[{"scopus_import":"1","_id":"21328","type":"conference","acknowledgement":"This work was done when Y. Z. was at the Institute of Science and Technology Austria. Y. Z. and\r\nM. M. are funded by the European Union (ERC, INF2, project number 101161364). Views and\r\nopinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them. The authors would like to acknowledge (in alphabetical order) discussions with Yatin Dandi, Leonardo Defilippis and Bruno Loureiro concerning their parallel work (Defilippis et al., 2025).","oa_version":"Published Version","intvolume":"       291","date_updated":"2026-02-19T09:03:53Z","publisher":"ML Research Press","arxiv":1,"department":[{"_id":"MaMo"}],"date_published":"2025-07-01T00:00:00Z","month":"07","language":[{"iso":"eng"}],"file_date_updated":"2026-02-19T09:03:43Z","oa":1,"day":"01","status":"public","alternative_title":["PMLR"],"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"OA_type":"gold","project":[{"_id":"911e6d1f-16d5-11f0-9cad-c5c68c6a1cdf","grant_number":"101161364","name":"Inference in High Dimensions: Light-speed Algorithms and Information Limits"}],"citation":{"chicago":"Kovačević, Filip, Zhang Yihan, and Marco Mondelli. “Spectral Estimators for Multi-Index Models: Precise Asymptotics and Optimal Weak Recovery.” In <i>Proceedings of 38th Conference on Learning Theory</i>, 291:3354–3404. ML Research Press, 2025.","apa":"Kovačević, F., Yihan, Z., &#38; Mondelli, M. (2025). Spectral estimators for multi-index models: Precise asymptotics and optimal weak recovery. In <i>Proceedings of 38th Conference on Learning Theory</i> (Vol. 291, pp. 3354–3404). Lyon, France: ML Research Press.","mla":"Kovačević, Filip, et al. “Spectral Estimators for Multi-Index Models: Precise Asymptotics and Optimal Weak Recovery.” <i>Proceedings of 38th Conference on Learning Theory</i>, vol. 291, ML Research Press, 2025, pp. 3354–404.","short":"F. Kovačević, Z. Yihan, M. Mondelli, in:, Proceedings of 38th Conference on Learning Theory, ML Research Press, 2025, pp. 3354–3404.","ieee":"F. Kovačević, Z. Yihan, and M. Mondelli, “Spectral estimators for multi-index models: Precise asymptotics and optimal weak recovery,” in <i>Proceedings of 38th Conference on Learning Theory</i>, Lyon, France, 2025, vol. 291, pp. 3354–3404.","ista":"Kovačević F, Yihan Z, Mondelli M. 2025. Spectral estimators for multi-index models: Precise asymptotics and optimal weak recovery. Proceedings of 38th Conference on Learning Theory. COLT: Conference on Learning Theory, PMLR, vol. 291, 3354–3404.","ama":"Kovačević F, Yihan Z, Mondelli M. Spectral estimators for multi-index models: Precise asymptotics and optimal weak recovery. In: <i>Proceedings of 38th Conference on Learning Theory</i>. Vol 291. ML Research Press; 2025:3354-3404."},"volume":291,"OA_place":"publisher","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","author":[{"last_name":"Kovačević","first_name":"Filip","id":"d0258e7b-50b8-11ef-ad56-8b9f537b6b1b","full_name":"Kovačević, Filip"},{"first_name":"Zhang","last_name":"Yihan","full_name":"Yihan, Zhang"},{"orcid":"0000-0002-3242-7020","id":"27EB676C-8706-11E9-9510-7717E6697425","full_name":"Mondelli, Marco","first_name":"Marco","last_name":"Mondelli"}],"title":"Spectral estimators for multi-index models: Precise asymptotics and optimal weak recovery","publication_status":"published","external_id":{"arxiv":["2502.01583"]},"abstract":[{"text":"Multi-index models provide a popular framework to investigate the learnability of functions with low-dimensional structure and, also due to their connections with neural networks, they have been object of recent intensive study. In this paper, we focus on recovering the subspace spanned by the signals via spectral estimators – a family of methods routinely used in practice, often as a warm-start for iterative algorithms. Our main technical contribution is a precise asymptotic characterization of the performance of spectral methods, when sample size and input dimension grow proportionally and the dimension p of the space to recover is fixed. Specifically, we locate the top-p eigenvalues of the spectral matrix and establish the overlaps between the corresponding eigenvectors (which give the spectral estimators) and a basis of the signal subspace. Our analysis unveils a phase transition phenomenon in which, as the sample complexity grows, eigenvalues escape from the bulk of the spectrum and, when that happens, eigenvectors recover directions of the desired subspace. The precise characterization we put forward enables the optimization of the data preprocessing, thus allowing to identify the spectral estimator that requires the minimal sample size for weak recovery.","lang":"eng"}],"article_processing_charge":"No","page":"3354-3404","publication":"Proceedings of 38th Conference on Learning Theory","publication_identifier":{"eissn":["2640-3498"]},"date_created":"2026-02-18T12:12:47Z","year":"2025","conference":{"start_date":"2025-06-30","name":"COLT: