arxiv: 2605.13146 · v1 · submitted 2026-05-13 · 📊 stat.ML · cs.CV· cs.LG

Recognition: 2 theorem links

· Lean Theorem

On Hallucinations in Inverse Problems: Fundamental Limits and Provable Assessment Methods

David Iagaru , Nina M. Gottschling , Anders C. Hansen , Josselin Garnier

Authors on Pith no claims yet

Pith reviewed 2026-05-14 18:10 UTC · model grok-4.3

classification 📊 stat.ML cs.CVcs.LG

keywords hallucinationsinverse problemsdeep learningimagingforward modelfaithfulness assessmentill-posed problemsreconstruction bounds

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The pith

Hallucinations in AI image reconstructions arise necessarily from the ill-posed inverse problem, with magnitude bounds set only by the forward model.

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

The paper establishes that hallucinations—realistic but incorrect details produced by deep networks in inverse imaging problems—are not merely model failures but can be forced by the fundamental difficulty of recovering a signal from incomplete measurements. It derives the precise conditions under which any reconstruction method must introduce hallucinations and supplies computable bounds on their size that use only the known forward model. Algorithms are then given to estimate the smallest hallucination magnitude possible for any input and to check how faithful the details in a given reconstruction are. A sympathetic reader would care because these tools allow reliability assessment in medical or scientific imaging without access to ground-truth data.

Core claim

We develop a theoretical framework showing that such hallucinations are not merely artifacts of particular models, but can arise from the ill-posed nature of the inverse problem itself. We derive necessary and sufficient conditions for hallucinations, together with computable bounds on their magnitude that depend only on the forward model. Building on this theory, we introduce algorithms to estimate the minimum hallucination magnitude achievable by any reconstruction model for a given input and to assess the faithfulness of reconstructed details by a given reconstruction model.

What carries the argument

Necessary and sufficient conditions for the occurrence of hallucinations, together with bounds on their magnitude that are computed solely from the forward model.

If this is right

Any reconstruction method, including modern generative models, is subject to the same hallucination limits fixed by the forward model.
The minimum hallucination magnitude achievable for any given input can be estimated by algorithm.
Faithfulness of individual details in a reconstruction can be assessed without ground-truth data.
The framework applies across distinct imaging tasks and supplies a principled way to quantify hallucinations.

Where Pith is reading between the lines

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

Reconstruction algorithms could be trained or regularized to approach the theoretical minimum hallucination bound derived from the forward model.
The same necessary-and-sufficient conditions may extend to other ill-posed inverse problems outside imaging, such as limited-angle tomography or sparse signal recovery.
Practitioners could compare the estimated minimum bound against the output of any chosen model to decide whether further measurements or a different method are required.

Load-bearing premise

The forward model is known exactly and the function spaces chosen for the signals permit derivation of the necessary and sufficient conditions without further data-dependent assumptions.

What would settle it

A reconstruction method that produces zero hallucinations on an input for which the derived conditions require positive hallucination, or that achieves a hallucination magnitude strictly below the bound computed from the forward model.

Figures

Figures reproduced from arXiv: 2605.13146 by Anders C. Hansen, David Iagaru, Josselin Garnier, Nina M. Gottschling.

**Figure 2.** Figure 2: Methods to anticipate and detect hallucinations in this paper. [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

**Figure 3.** Figure 3: Manual detail pasting method for linear forward models with additive noise, illustrated for inverse problems [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

**Figure 4.** Figure 4: The number of hallucinations increases with the worst-case kernel size: Super-resolution results for MNIST [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗

**Figure 5.** Figure 5: Illustration of Definition 2.5 and Theorem 3.2. The reconstructions are generated by a decoder trained [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗

**Figure 6.** Figure 6: Qualitative hallucinations examples in MRI acceleration. The first three column show the the base image [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗

**Figure 7.** Figure 7: Sharpness of the lower bound in Theorem 3.6. Both the errors and the diameters are measured in terms of [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗

