{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,27]],"date-time":"2026-04-27T21:40:30Z","timestamp":1777326030627,"version":"3.51.4"},"reference-count":96,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2025,1,27]],"date-time":"2025-01-27T00:00:00Z","timestamp":1737936000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,1,27]],"date-time":"2025-01-27T00:00:00Z","timestamp":1737936000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Found Comput Math"],"published-print":{"date-parts":[[2026,4]]},"DOI":"10.1007\/s10208-025-09693-y","type":"journal-article","created":{"date-parts":[[2025,1,27]],"date-time":"2025-01-27T15:13:51Z","timestamp":1737990831000},"page":"889-938","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Optimal Regularization for a Data Source"],"prefix":"10.1007","volume":"26","author":[{"given":"Oscar","family":"Leong","sequence":"first","affiliation":[]},{"given":"Eliza","family":"O\u2019 Reilly","sequence":"additional","affiliation":[]},{"given":"Yong Sheng","family":"Soh","sequence":"additional","affiliation":[]},{"given":"Venkat","family":"Chandrasekaran","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,1,27]]},"reference":[{"issue":"4","key":"9693_CR1","doi-asserted-by":"publisher","first-page":"2775","DOI":"10.1137\/140979861","volume":"26","author":"A Agarwal","year":"2016","unstructured":"A. Agarwal, A. Anandkumar, P. Jain, and P. Netrapalli. Learning sparsely used overcomplete dictionaries via alternating minimization. SIAM Journal on Optimization 26(4) (2016), pp. 2775\u20132799.","journal-title":"SIAM Journal on Optimization"},{"issue":"1","key":"9693_CR2","doi-asserted-by":"publisher","first-page":"575","DOI":"10.1109\/TIT.2016.2614684","volume":"63","author":"A Agarwal","year":"2017","unstructured":"A. Agarwal, A. Anandkumar, and P. Netrapalli. A clustering approach to learn sparsely-used overcomplete dictionaries. IEEE Transactions on Information Theory 63(1) (2017), pp. 575\u2013592.","journal-title":"IEEE Transactions on Information Theory"},{"issue":"11","key":"9693_CR3","doi-asserted-by":"publisher","first-page":"4311","DOI":"10.1109\/TSP.2006.881199","volume":"54","author":"M Aharon","year":"2006","unstructured":"M. Aharon, M. Elad, and A. Bruckstein. K-SVD: an algorithm for designing overcomplete dictionaries for sparse representation. IEEE Transactions on Signal Processing 54(11) (2006), pp. 4311\u20134322.","journal-title":"IEEE Transactions on Signal Processing"},{"key":"9693_CR4","unstructured":"S. Amari and H. Nagaoka. Methods of Information Geometry. Translations of mathematical monographs. American Mathematical Society, 2000. ISBN: 9780821843024. URL: https:\/\/books.google.com\/books?id=vc2FWSo7wLUC."},{"issue":"3","key":"9693_CR5","doi-asserted-by":"publisher","first-page":"224","DOI":"10.1093\/imaiai\/iau005","volume":"3","author":"D Amelunxen","year":"2014","unstructured":"D. Amelunxen, M. Lotz, M. B. McCoy, and J. A. Tropp. \u201cLiving on the edge: Phase transitions in convex programs with random data\u201d. Information and Inference: A Journal of the IMA 224\u2013294 (2014), 3(3).","journal-title":"Information and Inference: A Journal of the IMA"},{"key":"9693_CR6","unstructured":"S. Arora, R. Ge, T. Ma, and A. Moitra. Simple, Efficient, and Neural Algorithms for Sparse Coding. Conference on Learning Theory (2015)."},{"key":"9693_CR7","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1017\/S0962492919000059","volume":"28","author":"S Arridge","year":"2019","unstructured":"S. Arridge, P. Maass, O. \u00d6ktem, and C.-B. Sch\u00f6nlieb. \u201cSolving inverse problems using data-driven models\u201d. Acta Numerica 28 (2019), pp. 1\u2013174.","journal-title":"Acta Numerica"},{"key":"9693_CR8","doi-asserted-by":"crossref","unstructured":"B. Barak, J. A. Kelner, and D. Steurer. Dictionary Learning and Tensor Decomposition via the Sum-of-Squares Method. Proceedings of the Forty-seventh Annual ACM Symposium on Theory of Computing (2015), pp. 143\u2013151.","DOI":"10.1145\/2746539.2746605"},{"key":"9693_CR9","doi-asserted-by":"publisher","first-page":"1497","DOI":"10.1214\/009053605000000282","volume":"33","author":"PL Bartlett","year":"2005","unstructured":"P. L. Bartlett, O. Bousquet, and S. Mendelson. \u201cLocal Rademacher Complexities\u201d. Annals of Statistics 33 (2005), pp. 1497\u20131537.","journal-title":"Annals of