{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T14:35:00Z","timestamp":1787236500188,"version":"build-2736575974"},"reference-count":76,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","funder":[{"name":"Royal Society University Research Fellowship"},{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher","award":["R611675"],"award-info":[{"award-number":["R611675"]}],"id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000275","name":"Leverhulme Trust","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100000275","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100009059","name":"Pacific Institute for the Mathematical Sciences","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100009059","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004326","name":"Simon Fraser University","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100004326","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM Rev."],"published-print":{"date-parts":[[2025,3,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>Methods inspired by artificial intelligence (AI) are starting to fundamentally change computational science and engineering through breakthrough performance on challenging problems. However, the reliability and trustworthiness of such techniques is a major concern. In inverse problems in imaging, the focus of this paper, there is increasing empirical evidence that methods may suffer from hallucinations, i.e., false, but realistic-looking artifacts; instability, i.e., sensitivity to perturbations in the data; and unpredictable generalization, i.e., excellent performance on some images, but significant deterioration on others. This paper provides a theoretical foundation for these phenomena.\u00a0We give mathematical explanations for how and when such effects arise in arbitrary reconstruction methods, with several of our results taking the form of \u201cno free lunch\u201d theorems.\u00a0Specifically, we show that (i) methods that overperform on a single image can wrongly transfer details from one image to another, creating a hallucination; (ii) methods that overperform on two or more images can hallucinate or be unstable; (iii) optimizing the accuracy-stability tradeoff is generally difficult; (iv) hallucinations and instabilities, if they occur, are not rare events and may be encouraged by standard training; and (v) it may be impossible to construct optimal reconstruction maps for certain problems.\u00a0Our results trace these effects to the kernel of the forward operator whenever it is nontrivial, but also apply to the case when the forward operator is ill-conditioned. Based on these insights, our work aims to spur research into new ways to develop robust and reliable AI-based methods for inverse problems in imaging.<\/jats:p>","DOI":"10.1137\/23m1568739","type":"journal-article","created":{"date-parts":[[2025,2,6]],"date-time":"2025-02-06T13:00:56Z","timestamp":1738846856000},"page":"73-104","source":"Crossref","is-referenced-by-count":21,"title":["The Troublesome Kernel: On Hallucinations, No Free Lunches, and the Accuracy-Stability Tradeoff in Inverse Problems"],"prefix":"10.1137","volume":"67","author":[{"ORCID":"https:\/\/orcid.org\/0009-0004-0275-7522","authenticated-orcid":true,"given":"Nina M.","family":"Gottschling","sequence":"first","affiliation":[{"name":"University of Cambridge, Cambridge CB3 0WA, UK."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4045-9775","authenticated-orcid":true,"given":"Vegard","family":"Antun","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Oslo, Blindern, 0316 Oslo, Norway."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Anders C.","family":"Hansen","sequence":"additional","affiliation":[{"name":"University of Cambridge, Cambridge CB3 0WA, UK."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ben","family":"Adcock","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6, Canada."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2025,2,6]]},"reference":[{"key":"ref1","unstructured":"B. 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