{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/the-troublesome-kernel-why-deep-learning-for","title":"The troublesome kernel -- On hallucinations, no free lunches and the accuracy-stability trade-off in inverse problems","arxiv_id":"2001.01258","date":"2020-01-05","proceeding":null,"authors":["Nina M. Gottschling","Vegard Antun","Anders C. Hansen","Ben Adcock"],"abstract":"Methods inspired by Artificial Intelligence (AI) are starting to fundamentally change computational science and engineering through breakthrough performances on challenging problems. However, 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. We give mathematical explanations for how and when such effects arise in arbitrary reconstruction methods, with several of our results taking the form of `no free lunch' theorems. Specifically, 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 trade-off is generally difficult, (iv) hallucinations and instabilities, if they occur, are not rare events, and may be encouraged by standard training, (v) it may be impossible to construct optimal reconstruction maps for certain problems. Our 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.","url_abs":"https://arxiv.org/abs/2001.01258v4","url_pdf":"https://arxiv.org/pdf/2001.01258v4.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"the-troublesome-kernel-why-deep-learning-for","repo_url":"https://github.com/vegarant/troublesome_kernel","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"pytorch","reach":null}],"tasks":[{"task_slug":"hallucination","task_name":"Hallucination"},{"task_slug":"image-classification","task_name":"Image Classification"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2001.01258","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2001.01258"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-24T18:15:14+00:00","read_at_is":"when the build read Syntology's graph, not when any sample ran","claim":"Per-sample execution status on synthesized fixtures; not a correctness claim about the paper. Samples come from repositories linked to the paper, official or community; repo_kind says which.","repos":[{"provenance":"external:paperswithcode_snapshot_2025-07-28","url":"https://github.com/vegarant/troublesome_kernel","reach":null}],"summary":{"ran_honours":1},"by_repo_kind":{"official":{"samples":1,"ran":1,"repositories":1}},"repo_kind_vocabulary":{"official":"The archive marks this repository official for the paper","named_in_paper":"The archive records that the paper mentions this repository; it is not marked official","listed":"In the archive's code links for this paper, not marked official and not recorded as mentioned in the paper","found_in_text":"Syntology found this repository in the paper's own text; whether it is the authors' implementation is not asserted","community":"Not in the archive's code links for this paper; a community repository Syntology harvested"},"n_pointer_only_for_licence":1,"samples":[{"code_sha256_prefix":"bd9b4f62ae1d8205","entry":"loss_func","repo":"vegarant/troublesome_kernel","repo_kind":"official","path":"ellipses/script_train_fourier_unet_it_jit-nojit.py","file_url":"https://github.com/vegarant/troublesome_kernel/blob/HEAD/ellipses/script_train_fourier_unet_it_jit-nojit.py","link_basis":"first_harvest_node","language":"python","status":"ran_honours","verification_level":1,"contract_check":"HONOURS","metamorphic_tier":"invariant","behaviour_fingerprint":true,"licence":"NONE","inline_ok":false,"mcp_get_code":{"code_sha256":"bd9b4f62ae1d8205"}}]},"arxiv_metadata":null,"syntology_extracted_results":null}