On Hallucinations in Inverse Problems: Fundamental Limits and Provable Assessment Methods
David Iagaru, Nina M. Gottschling, Anders C. Hansen, Josselin Garnier
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.
AI Impact Assessments
(1 model)Scientific Impact Assessment
1. Core Contribution
This paper provides a rigorous mathematical framework for understanding AI hallucinations in inverse problems, moving beyond empirical observations to provable theoretical guarantees. The core contributions are:
The key insight is that hallucinations are not merely artifacts of poor models but are fundamentally linked to the ill-posedness of the inverse problem itself—specifically, the size of feasible sets determines inevitable hallucination magnitudes.
2. Methodological Rigor
The theoretical development is sound and carefully structured. The progression from Definition 2.5 (formalizing hallucination as detail transfer) through necessary conditions (Proposition 3.1), necessary and sufficient conditions (Theorem 3.2), upper bounds (Proposition 3.5), and inevitability results (Theorem 3.6) is logical and well-motivated.
Strengths in rigor:
Potential concerns:
3. Potential Impact
Direct applications: The framework applies to medical imaging (MRI reconstruction), remote sensing (Sentinel-2 super-resolution), and potentially any imaging inverse problem. The decoder-agnostic method is particularly valuable because it can assess *any* reconstruction method's fundamental limitations without ground truth.
Broader implications:
Limitations on impact:
4. Timeliness & Relevance
This paper is highly timely. AI hallucinations are a central concern across fields, and the deployment of generative models (diffusion models, etc.) for inverse problems has exploded. The paper directly addresses the growing gap between AI capability and trustworthiness in safety-critical applications like medical imaging. The reference to the 2023 Cambridge Dictionary word of the year and the Financial Times headline underscores the societal relevance. The connection to the 2020 FastMRI challenge, where hallucinations were observed among top submissions, grounds the work in real practical concerns.
5. Strengths & Limitations
Key strengths:
Notable weaknesses:
6. Additional Observations
The paper builds substantially on [21, 22], extending their linear-model results to nonlinear settings and adding computational algorithms with convergence guarantees. The progression from sufficient conditions in [21] to necessary and sufficient conditions here represents genuine theoretical advancement. The code repository availability supports reproducibility, though the experimental scale is moderate.
Generated May 14, 2026
Comparison History (71)
Paper 2 has higher potential impact due to its broad, timely relevance to trustworthy AI in high-stakes imaging (medicine, remote sensing), and its field-spanning theoretical framing: hallucinations as inherent to ill-posed inverse problems, with necessary/sufficient conditions and computable, model-independent bounds tied to the forward operator. The proposed assessment algorithms enable practical deployment and benchmarking without ground truth, amplifying real-world applicability. Paper 1 is novel and rigorous for semi-supervised conditional diffusion, but its impact is more contained within generative modeling, whereas Paper 2’s insights and tools can influence inverse problems, uncertainty quantification, and AI safety across modalities.
Paper 1 is likely to have higher impact: it offers a fundamentally new, broadly applicable theory that attributes hallucinations to ill-posedness, with necessary/sufficient conditions and computable, model-agnostic bounds plus practical assessment algorithms usable without ground truth—highly timely for medical/remote-sensing imaging and generative reconstruction. The combination of foundational limits + actionable evaluation tools can influence multiple application domains and standards for reliability. Paper 2 is novel and rigorous for performative prediction, but its core impact is more concentrated in ML theory/deployment dynamics, whereas Paper 1 spans theory, diagnostics, and high-stakes real-world imaging.
Paper 2 has higher estimated impact due to strong novelty and breadth: it offers a unifying probabilistic theory connecting and explaining multiple high-profile sub-quadratic sequence architectures (linear attention, GLA, Mamba-2/SSD, DeltaNet), and introduces a principled Bayesian/covariance mechanism that yields testable predictions and robustness gains, including integration with large pretrained models. Its applications span essentially all long-context sequence modeling domains (NLP, time series, control), making it timely and widely relevant. Paper 1 is rigorous and important for imaging, but its reach is more domain-specific.