Statistics"},{"key":"9693_CR10","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1017\/S0962492918000016","volume":"27","author":"M Benning","year":"2018","unstructured":"M. Benning and M. Burger. \u201cModern regularization methods for inverse problems\u201d. Acta Numerica 27 (2018), pp. 1\u2013111.","journal-title":"Acta Numerica"},{"issue":"23","key":"9693_CR11","doi-asserted-by":"publisher","first-page":"5987","DOI":"10.1109\/TSP.2013.2273443","volume":"61","author":"B Bhaskar","year":"2013","unstructured":"B. Bhaskar and B. Recht. Atomic norm denoising with applications to line spectral estimation. IEEE Transactions on Signal Processing 61(23) (2013), pp. 5987\u20135999.","journal-title":"IEEE Transactions on Signal Processing"},{"key":"9693_CR12","unstructured":"A. Bora, A. Jalal, E. Price, and A. Dimakis. Compressed sensing using generative models. International Conference on Machine Learning (2017)."},{"key":"9693_CR13","doi-asserted-by":"crossref","unstructured":"A. Braides. A Handbook of $$\\Gamma $$-Convergence. Handbook of Differential Equations: Stationary Partial Differential Equations Volume 3 (2007).","DOI":"10.1016\/S1874-5733(06)80006-9"},{"key":"9693_CR14","doi-asserted-by":"publisher","first-page":"492","DOI":"10.1137\/090769521","volume":"3","author":"K Bredies","year":"2011","unstructured":"K. Bredies, K. Kunisch, and T. Pock. \u201cTotal generalized variation\u201d. SIAM Journal on Imaging Sciences 3 (2011), pp. 492\u2013526.","journal-title":"SIAM Journal on Imaging Sciences"},{"key":"9693_CR15","doi-asserted-by":"publisher","first-page":"757","DOI":"10.1007\/s00454-018-9992-1","volume":"59","author":"S Bubeck","year":"2018","unstructured":"S. Bubeck, R. Eldan, and J. Lehec. Sampling from a log-concave distribution with projected Langevin Monte Carlo. Discrete and Computational Geometry 59 (2018), pp. 757\u2013783.","journal-title":"Discrete and Computational Geometry"},{"issue":"6","key":"9693_CR16","doi-asserted-by":"publisher","first-page":"717","DOI":"10.1007\/s10208-009-9045-5","volume":"9","author":"EJ Cand\u00e8s","year":"2009","unstructured":"E. J. Cand\u00e8s and B. Recht. Exact matrix completion via convex optimization. Foundations of Computational Mathematics 9(6) (2009), pp. 717\u2013772.","journal-title":"Foundations of Computational Mathematics"},{"issue":"8","key":"9693_CR17","doi-asserted-by":"publisher","first-page":"1207","DOI":"10.1002\/cpa.20124","volume":"59","author":"EJ Cand\u00e8s","year":"2006","unstructured":"E. J. Cand\u00e8s, J. K. Romberg, and T. Tao. \u201cStable signal recovery from incomplete and inaccurate measurements\u201d. Communications on Pure and Applied Mathematics 59(8) (2006), pp. 1207\u20131223.","journal-title":"Communications on Pure and Applied Mathematics"},{"key":"9693_CR18","doi-asserted-by":"publisher","first-page":"167","DOI":"10.1007\/s002110050258","volume":"76","author":"A Chambolle","year":"1997","unstructured":"A. Chambolle and P.-L. Lions. Image recovery via total variation minimization and related problems. Numerische Mathematik 76 (1997), pp. 167\u2013188.","journal-title":"Numerische Mathematik"},{"issue":"13","key":"9693_CR19","doi-asserted-by":"publisher","first-page":"E1181","DOI":"10.1073\/pnas.1302293110","volume":"110","author":"V Chandrasekaran","year":"2013","unstructured":"V. Chandrasekaran and M. I. Jordan. Computational and statistical tradeoffs via convex relaxation. Proceedings of the National Academy of Sciences 110(13) (2013), E1181\u2013E1190.","journal-title":"Proceedings of the National Academy of Sciences"},{"key":"9693_CR20","doi-asserted-by":"publisher","first-page":"805","DOI":"10.1007\/s10208-012-9135-7","volume":"12","author":"V Chandrasekaran","year":"2012","unstructured":"V. Chandrasekaran, B. Recht, P. A. Parrilo, and A. S. Willsky. The convex geometry of linear inverse problems. Foundations of Computational Mathematics 12 (2012), pp. 805\u2013849.","journal-title":"Foundations of Computational Mathematics"},{"key":"9693_CR21","unstructured":"N. S. Chatterji and P. L. Bartlett. Alternating minimization for dictionary learning: Local Convergence Guarantees. Advances in Neural Information Processing Systems (NeurIPS) (2017)."