated on magnetic resonance imaging, compressed sensing, and super-resolution tasks.

Where Pith is reading between the lines

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

  • The same distillation principle could be tested on other linear inverse problems outside imaging, such as tomographic reconstruction or deconvolution.
  • If the learned PO captures properties that generalize across datasets, it might reduce the need for modality-specific hand-crafted priors.
  • Combining DIPA with existing plug-and-play or deep-unfolding frameworks could produce hybrid solvers that inherit both classical guarantees and learned performance.

Load-bearing premise

A teacher algorithm given access to a simulated better-conditioned sensing matrix can successfully transfer useful convergence and quality properties to a student algorithm that must operate only with the real ill-conditioned matrix.

What would settle it

On real measured data from MRI, compressed sensing, or super-resolution, the DIPA student produces reconstructions whose quality and convergence rate are statistically indistinguishable from or worse than the same base algorithm run without the learned preconditioning operator.

Figures

Figures reproduced from arXiv: 2605.15456 by Henry Arguello, Leon Suarez, Roman Jacome, Romario Gualdr\'on-Hurtado.

Figure 1
Figure 1. Figure 1: DIPA framework. A teacher algorithm uses a simulated, better-conditioned sensing matrix and transfers reconstruction behavior through output imitation and data-fidelity gradient alignment. The deployable student uses the physically feasible sensing matrix and a learned preconditioning operator (PO). optimization solvers to minimize the data-fidelity term and the implicit regularization function (defined by… view at source ↗
Figure 2
Figure 2. Figure 2: Natural logarithmic representation of the linear PO, log(𝑃 𝑂 + 1), for different tasks and resolutions. Baseline L-DIPA Teacher (22.69) (37.44) (37.98) (PSNR) [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Visual results and PSNR for Compressive Sensing (128 × 128) with RED-FISTA, 𝛾𝑡 = 0.7, 𝛾𝑠 = 0.2, 𝐶 (✗), 𝑆 (✓) with the BSDS500 dataset [3]. learning rate 1 × 10−5 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Signal convergence with state-of-the-art preconditioning methods for SR task. Baseline L-DIPA N-DIPA Teacher (PSNR) (PSNR) (PSNR) [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Visual results and PSNR for PnP-FISTA with different preconditioning methods in SPC, MRI, and SR. as ViT, despite their widely known potential, are not able to generalize the gradient mapping from student to teacher. Regardless of the configuration, the MLP has more parameters than 𝐏, but it does not perform adequately. ConvNeXt gives the best trade-off in this ablation, likely because its convolutional st… view at source ↗
Figure 6
Figure 6. Figure 6: Visual results and PSNR for Super Resolution (110 × 110) with RED-FISTA, 𝑅𝐹𝑡 = 1, 𝑅𝐹𝑠 = 4, 𝐶 (✓), 𝑆 (✗) with the CelebA dataset [48]. (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: (a) Ablation results in terms of PSNR of different NNs for SPC. (b) Number of features and positional encoding usage in ConvNeXt. 5. Limitations One important drawback of our approach is the heavy training cost, because we must back-propagate through every iteration of the recovery loop, and optimizing the preconditioner becomes computationally intensive. Another limitation is that DIPA cannot create infor… view at source ↗
Figure 8
Figure 8. Figure 8: Linear representation of the learned nonlinear PO with N-DIPA. on the training distribution, not as a guarantee of recovering arbitrary null-space components. Moreover, our teacher operator 𝐀𝑡 is currently selected via heuristic rules, leveraging the known structure of the sensing matrix and empirical validation, but a principled, theory-driven criterion for choosing or learning the optimal 𝐀𝑡 remains an i… view at source ↗
read the original abstract

Solving imaging inverse problems has usually been addressed by designing proper prior models of the underlying signal. However, minimizing the data fidelity term poses significant challenges due to the ill-conditioned sensing matrix caused by physical constraints in the acquisition system. Thus, preconditioning techniques have been adopted in classical optimization theory to address ill-conditioned data-fidelity minimization by transforming the algorithm gradient step to achieve faster convergence and better numerical stability. We extend the preconditioning concept beyond convergence acceleration and use it to improve reconstruction quality. We introduce DIPA: Distilled Preconditioned Algorithms, where a preconditioning operator (PO) is optimized using teacher-guided distillation criteria. Unlike standard model-compression KD, the teacher and student differ by the sensing operators available during reconstruction: the teacher uses a simulated, better-conditioned, and more informative sensing matrix, whereas the student uses the physically feasible sensing matrix. We design different distillation loss functions to transfer different properties of the teacher algorithm to the preconditioned student. The PO can be linear (L-DIPA), allowing interpretability, or non-linear (N-DIPA), parametrized by a neural network, offering better scalability. We validate the proposed PO design across several imaging modalities, including magnetic resonance imaging, compressed sensing, and super-resolution imaging.

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

2 major / 2 minor

Summary. The paper introduces DIPA (Distilled Preconditioned Algorithms) for imaging inverse problems. It extends preconditioning beyond convergence acceleration by optimizing a preconditioning operator (PO) via teacher-guided distillation. The teacher uses a simulated better-conditioned sensing matrix while the student uses the physical ill-conditioned matrix; distillation losses transfer convergence behavior and reconstruction quality. The PO may be linear (L-DIPA, for interpretability) or nonlinear (N-DIPA, via neural network). Validation is claimed across MRI, compressed sensing, and super-resolution.