**Figure 8.** Figure 8: Examples of significant detail drawing following the method described in Figure 3. A red arrow points at [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗

**Figure 9.** Figure 9: Quantities of Definition 2.5 and Theorem 3.2 for super-resolution of Sentinel-2 data. All the reconstruc [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗

**Figure 10.** Figure 10: Patch-wise application of the decoder agnostic method of Figure 2. Dark areas of the image correspond [PITH_FULL_IMAGE:figures/full_fig_p025_10.png] view at source ↗

**Figure 11.** Figure 11: Sharpness of the lower bound in Theorem 3.6. Both the errors and the diameters are normalised by the [PITH_FULL_IMAGE:figures/full_fig_p026_11.png] view at source ↗

read the original abstract

Artificial intelligence (AI) has transformed imaging inverse problems, from medical diagnostics to Earth observation. Yet deep neural networks can produce hallucinations, realistic-looking but incorrect details, undermining their reliability, especially when ground truth data is unavailable. We develop a theoretical framework showing that such hallucinations are not merely artifacts of particular models, but can arise from the ill-posed nature of the inverse problem itself. We derive necessary and sufficient conditions for hallucinations, together with computable bounds on their magnitude that depend only on the forward model. Building on this theory, we introduce algorithms to: (1) estimate the minimum hallucination magnitude achievable by any reconstruction model for a given input; (2) assess the faithfulness of reconstructed details by a given reconstruction model. Experiments across three imaging tasks demonstrate that our approach applies broadly, including to modern generative models, and provides a principled way to quantify and evaluate AI hallucinations.

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

The paper gives nec-and-suff conditions for hallucinations plus magnitude bounds claimed to depend only on the forward model, but the signal-space choice looks like a real caveat that needs checking.

read the letter

The headline result is that hallucinations in inverse-problem reconstructions are not just model artifacts but can be characterized by necessary and sufficient conditions tied to ill-posedness, with computable bounds that the authors say depend only on the forward operator. They also supply two practical algorithms: one to estimate the smallest hallucination magnitude any reconstructor could produce for a given input, and one to score how faithful a specific model's details are. That framing is new relative to the usual empirical checks in the literature, and the experiments on three imaging tasks (including modern generative models) show the approach can be applied without ground truth. The work is therefore useful for anyone who needs a principled way to quantify reliability when data is scarce, such as in medical or remote-sensing settings. The soft spot is the one flagged in the stress-test note. Defining what counts as an incorrect detail requires some structure on the signal space (a topology or seminorm that separates plausible from implausible features), and that structure sits outside the forward operator. If the derived conditions or bounds shift when you change the space in a reasonable way, the claim that everything depends only on the forward model does not fully hold. The abstract does not spell out invariance under such changes, so the proofs would need a careful look for hidden assumptions there. The experiments appear to fix specific tasks, which may mask how sensitive the numbers are to the choice of space. This is for readers working on theoretical foundations of inverse problems and AI reliability in imaging. A serious thinker who wants to see whether the math really closes the loop on hallucinations without extra data-dependent priors would get value from it. I would send it to peer review because the core idea is sharp enough to deserve referee time, even if the assumptions around the signal space need tightening.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a theoretical framework for hallucinations in AI-driven solutions to imaging inverse problems. It derives necessary and sufficient conditions for hallucinations and computable bounds on their magnitude that depend only on the forward model. Algorithms are introduced to estimate the minimum hallucination magnitude achievable by any reconstruction model and to assess faithfulness of reconstructed details for a given model. Experiments on three imaging tasks, including modern generative models, demonstrate applicability.