},{"issue":"6","key":"9693_CR22","doi-asserted-by":"publisher","first-page":"2524","DOI":"10.1137\/100814664","volume":"49","author":"M Dashti","year":"2011","unstructured":"M. Dashti and A. M. Stuart. Uncertainty quantification and weak approximation of an elliptic inverse problem. SIAM Journal on Numerical Analysis 49(6) (2011), pp. 2524\u20132542.","journal-title":"SIAM Journal on Numerical Analysis"},{"issue":"11","key":"9693_CR23","doi-asserted-by":"publisher","first-page":"1413","DOI":"10.1002\/cpa.20042","volume":"57","author":"I Daubechies","year":"2004","unstructured":"I. Daubechies, M. Defrise, and C. de Mol. An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Communications on Pure and Applied Mathematics 57.11 (2004), pp. 1413\u20131457.","journal-title":"Communications on Pure and Applied Mathematics"},{"issue":"6","key":"9693_CR24","doi-asserted-by":"publisher","first-page":"797","DOI":"10.1002\/cpa.20132","volume":"59","author":"D Donoho","year":"2006","unstructured":"D. Donoho. For most large underdetermined systems of linear equations the minimal l1-norm solution is also the sparsest solution. Communications on Pure and Applied Mathematics 59(6) (2006), pp. 797\u2013829.","journal-title":"Communications on Pure and Applied Mathematics"},{"key":"9693_CR25","first-page":"1","volume":"20","author":"A Durmus","year":"2019","unstructured":"A. Durmus, S. Majewski, and B. Miasojedow. Analysis of Langevin Monte Carlo via Convex Optimization. Journal of Machine Learning Research 20 (2019), pp. 1\u201346.","journal-title":"Journal of Machine Learning Research"},{"key":"9693_CR26","first-page":"42","volume":"20","author":"R Dwivedi","year":"2019","unstructured":"R. Dwivedi, Y. Chen, M. J. Wainwright, and B. Yu. \u201cLog-concave sampling: Metropolis-Hastings algorithms are fast\u201d. Journal of Machine Learning Research 20 (2019), p. 42.","journal-title":"Journal of Machine Learning Research"},{"issue":"6","key":"9693_CR27","doi-asserted-by":"publisher","first-page":"1557","DOI":"10.1137\/0524088","volume":"24","author":"PPB Eggermont","year":"1993","unstructured":"P. P. B. Eggermont. \u201cMaximum entropy regularization of Fredholm integral equations of the first kind\u201d. SIAM Journal on Mathematical Analysis 24(6) (1993), pp. 1557\u20131576.","journal-title":"SIAM Journal on Mathematical Analysis"},{"key":"9693_CR28","doi-asserted-by":"crossref","unstructured":"M. Elad. Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing. Springer (2010).","DOI":"10.1007\/978-1-4419-7011-4"},{"key":"9693_CR29","doi-asserted-by":"publisher","first-page":"1348","DOI":"10.1198\/016214501753382273","volume":"96","author":"J Fan","year":"2001","unstructured":"J. Fan and R. Li. Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association 96 (2001), pp. 1348\u20131360.","journal-title":"Journal of the American Statistical Association"},{"key":"9693_CR30","unstructured":"M. Fazel. Matrix Rank Minimization with Applications. Ph.D. Thesis, Department of Electrical Engineering, Stanford University (2002)."},{"key":"9693_CR31","doi-asserted-by":"crossref","unstructured":"S. Foucart and M.-J. Lai. Sparsest Solutions of Underdetermined Linear Systems via $$\\ell _q$$ -minimization for $$0 < q \\leqslant 1$$. Applied and Computational Harmonic Analysis 26 (2009), pp. 395\u2013407.","DOI":"10.1016\/j.acha.2008.09.001"},{"issue":"3","key":"9693_CR32","doi-asserted-by":"publisher","first-page":"366","DOI":"10.1109\/TCI.2018.2840334","volume":"4","author":"C Garcia-Cardona","year":"2018","unstructured":"C. Garcia-Cardona and B. Wohlberg. \u201cConvolutional dictionary learning: A comparative review and new algorithms\u201d. IEEE Transactions on Computational Imaging 4(3) (2018), pp. 366\u2013381.","journal-title":"IEEE Transactions on Computational Imaging"},{"key":"9693_CR33","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781107341029","volume-title":"Geometric Tomography","author":"RJ Gardner","year":"2006","unstructured":"R. J. Gardner. \u201cGeometric Tomography\u201d. Cambridge: Cambridge University Press (2006)."},{"issue":"2","key":"9693_CR34","doi-asserted-by":"publisher","first-page":"163","DOI":"10.1214\/ss\/1028905934","volume":"13","author":"A Gelman","year":"1998","unstructured":"A. Gelman and X.