Conference on Learning Theory","location":"Lyon, France","end_date":"2025-07-04"},"ddc":["000"],"has_accepted_license":"1","file":[{"date_created":"2026-02-19T09:03:43Z","content_type":"application/pdf","file_name":"2025_LearningTheory_Kovacevic.pdf","creator":"dernst","success":1,"file_id":"21339","relation":"main_file","checksum":"19aa70ab4f57fb9067b6ebb99a5fd6f0","access_level":"open_access","date_updated":"2026-02-19T09:03:43Z","file_size":844611}],"quality_controlled":"1","corr_author":"1"},{"acknowledgement":"We thank Junhyung Park for valuable feedback on the manuscript. AT was supported by a PhD fellowship from the Swiss Data Science Center. TW was supported by the SNF Grant 204439. This work was done in part while TW and FY were visiting the Simons Institute for the Theory of\r\nComputing.","type":"conference","oa_version":"Preprint","intvolume":"       258","_id":"20300","scopus_import":"1","oa":1,"department":[{"_id":"MaMo"}],"arxiv":1,"publisher":"ML Research Press","date_updated":"2025-09-09T07:00:34Z","language":[{"iso":"eng"}],"month":"05","date_published":"2025-05-01T00:00:00Z","OA_type":"green","status":"public","day":"01","alternative_title":["PMLR"],"citation":{"chicago":"Wegel, Tobias, Filip Kovačević, Alexandru Ţifrea, and Fanny Yang. “Learning Pareto Manifolds in High Dimensions: How Can Regularization Help?” In <i>The 28th International Conference on Artificial Intelligence and Statistics</i>, 258:4591–99. ML Research Press, 2025.","apa":"Wegel, T., Kovačević, F., Ţifrea, A., &#38; Yang, F. (2025). Learning Pareto manifolds in high dimensions: How can regularization help? In <i>The 28th International Conference on Artificial Intelligence and Statistics</i> (Vol. 258, pp. 4591–4599). Mai Khao, Thailand: ML Research Press.","mla":"Wegel, Tobias, et al. “Learning Pareto Manifolds in High Dimensions: How Can Regularization Help?” <i>The 28th International Conference on Artificial Intelligence and Statistics</i>, vol. 258, ML Research Press, 2025, pp. 4591–99.","short":"T. Wegel, F. Kovačević, A. Ţifrea, F. Yang, in:, The 28th International Conference on Artificial Intelligence and Statistics, ML Research Press, 2025, pp. 4591–4599.","ieee":"T. Wegel, F. Kovačević, A. Ţifrea, and F. Yang, “Learning Pareto manifolds in high dimensions: How can regularization help?,” in <i>The 28th International Conference on Artificial Intelligence and Statistics</i>, Mai Khao, Thailand, 2025, vol. 258, pp. 4591–4599.","ama":"Wegel T, Kovačević F, Ţifrea A, Yang F. Learning Pareto manifolds in high dimensions: How can regularization help? In: <i>The 28th International Conference on Artificial Intelligence and Statistics</i>. Vol 258. ML Research Press; 2025:4591-4599.","ista":"Wegel T, Kovačević F, Ţifrea A, Yang F. 2025. Learning Pareto manifolds in high dimensions: How can regularization help? The 28th International Conference on Artificial Intelligence and Statistics. AISTATS: Conference on Artificial Intelligence and Statistics, PMLR, vol. 258, 4591–4599."},"OA_place":"repository","volume":258,"author":[{"last_name":"Wegel","first_name":"Tobias","full_name":"Wegel, Tobias"},{"first_name":"Filip","last_name":"Kovačević","id":"d0258e7b-50b8-11ef-ad56-8b9f537b6b1b","full_name":"Kovačević, Filip"},{"first_name":"Alexandru","last_name":"Ţifrea","full_name":"Ţifrea, Alexandru"},{"first_name":"Fanny","last_name":"Yang","full_name":"Yang, Fanny"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","abstract":[{"lang":"eng","text":"Simultaneously addressing multiple objectives is becoming increasingly important in modern machine learning. At the same time, data is often high-dimensional and costly to label. For a single objective such as prediction risk, conventional regularization techniques are known to improve generalization when the data exhibits low-dimensional structure like sparsity. However, it is largely unexplored how to leverage this structure in the context of multi-objective learning (MOL) with multiple competing objectives. In this work, we discuss how the application of vanilla regularization approaches can fail, and propose a two-stage MOL framework that can successfully leverage low-dimensional structure. We demonstrate its effectiveness experimentally for multi-distribution learning and fairness-risk trade-offs."}],"external_id":{"arxiv":["2503.08849"]},"publication_status":"published","page":"4591-4599","article_processing_charge":"No","title":"Learning Pareto manifolds in high dimensions: How can regularization help?","year":"2025","date_created":"2025-09-07T22:01:35Z","publication":"The 28th International Conference on Artificial Intelligence and Statistics","publication_identifier":{"eissn":["2640-3498"]},"quality_controlled":"1","conference":{"start_date":"2025-05-03","name":"AISTATS: Conference on Artificial Intelligence and Statistics","location":"Mai Khao, Thailand","end_date":"2025-05-05"},"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2503.08849","open_access":"1"}]}]