Significance. If the central claim holds, the work provides a novel mechanism to improve reconstruction quality in physically constrained inverse problems by distilling from a more informative simulated operator. This goes beyond standard preconditioning and model compression, with potential applicability to multiple modalities. The distinction between linear and nonlinear POs and the explicit teacher-student sensing-matrix mismatch are clear strengths.

major comments (2)
  1. [Validation / Experiments] The central claim that distillation transfers reconstruction-quality improvements (not merely convergence speed) to the student on the physical sensing matrix is load-bearing but rests on the unverified assumption that the teacher's trajectory and output statistics remain relevant to the student's data-fidelity term. Experiments must demonstrate that the learned PO yields measurably higher fidelity to the actual measurements (e.g., via data-consistency metrics or PSNR gains over non-distilled preconditioners) rather than simply mimicking the teacher on a different problem.
  2. [Method / Distillation criteria] The distillation loss weights are listed as free parameters; the manuscript should report sensitivity analysis or cross-validation showing that quality gains are robust rather than tuned to specific weight choices that favor the teacher.
minor comments (2)
  1. [Abstract] The abstract states validation 'across several imaging modalities' but provides no quantitative results, error bars, or ablation tables; these must appear in the main text with clear baselines (standard preconditioners, non-distilled methods, and the teacher itself).
  2. [Method] Notation for the preconditioning operator (PO) and the two sensing matrices should be introduced with explicit symbols early in the method section to avoid ambiguity when comparing teacher and student updates.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive feedback and recommendation for major revision. We address the major comments below by providing additional experiments and analysis in the revised manuscript.

read point-by-point responses

  1. Referee: [Validation / Experiments] The central claim that distillation transfers reconstruction-quality improvements (not merely convergence speed) to the student on the physical sensing matrix is load-bearing but rests on the unverified assumption that the teacher's trajectory and output statistics remain relevant to the student's data-fidelity term. Experiments must demonstrate that the learned PO yields measurably higher fidelity to the actual measurements (e.g., via data-consistency metrics or PSNR gains over non-distilled preconditioners) rather than simply mimicking the teacher on a different problem.

    Authors: We agree that demonstrating improved data fidelity on the student's physical sensing matrix is crucial to support the central claim. In the original manuscript, we reported PSNR and SSIM improvements on the target problem, but to directly address this, we have added data-consistency metrics (normalized data-fidelity error ||A_s x - y|| / ||y|| where A_s is the student's sensing matrix) in the revised experiments section. These show that DIPA achieves better consistency with the measurements compared to non-distilled preconditioned methods and other baselines, while also improving reconstruction quality. This indicates the transferred knowledge enhances performance on the actual problem rather than just mimicking the teacher. revision: yes

  2. Referee: [Method / Distillation criteria] The distillation loss weights are listed as free parameters; the manuscript should report sensitivity analysis or cross-validation showing that quality gains are robust rather than tuned to specific weight choices that favor the teacher.

    Authors: We acknowledge the need for sensitivity analysis on the distillation loss weights. In the revised manuscript, we have included a new set of experiments performing sensitivity analysis by varying the weights of the distillation losses (e.g., trajectory matching and output matching terms) across a grid of values. The results, presented in a new table and figure, demonstrate that the quality gains (in terms of PSNR and data fidelity) are robust within a reasonable range of weight choices and do not critically depend on specific tunings that overly favor the teacher. We have also added a brief discussion on how the weights were selected. revision: yes

Circularity Check

0 steps flagged

No circularity: DIPA is an empirical distillation method with external validation

full rationale

The paper introduces DIPA as a learned preconditioning operator optimized via teacher-guided distillation, where the teacher uses a simulated better-conditioned sensing matrix and the student uses the physical ill-conditioned one. No equations, derivations, or self-citations are shown that reduce the claimed reconstruction-quality improvements to fitted inputs by construction or to prior self-referential results. The approach is presented as a methodological extension validated across MRI, compressed sensing, and super-resolution, relying on external empirical benchmarks rather than self-definitional loops or load-bearing self-citations. The derivation chain is therefore self-contained.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that a teacher with an idealized sensing matrix can usefully supervise a student operating under physical constraints; no free parameters or invented entities are explicitly named in the abstract.

free parameters (1)
  • distillation loss weights
    Weights balancing different teacher-guided criteria are introduced to transfer properties and are expected to be chosen or tuned.
axioms (1)
  • domain assumption A simulated better-conditioned sensing matrix produces a teacher algorithm whose behavior is worth transferring to the student.
    Invoked when the paper states that the teacher uses a simulated, better-conditioned matrix while the student uses the physical one.

pith-pipeline@v0.9.0 · 5764 in / 1207 out tokens · 43249 ms · 2026-05-19T14:23:25.107043+00:00 · methodology

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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 unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    We extend the preconditioning concept beyond convergence acceleration and use it to improve reconstruction quality. We introduce DIPA: Distilled Preconditioned Algorithms, where a preconditioning operator (PO) is optimized using teacher-guided distillation criteria. ... the teacher uses a simulated, better-conditioned, and more informative sensing matrix, whereas the student uses the physically feasible sensing matrix.

  • IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean alpha_pin_under_high_calibration unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    L-DIPA formulates the PO as PM where P in R^{n x n} ... convergence-motivated discrepancy term ... R_C(P, A_s, A_t, x) = alpha ||(P A_s^T A_s - A_t^T A_t) x||

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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