-L. Meng. \u201cSimulating normalizing constants: from importance sampling to bridge sampling to path sampling\u201d. Statistical Science 13.2 (1998), pp. 163\u2013185.","journal-title":"Statistical Science"},{"issue":"8","key":"9693_CR35","doi-asserted-by":"publisher","first-page":"085001","DOI":"10.1088\/1361-6420\/ab7d2a","volume":"36","author":"M Giordano","year":"2020","unstructured":"M. Giordano and R. Nickl. \u201cConsistency of Bayesian inference with Gaussian process priors in an elliptic inverse problem\u201d. Inverse Problems 36(8) (2020), p. 085001.","journal-title":"Inverse Problems"},{"issue":"6","key":"9693_CR36","doi-asserted-by":"publisher","first-page":"3469","DOI":"10.1109\/TIT.2015.2424238","volume":"61","author":"R Gribonval","year":"2015","unstructured":"R. Gribonval, R. Jenatton, F. Bach, M. Kleinsteuber, and M. Seibert. Sample complexity of dictionary learning and other matrix factorizations. IEEE Transactions on Information Theory 61(6) (2015), pp. 3469\u20133486.","journal-title":"IEEE Transactions on Information Theory"},{"issue":"1","key":"9693_CR37","doi-asserted-by":"publisher","first-page":"385","DOI":"10.1214\/11-AOS959","volume":"40","author":"A Guntuboyina","year":"2012","unstructured":"A. Guntuboyina. \u201cOptimal rates of convergence for convex set estimation from support functions\u201d. Annals of Statistics 40(1) (2012), pp. 385\u2013411.","journal-title":"Annals of Statistics"},{"issue":"1","key":"9693_CR38","doi-asserted-by":"publisher","first-page":"306","DOI":"10.1137\/21M1414978","volume":"4","author":"A Habring","year":"2022","unstructured":"A. Habring and M. Holler. A generative variational model for inverse problems in imaging. SIAM Journal on Mathematics of Data Science 4(1) (2022), pp. 306\u2013335.","journal-title":"SIAM Journal on Mathematics of Data Science"},{"key":"9693_CR39","doi-asserted-by":"publisher","first-page":"1001","DOI":"10.1007\/s00010-020-00720-7","volume":"94","author":"G Hansen","year":"2020","unstructured":"G. Hansen, I. Herburt, H. Martini, and M. Moszynska. \u201cStarshaped sets\u201d. Aequationes Mathematicae 94 (2020), pp. 1001\u20131092.","journal-title":"Aequationes Mathematicae"},{"key":"9693_CR40","doi-asserted-by":"publisher","DOI":"10.1201\/b18401","volume-title":"Statistical Learning with Sparsity: The Lasso and Generalizations","author":"T Hastie","year":"2015","unstructured":"T. Hastie, R. Tibshirani, and M. Wainwright. Statistical Learning with Sparsity: The Lasso and Generalizations. Chapman & Hall\/CRC, 2015."},{"issue":"3","key":"9693_CR41","first-page":"209","volume":"41","author":"T Hirose","year":"1965","unstructured":"T. Hirose. \u201cOn the convergence theorem for star-shaped sets in $${\\rm E}^n$$\u201d. Proc. Japan Acad. 41.3 (1965), pp. 209\u2013211.","journal-title":"Proc. Japan Acad."},{"issue":"5","key":"9693_CR42","doi-asserted-by":"publisher","first-page":"2626","DOI":"10.1214\/11-AOS920","volume":"39","author":"BT Knapik","year":"2011","unstructured":"B. T. Knapik, A. W. van der Vaart, and J. H. van Zanten. \u201cBayesian Inverse Problems with Gaussian Priors\u201d. The Annals of Statistics 39(5) (2011), pp. 2626\u20132657.","journal-title":"The Annals of Statistics"},{"key":"9693_CR43","unstructured":"E. Kobler, A. Effland, K. Kunisch, and T. Pock. \u201cTotal Deep Variation: A Stable Regularizer for Inverse Problems\u201d. arXiv preprint[SPACE]arXiv:2006.08789 (2020)."},{"issue":"4","key":"9693_CR44","doi-asserted-by":"publisher","first-page":"985","DOI":"10.1137\/S009753970544727X","volume":"35","author":"L Lov\u00e1sz","year":"2006","unstructured":"L. Lov\u00e1sz and S. Vempala. \u201cHit-and-run from a corner\u201d. SIAM Journal on Computing 35(4) (2006), pp. 985\u20131005.","journal-title":"SIAM Journal on Computing"},{"issue":"3","key":"9693_CR45","doi-asserted-by":"publisher","first-page":"307","DOI":"10.1002\/rsa.20135","volume":"30","author":"L Lov\u00e1sz","year":"2007","unstructured":"L. Lov\u00e1sz and S. Vempala. The geometry of logconcave functions and sampling algorithms. Random Structures & Algorithms 30(3) (2007), pp. 307\u2013358.","journal-title":"Random Structures & Algorithms"},{"key":"9693_CR46","unstructured":"S. Lunz, O. \u00d6ktem, and C.-B. Sch\u00f6nlieb. Adversarial Regularizers in Inverse Problems. Advances in Neural Information Processing Systems (NeurIPS) 31 (2018)."},{"key":"9693_CR47","doi-asserted-by":"publisher","unstructured":"E. Lutwak. Centroid Bodies and Dual Mixed Volumes. Proceedings of the London Mathematical Society s3-60.2 (1990), pp. 365\u2013391. https:\/\/doi.org\/10.1112\/plms\/s3-60.2.365.eprint: https:\/\/londmathsoc.onlinelibrary.wiley.com\/doi\/pdf\/10.1112\/plms\/s3-60.2.365. URL: https:\/\/londmathsoc.onlinelibrary.wiley.com\/doi\/abs\/10.1112\/plms\/s3-60.2.365.","DOI":"10.1112\/plms\/s3-60.2.365."