[{"tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"publication_identifier":{"isbn":["9798331339678"]},"citation":{"short":"L. Iurada, S. Bombari, T. Tommasi, M. Mondelli, in:, 14th International Conference on Learning Representations, OpenReview, 2026, pp. 145275–145314.","chicago":"Iurada, Leonardo, Simone Bombari, Tatiana Tommasi, and Marco Mondelli. “A Law of Data Reconstruction for Random Features (and Beyond).” In <i>14th International Conference on Learning Representations</i>, 2026:145275–314. OpenReview, 2026.","ista":"Iurada L, Bombari S, Tommasi T, Mondelli M. 2026. A law of data reconstruction for random features (and beyond). 14th International Conference on Learning Representations. ICLR: International Conference on Learning Representations  vol. 2026, 145275–145314.","mla":"Iurada, Leonardo, et al. “A Law of Data Reconstruction for Random Features (and Beyond).” <i>14th International Conference on Learning Representations</i>, vol. 2026, OpenReview, 2026, pp. 145275–314.","apa":"Iurada, L., Bombari, S., Tommasi, T., &#38; Mondelli, M. (2026). A law of data reconstruction for random features (and beyond). In <i>14th International Conference on Learning Representations</i> (Vol. 2026, pp. 145275–145314). Rio de Janeiro, Brazil: OpenReview.","ieee":"L. Iurada, S. Bombari, T. Tommasi, and M. Mondelli, “A law of data reconstruction for random features (and beyond),” in <i>14th International Conference on Learning Representations</i>, Rio de Janeiro, Brazil, 2026, vol. 2026, pp. 145275–145314.","ama":"Iurada L, Bombari S, Tommasi T, Mondelli M. A law of data reconstruction for random features (and beyond). In: <i>14th International Conference on Learning Representations</i>. Vol 2026. OpenReview; 2026:145275-145314."},"oa":1,"OA_place":"publisher","OA_type":"gold","day":"26","file":[{"access_level":"open_access","success":1,"content_type":"application/pdf","creator":"cchlebak","checksum":"a3dcba6649b7496124fa0ae68bf2c4cb","file_size":7869510,"file_name":"2026_ICLR_Iurada.pdf","file_id":"22907","relation":"main_file","date_created":"2026-09-11T12:07:48Z","date_updated":"2026-09-11T12:07:48Z"}],"file_date_updated":"2026-09-11T12:07:48Z","month":"01","conference":{"location":"Rio de Janeiro, Brazil","end_date":"2026-04-27","start_date":"2026-04-23","name":"ICLR: International Conference on Learning Representations "},"user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","corr_author":"1","arxiv":1,"oa_version":"Published Version","ddc":["000"],"date_updated":"2026-09-11T12:26:43Z","date_created":"2026-09-09T13:31:46Z","department":[{"_id":"GradSch"},{"_id":"MaMo"}],"type":"conference","related_material":{"record":[{"id":"22857","status":"for_moderation","relation":"dissertation_contains"}]},"project":[{"_id":"911e6d1f-16d5-11f0-9cad-c5c68c6a1cdf","grant_number":"101161364","name":"Inference in High Dimensions: Light-speed Algorithms and Information Limits"},{"name":"Trustworthy Deep Learning Theory: Private Over-Parameterized Models and Robust LLMs","_id":"92099302-16d5-11f0-9cad-f9a785f54fbd"}],"page":"145275-145314","external_id":{"arxiv":["2509.22214"]},"quality_controlled":"1","das_tickbox":"0","_id":"22894","publisher":"OpenReview","article_processing_charge":"No","publication":"14th International Conference on Learning Representations","volume":2026,"acknowledgement":"M.M. is funded by the European Union (ERC, INF2\r\n, project number 101161364). S.B. was supported\r\nby a Google PhD fellowship. L.I. acknowledges the grant received from the European Union NextGenerationEU (Piano Nazionale di Ripresa E Resilienza (PNRR)) DM 351 on Trustworthy AI. T.T. &\r\nL.I. acknowledge the EU project ELSA - European Lighthouse on Secure and Safe AI. This study was\r\ncarried out within the FAIR - Future Artificial Intelligence Research and received funding from the\r\nEuropean Union Next-GenerationEU (PIANO NAZIONALE DI RIPRESA E RESILIENZA (PNRR)\r\n– MISSIONE 4 COMPONENTE 2, INVESTIMENTO 1.3 – D.D. 1555 11/10/2022, PE00000013).\r\nThis manuscript reflects only the authors’ views and opinions, neither the European Union nor the\r\nEuropean Commission can be considered responsible for them. The authors would like to thank\r\nYizhe Zhu for helpful discussions.","intvolume":"      2026","language":[{"iso":"eng"}],"status":"public","abstract":[{"text":"Large-scale deep learning models are known to memorize parts of the training\r\nset. In machine learning theory, memorization is often framed as interpolation or\r\nlabel fitting, and classical results show that this can be achieved when the number\r\nof parameters p in the model is larger than the number of training samples n. In\r\nthis work, we consider memorization from the perspective of data reconstruction,\r\ndemonstrating that this can be achieved when p is larger than dn, where d is\r\nthe dimensionality of the data. More specifically, we show that, in the random\r\nfeatures model, when p ≫ dn, the subspace spanned by the training samples in\r\nfeature space gives sufficient information to identify the individual samples in input\r\nspace. Our analysis suggests an optimization method to reconstruct the dataset\r\nfrom the model parameters, and we demonstrate that this method performs well on\r\nvarious architectures (random features, two-layer fully-connected and deep residual\r\nnetworks). Our results reveal a law of data reconstruction, according to which the\r\nentire training dataset can be recovered as p exceeds the threshold dn.\r\n","lang":"eng"}],"publication_status":"published","year":"2026","has_accepted_license":"1","title":"A law of data reconstruction for random features (and beyond)","author":[{"full_name":"Iurada, Leonardo","first_name":"Leonardo","last_name":"Iurada"},{"id":"ca726dda-de17-11ea-bc14-f9da834f63aa","full_name":"Bombari, Simone","first_name":"Simone","last_name":"Bombari"},{"full_name":"Tommasi, Tatiana","first_name":"Tatiana","last_name":"Tommasi"},{"full_name":"Mondelli, Marco","id":"27EB676C-8706-11E9-9510-7717E6697425","last_name":"Mondelli","first_name":"Marco","orcid":"0000-0002-3242-7020"}],"date_published":"2026-01-26T00:00:00Z"}]