When Rough Data Helps: A Phase Transition in Convergence Rates for Kernel Recovery in Integral Operators

Mathematics > Numerical Analysis [Submitted on 16 Jun 2026] Title:When Rough Data Helps: A Phase Transition in Convergence Rates for Kernel Recovery in Integral Operators View PDF HTML (experimental)Abstract:Learning kernels in operators from data is a fundamental task that arises in nonlocal continuum mechanics, operator learning, and interacting particle systems. A central question is how the roughness of input data impacts the accuracy of kernel recovery. We quantify the roughness of the input data via its spectral decay exponent and analyze how it determines the degree of ill-posedness of the inverse problem and, consequently, the convergence rates of the Tikhonov-regularized estimator in the small-noise limit. Within this framework, we identify a phase transition between an under-rough regime, in which rougher data improves recovery, and an over-rough regime, in which further roughening leads to slower rates. These theoretical findings are supported by numerical experiments ranging from idealized settings to more realistic configurations, with quantitative agreement in the former and broad consistency of the main trends in the latter. Current browse context: math.NA References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.