},{"issue":"2","key":"9693_CR48","doi-asserted-by":"publisher","first-page":"531","DOI":"10.2140\/pjm.1975.58.531","volume":"58","author":"E Lutwak","year":"1975","unstructured":"E. Lutwak. Dual mixed volumes. Pacific Journal of Mathematics 58.2 (1975), pp. 531\u2013538.","journal-title":"Pacific Journal of Mathematics"},{"issue":"2\u20133","key":"9693_CR49","doi-asserted-by":"publisher","first-page":"85","DOI":"10.1561\/0600000058","volume":"8","author":"J Mairal","year":"2014","unstructured":"J. Mairal, F. Bach, and J. Ponce. Sparse modeling for image and vision processing. Foundations and Trends in Computer Graphics and Vision 8(2\u20133) (2014), pp. 85\u2013283.","journal-title":"Foundations and Trends in Computer Graphics and Vision"},{"issue":"1","key":"9693_CR50","doi-asserted-by":"publisher","first-page":"289","DOI":"10.1109\/TSP.2008.2007606","volume":"57","author":"H Mohimani","year":"2008","unstructured":"H. Mohimani, M. Babaie-Zadeh, and C. Jutten. A fast approach for overcomplete sparse decomposition based on smoothed $$\\ell _0$$ norm. IEEE Transactions on Signal Processing 57(1) (2008), pp. 289\u2013301.","journal-title":"IEEE Transactions on Signal Processing"},{"issue":"5","key":"9693_CR51","doi-asserted-by":"publisher","first-page":"1045","DOI":"10.1002\/cpa.21942","volume":"74","author":"F Monard","year":"2021","unstructured":"F. Monard, R. Nickl, and G. P. Paternain. Consistent inversion of noisy non-abelian X-Ray transforms. Communications on Pure and Applied Mathematics 74(5) (2021), pp. 1045\u20131099.","journal-title":"Communications on Pure and Applied Mathematics"},{"issue":"2","key":"9693_CR52","first-page":"617","volume":"47","author":"H Nyquist","year":"1928","unstructured":"H. Nyquist. \u201cCertain topics in telegraph transmission theory\u201d. Trans. AIEE. 47.2 (1928), pp. 617\u2013644.","journal-title":"Trans. AIEE."},{"issue":"6583","key":"9693_CR53","doi-asserted-by":"publisher","first-page":"607","DOI":"10.1038\/381607a0","volume":"381","author":"BA Olshausen","year":"1996","unstructured":"B. A. Olshausen and D. J. Field. \u201cEmergence of simple-cell receptive field properties by learning a sparse code for natural images\u201d. Nature 381(6583) (1996), pp. 607\u2013609.","journal-title":"Nature"},{"issue":"23","key":"9693_CR54","doi-asserted-by":"publisher","first-page":"3311","DOI":"10.1016\/S0042-6989(97)00169-7","volume":"37","author":"BA Olshausen","year":"1997","unstructured":"B. A. Olshausen and D. J. Field. Sparse coding with an overcomplete basis set: A strategy employed by V1? Vision in Research 37(23) (1997), pp. 3311\u20133325.","journal-title":"Vision in Research"},{"key":"9693_CR55","doi-asserted-by":"publisher","first-page":"965","DOI":"10.1007\/s10208-015-9278-4","volume":"16","author":"S Oymak","year":"2016","unstructured":"S. Oymak and B. Hassibi. Sharp MSE bounds for proximal denoising. Foundations of Computational Mathematics 16 (2016), pp. 965\u20131029.","journal-title":"Foundations of Computational Mathematics"},{"key":"9693_CR56","doi-asserted-by":"crossref","unstructured":"S. Oymak, C. Thrampoulidis, and B. Hassibi. The squared-error of generalized lasso: A precise analysis. 51st Annual Allerton Conference on Communication, Control, and Computing (Allerton) (2013).","DOI":"10.1109\/Allerton.2013.6736635"},{"key":"9693_CR57","doi-asserted-by":"crossref","unstructured":"V. Papyan, Y. Romano, J. Sulam, and M. Elad. Convolutional dictionary learning via local processing. International Conference on Computer Vision (ICCV) (2017), pp. 5296\u20135304.","DOI":"10.1109\/ICCV.2017.566"},{"key":"9693_CR58","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1016\/j.acha.2022.05.003","volume":"61","author":"K Pieper","year":"2022","unstructured":"K. Pieper and A. Petrosyan. \u201cNonconvex regularization for sparse neural networks\u201d. Applied and Computational Harmonic Analysis 61 (2022), pp. 25\u201356.","journal-title":"Applied and Computational Harmonic Analysis"},{"key":"9693_CR59","volume-title":"Information and information stability of random variables and processes","author":"MS Pinsker","year":"1964","unstructured":"M. S. Pinsker. \u201cInformation and information stability of random variables and processes\u201d. Holden-Day, Inc., San Francisco, Calif.