Paper 1 addresses AI hallucinations in inverse problems, a critically timely and high-stakes issue given AI's expanding role in medical imaging and Earth observation. Its theoretical framework with provable bounds and model-agnostic assessment tools directly tackles reliability concerns for generative models, offering broad, immediate practical impact across safety-critical domains. Paper 2's LatentFlow is elegant and general for conditioning stochastic processes, but its impact is more specialized within computational statistics/simulation. Paper 1's combination of fundamental theory, practical algorithms, and urgent relevance to trustworthy AI gives it wider reach and greater timeliness.
Paper 1 addresses a fundamental and urgent problem—hallucinations in AI-based inverse problems—that affects critical applications like medical imaging and remote sensing. It provides theoretical foundations (necessary and sufficient conditions, computable bounds) showing hallucinations are inherent to ill-posed problems, not just model artifacts. This reframes a widely discussed practical concern with rigorous theory and actionable algorithms. Its breadth of applicability across imaging tasks and relevance to the trustworthiness of AI systems gives it broader impact potential. Paper 2 makes solid theoretical and methodological contributions to deep GPs on DAGs, but addresses a more specialized audience.
Paper 2 likely has higher impact due to broader, timely relevance: hallucinations affect many high-stakes inverse problems (medical imaging, remote sensing) and the paper offers fundamental, model-agnostic limits plus practical assessment algorithms usable without ground truth. This combination of necessary/sufficient conditions, computable forward-model–only bounds, and evaluation tools can influence theory, benchmarking, and regulation across ML and imaging communities. Paper 1 is methodologically strong and novel for structured deep GP inference on DAGs, but its audience and applicability are comparatively narrower and more specialized.
Paper 2 has higher potential impact due to a more broadly applicable, novel framework: extending causal inference beyond i.i.d. via symmetry/geometric structure, tying together group theory, ergodic theory, geometric deep learning, and Bayesian inference. This can influence multiple fields (statistics, ML, causal inference, spatial/network science, genomics) and opens new model classes and estimators. Paper 1 is timely and rigorous with strong practical relevance for imaging reliability, but its impact is more domain-focused (inverse problems/imaging) and less cross-cutting than a general causal modeling paradigm.
Paper 1 offers higher scientific impact by establishing fundamental theoretical limits on AI hallucinations in inverse problems. While Paper 2 presents a strong algorithmic advance for PDEs, Paper 1 addresses a critical, widespread bottleneck in AI reliability. By proving that hallucinations stem from the ill-posed nature of the problems rather than just model flaws, and providing computable bounds, Paper 1 provides foundational knowledge. Its applications span high-stakes domains like medical imaging, giving it broader cross-disciplinary relevance and timeliness compared to Paper 2's more specialized functional regression framework.
Paper 1 likely has higher impact: it tackles a widely recognized, cross-cutting reliability barrier (hallucinations) with a fundamental, model-agnostic theory for inverse problems, providing necessary/sufficient conditions and computable bounds tied only to the forward operator. This combination of conceptual novelty, provable assessment methods, and broad applicability across medical/remote-sensing imaging and generative models is timely and could influence evaluation standards and regulation. Paper 2 is strong methodologically but is more specialized to function-space Bayesian sampling and competes within a crowded posterior-sampling landscape.
Paper 2 addresses the fundamental and widely-recognized problem of hallucinations in AI-based inverse problems, which has enormous implications for medical imaging, remote sensing, and other safety-critical applications. Its theoretical framework establishing that hallucinations are inherent to ill-posed problems (not just model artifacts) is a profound insight. The practical algorithms for quantifying hallucination magnitude are immediately applicable across many domains. Paper 1, while technically novel in bringing generative modeling to metric graphs, addresses a more niche mathematical setting with narrower applicability. Paper 2's breadth of impact, timeliness given AI reliability concerns, and cross-disciplinary relevance give it significantly higher potential impact.