Significance. If the central derivations hold, the work supplies a principled, forward-model-only approach to quantifying hallucinations in ill-posed inverse problems. This would be significant for reliability assessment in applications such as medical imaging where ground truth is unavailable, moving beyond purely empirical checks.

major comments (2)

[Abstract] Abstract: The claim that bounds 'depend only on the forward model' is load-bearing for the central contribution, yet the definition of a hallucination as an 'incorrect detail' requires a precise signal space X (e.g., a Banach space with topology or seminorm) that is external to the forward operator A. Without an explicit invariance argument under reasonable changes to X, the 'only on the forward model' statement does not hold.
[Theory (conditions derivation)] The necessary-and-sufficient conditions (stated in the abstract) appear to presuppose a fixed admissible-signal set without data-dependent assumptions. If the proofs rely on any particular choice of X that is not shown to be canonical or invariant, the conditions risk being non-unique and the computable bounds may change with that choice.

minor comments (2)

Clarify in the introduction whether the three imaging tasks are standard benchmarks or custom, and list the forward operators explicitly.
Ensure all function-space notation (e.g., norms used to quantify hallucination magnitude) is defined at first use.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough and constructive review. The comments raise important points about the precise dependence of our results on the forward operator alone. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

Referee: [Abstract] Abstract: The claim that bounds 'depend only on the forward model' is load-bearing for the central contribution, yet the definition of a hallucination as an 'incorrect detail' requires a precise signal space X (e.g., a Banach space with topology or seminorm) that is external to the forward operator A. Without an explicit invariance argument under reasonable changes to X, the 'only on the forward model' statement does not hold.

Authors: We thank the referee for this observation. In the manuscript the signal space X is the domain on which the forward operator A is defined, and a hallucination is any nonzero component lying in ker(A). The necessary-and-sufficient condition is therefore simply that A is not injective, while the computable bounds are expressed via the operator norm of the pseudo-inverse on the orthogonal complement of the kernel (or, equivalently, distances in the quotient space X/ker(A)). These quantities are invariant under equivalent renormings of X. We will revise the abstract to state this invariance explicitly and add a short remark in Section 2 clarifying that the results hold for any norm on X that makes A continuous. revision: partial
Referee: [Theory (conditions derivation)] The necessary-and-sufficient conditions (stated in the abstract) appear to presuppose a fixed admissible-signal set without data-dependent assumptions. If the proofs rely on any particular choice of X that is not shown to be canonical or invariant, the conditions risk being non-unique and the computable bounds may change with that choice.

Authors: The admissible-signal set is exactly the fiber A^{-1}(y) determined by the forward model A and the observed measurement y; no external or data-independent set is assumed. The necessary-and-sufficient condition for hallucinations reduces to non-injectivity of A, and the magnitude bounds follow from the spectral properties of A alone. We will add a paragraph in the theory section demonstrating that the conditions and bounds remain unchanged under any continuous linear isomorphism of X that preserves the kernel and range of A, thereby establishing canonicity with respect to the forward operator. revision: partial

Circularity Check

0 steps flagged

No significant circularity; bounds and conditions derived from forward model alone

full rationale

The paper's central claims derive necessary and sufficient conditions for hallucinations plus magnitude bounds that depend only on the forward model, without reducing to self-referential definitions, fitted parameters renamed as predictions, or load-bearing self-citations. The abstract and described framework present these as arising directly from the ill-posed inverse problem structure, with no equations or steps shown that collapse by construction to inputs. This is consistent with a low circularity score; the derivation is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on standard domain assumptions from inverse problems theory; no free parameters, new entities, or ad-hoc axioms are mentioned in the abstract.

axioms (1)

domain assumption The inverse problem is ill-posed
This is invoked as the source of inevitable hallucinations.

pith-pipeline@v0.9.0 · 5465 in / 1040 out tokens · 40569 ms · 2026-05-14T18:10:48.341846+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 unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We derive necessary and sufficient conditions for hallucinations, together with computable bounds on their magnitude that depend only on the forward model.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Any decoder ϕ ... does not hallucinate by transferring details x_det with a magnitude that is larger than the worst-case kernel size.

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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