-London, Amsterdam (1964)."},{"key":"9693_CR60","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-5254-2","volume-title":"Convergence of Stochastic Processes","author":"D Pollard","year":"1984","unstructured":"D. Pollard. Convergence of Stochastic Processes. Springer-Verlag, 1984."},{"key":"9693_CR61","unstructured":"P. Rebeschini. Lecture 5: Covering Numbers Bounds for Rademacher Complexity. Chaining. (2021)."},{"issue":"3","key":"9693_CR62","doi-asserted-by":"publisher","first-page":"471","DOI":"10.1137\/070697835","volume":"52","author":"B Recht","year":"2010","unstructured":"B. Recht, M. Fazel, and P. A. Parrilo. Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. SIAM Review 52(3) (2010), pp. 471\u2013501.","journal-title":"SIAM Review"},{"issue":"1","key":"9693_CR63","doi-asserted-by":"publisher","first-page":"52","DOI":"10.1109\/TCI.2018.2880326","volume":"5","author":"ET Reehorst","year":"2019","unstructured":"E. T. Reehorst and P. Schniter. \u201cRegularization by denoising: clarifications and new interpretations\u201d. IEEE Transactions on Computational Imaging 5(1) (2019), pp. 52\u201367.","journal-title":"IEEE Transactions on Computational Imaging"},{"key":"9693_CR64","doi-asserted-by":"crossref","unstructured":"G. O. Roberts and R. L. Tweedie. Exponential Convergence of Langevin Distributions and Their Discrete Approximations. Bernoulli (1996), pp. 341\u2013363.","DOI":"10.2307\/3318418"},{"issue":"1","key":"9693_CR65","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1093\/biomet\/83.1.95","volume":"83","author":"GO Roberts","year":"1996","unstructured":"G. O. Roberts and R. L. Tweedie. Geometric convergence and central limit theorems for multidimensional hastings and metropolis algorithms. Biometrika 83(1) (1996), pp. 95\u2013110.","journal-title":"Biometrika"},{"issue":"2","key":"9693_CR66","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1137\/1035044","volume":"35","author":"RT Rockafellar","year":"1993","unstructured":"R. T. Rockafellar. Lagrange multipliers and optimality. SIAM Review 35.2 (1993), pp. 183\u2013238.","journal-title":"SIAM Review"},{"issue":"4","key":"9693_CR67","doi-asserted-by":"publisher","first-page":"1804","DOI":"10.1137\/16M1102884","volume":"10","author":"Y Romano","year":"2017","unstructured":"Y. Romano, M. Elad, and P. Milanfar. \u201cThe little engine that could: Regularization by denoising (RED)\u201d. SIAM Journal on Imaging Sciences 10.4 (2017), pp. 1804\u20131844.","journal-title":"SIAM Journal on Imaging Sciences"},{"key":"9693_CR68","doi-asserted-by":"crossref","unstructured":"A. Rubinov. Radiant sets and their gauges. Quasidifferentiability and Related Topics. Nonconvex Optimization and Its Applications 43 (2000), pp. 235\u2013261.","DOI":"10.1007\/978-1-4757-3137-8_10"},{"issue":"1\u20134","key":"9693_CR69","doi-asserted-by":"publisher","first-page":"259","DOI":"10.1016\/0167-2789(92)90242-F","volume":"60","author":"LI Rudin","year":"1992","unstructured":"L. I. Rudin, S. Osher, and E. Fatemi. \u201cNonlinear total variation based noise removal algorithms\u201d. Physica D: nonlinear phenomena 60(1-4) (1992), pp. 259\u2013268.","journal-title":"Physica D: nonlinear phenomena"},{"issue":"1","key":"9693_CR70","doi-asserted-by":"publisher","first-page":"22","DOI":"10.1016\/j.acha.2016.08.002","volume":"45","author":"K Schnass","year":"2016","unstructured":"K. Schnass. Convergence radius and sample complexity of ITKM algorithms for dictionary learning. Applied and Computational Harmonic Analysis 45(1) (2016), pp. 22\u201358.","journal-title":"Applied and Computational Harmonic Analysis"},{"issue":"3","key":"9693_CR71","doi-asserted-by":"publisher","first-page":"464","DOI":"10.1016\/j.acha.2014.01.005","volume":"37","author":"K Schnass","year":"2014","unstructured":"K. Schnass. On the identifiability of overcomplete dictionaries via the minimisation principle underlying K-SVD. Applied and Computational Harmonic Analysis 37(3) (2014), pp. 464\u2013491.","journal-title":"Applied and Computational Harmonic Analysis"},{"key":"9693_CR72","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139003858","volume-title":"Convex Bodies: The Brunn-Minkowski Theory","author":"R Schneider","year":"2013","unstructured":"R. Schneider. \u201cConvex Bodies: The Brunn\u2013Minkowski Theory\u201d. Cambridge: Cambridge University Press. (2013)."},{"issue":"17","key":"9693_CR73","doi-asserted-by":"publisher","first-page":"4481","DOI":"10.1109\/TSP.2017.2711501","volume":"65","author":"I Selesnick","year":"2017","unstructured":"I. Selesnick. Sparse regularization via convex analysis. IEEE Transactions on Signal Processing 65(17) (2017), pp. 4481\u20134494.","journal-title":"IEEE Transactions on Signal Processing"},{"key":"9693_CR74","doi-asserted-by":"crossref","unstructured":"P. Shah, B. N. Bhaskar, G. Tang, and B. Recht. Linear system identification via atomic norm regularization. Proceedings of the 51st Annual Conference on Decision and Control (2012).","DOI":"10.1109\/CDC.2012.6426006"},{"key":"9693_CR75","doi-asserted-by":"crossref","unstructured":"S. Shalev-Shwartz and S. Ben-David. Understanding Machine Learning: From Theory to Algorithms. USA: Cambridge University Press, 2014. ISBN: 1107057132.","DOI":"10.1017\/CBO9781107298019"},{"key":"9693_CR76","doi-asserted-by":"publisher","first-page":"375","DOI":"10.1007\/s10208-018-9386-z","volume":"19","author":"YS Soh","year":"2019","unstructured":"Y. S. Soh and V. Chandrasekaran. Learning Semidefinite Regularizers. Foundations of Computational Mathematics 19 (2019), pp. 375\u2013434.","journal-title":"Foundations of Computational Mathematics"},{"key":"9693_CR77","doi-asserted-by":"publisher","first-page":"117","DOI":"10.1515\/advgeom-2012-0030","volume":"13","author":"G S\u00f3jka","year":"2013","unstructured":"G. S\u00f3jka. Metrics in the family of star bodies. Advances in Geometry 13 (2013), pp. 117\u2013144.","journal-title":"Advances in Geometry"},{"issue":"37","key":"9693_CR78","first-page":"1","volume":"23","author":"DA Spielman","year":"2012","unstructured":"D. A. Spielman, H. Wang, and J. Wright. Exact recovery of sparsely-used dictionaries. Conference on Learning Theory 23(37) (2012), pp. 1\u201318.","journal-title":"Conference on Learning Theory"},{"key":"9693_CR79","doi-asserted-by":"publisher","first-page":"055015","DOI":"10.1088\/1361-6420\/ab6f43","volume":"36","author":"B Sprungk","year":"2020","unstructured":"B. Sprungk. On the Local Lipschitz Stability of Bayesian Inverse Problems. Inverse Problems 36 (2020), p. 055015.","journal-title":"Inverse Problems"},{"key":"9693_CR80","doi-asserted-by":"publisher","first-page":"451","DOI":"10.1017\/S0962492910000061","volume":"19","author":"AM Stuart","year":"2010","unstructured":"A. M. Stuart. Inverse problems: a Bayesian perspective. Acta Numerica 19 (2010), pp. 451\u2013559.","journal-title":"Acta Numerica"},{"issue":"2","key":"9693_CR81","doi-asserted-by":"publisher","first-page":"853","DOI":"10.1109\/TIT.2016.2632162","volume":"63","author":"J Sun","year":"2016","unstructured":"J. Sun, Q. Qu, and J. Wright. Complete dictionary recovery over the sphere I: Overview and the geometric picture. IEEE Transactions on Information Theory 63(2) (2016), pp. 853\u2013884.","journal-title":"IEEE Transactions on Information Theory"},{"issue":"2","key":"9693_CR82","doi-asserted-by":"publisher","first-page":"885","DOI":"10.1109\/TIT.2016.2632149","volume":"63","author":"J Sun","year":"2016","unstructured":"J. Sun, Q. Qu, and J. Wright. Complete dictionary recovery over the sphere ii: Recovery by Riemannian trust-region method. IEEE Transactions on Information Theory 63(2) (2016), pp. 885\u2013914.","journal-title":"IEEE Transactions on Information Theory"},{"issue":"3","key":"9693_CR83","doi-asserted-by":"publisher","first-page":"329","DOI":"10.1016\/j.acha.2011.07.001","volume":"32","author":"Q Sun","year":"2012","unstructured":"Q. Sun. \u201cRecovery of sparsest signals via $$\\ell ^q$$-minimization\u201d. Applied and Computational Harmonic Analysis 32(3) (2012), pp. 329\u2013341.","journal-title":"Applied and Computational Harmonic Analysis"},{"issue":"11","key":"9693_CR84","doi-asserted-by":"publisher","first-page":"7465","DOI":"10.1109\/TIT.2013.2277451","volume":"59","author":"G Tang","year":"2013","unstructured":"G. Tang, B. N. Bhaskar, P. Shah, and B. Recht. Compressed Sensing off the Grid. IEEE Transactions on Information Theory 59(11) (2013), pp. 7465\u20137490.","journal-title":"IEEE Transactions on Information Theory"},{"key":"9693_CR85","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1111\/j.2517-6161.1996.tb02080.x","volume":"58","author":"R Tibshirani","year":"1994","unstructured":"R. Tibshirani. Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society, Series B 58 (1994), pp. 267\u2013288.","journal-title":"Journal of the Royal Statistical Society, Series B"},{"issue":"5","key":"9693_CR86","first-page":"176","volume":"39","author":"AN Tikhonov","year":"1943","unstructured":"A. N. Tikhonov. On stability of inverse problems. Dokl. Akad. Nauk SSSR 39.5 (1943), pp. 176\u2013179.","journal-title":"Dokl. Akad. Nauk SSSR"},{"key":"9693_CR87","doi-asserted-by":"crossref","unstructured":"Y. Traonmilin, R. Gribonval, and S. Vaiter. A theory of optimal convex regularization for low-dimensional recovery. Information and Inference: A Journal of the IMA 13(2) (June 2024), iaae013.","DOI":"10.1093\/imaiai\/iaae013"},{"key":"9693_CR88","first-page":"3259","volume":"12","author":"D Vainsencher","year":"2011","unstructured":"D. Vainsencher, S. Mannor, and A. M. Bruckstein. The sample complexity of dictionary learning. Journal of Machine Learning Research (JMLR) 12 (2011), pp. 3259\u20133281.","journal-title":"Journal of Machine Learning Research (JMLR)"},{"issue":"2","key":"9693_CR89","first-page":"573","volume":"52","author":"S Vempala","year":"2005","unstructured":"S. Vempala. Geometric random walks: A survey. Combinatorial and Computational Geometry 52(2) (2005), pp. 573\u2013612.","journal-title":"Combinatorial and Computational Geometry"},{"key":"9693_CR90","doi-asserted-by":"crossref","unstructured":"S. V. Venkatakrishnan, C. A. Bouman, and B. Wohlberg. Plug-and-play priors for model based reconstruction. 2013 IEEE Global Conference on Signal and Information Processing. IEEE. 2013, pp. 945\u2013948.","DOI":"10.1109\/GlobalSIP.2013.6737048"},{"key":"9693_CR91","volume-title":"High-Dimensional Probability: An Introduction with Applications in Data Science","author":"R Vershynin","year":"2020","unstructured":"R. Vershynin. High-Dimensional Probability: An Introduction with Applications in Data Science. Cambridge University Press, 2020."},{"key":"9693_CR92","doi-asserted-by":"publisher","DOI":"10.1090\/gsm\/058","volume-title":"Topics in Optimal Transport","author":"C Villani","year":"2003","unstructured":"C. Villani. Topics in Optimal Transport. Providence, RI: American Mathematical Society, 2003."},{"issue":"6","key":"9693_CR93","doi-asserted-by":"publisher","first-page":"2164","DOI":"10.1214\/14-AOS1238","volume":"42","author":"Z Wang","year":"2014","unstructured":"Z. Wang, H. Liu, and T. Zhang. Optimal computational and statistical rates of convergence for sparse nonconvex learning problems. Annals of Statistics 42(6) (2014), pp. 2164\u20132201.","journal-title":"Annals of Statistics"},{"issue":"1","key":"9693_CR94","doi-asserted-by":"publisher","first-page":"301","DOI":"10.1109\/TIP.2015.2495260","volume":"25","author":"B Wohlberg","year":"2015","unstructured":"B. Wohlberg. Efficient algorithms for convolutional sparse representations. IEEE Transactions on Image Processing 25(1) (2015), pp. 301\u2013315.","journal-title":"IEEE Transactions on Image Processing"},{"key":"9693_CR95","first-page":"1","volume":"18","author":"Q Yao","year":"2018","unstructured":"Q. Yao and J. T. Kwok. Efficient learning with a family of nonconvex regularizers by redistributing nonconvexity. Journal of Machine Learning Research (JMLR) 18 (2018), pp. 1\u201352.","journal-title":"Journal of Machine Learning Research (JMLR)"},{"issue":"2","key":"9693_CR96","doi-asserted-by":"publisher","first-page":"894","DOI":"10.1214\/09-AOS729","volume":"38","author":"C-H Zhang","year":"2010","unstructured":"C.-H. Zhang. \u201cNearly unbiased variable selection under minimax concave penalty\u201d. Annals of Statistics 38(2) (2010), pp. 894\u2013942.","journal-title":"Annals of Statistics"}],"container-title":["Foundations of Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-025-09693-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10208-025-09693-y","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-025-09693-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,27]],"date-time":"2026-04-27T21:02:21Z","timestamp":1777323741000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10208-025-09693-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,27]]},"references-count":96,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,4]]}},"alternative-id":["9693"],"URL":"https:\/\/doi.org\/10.1007\/s10208-025-09693-y","relation":{},"ISSN":["1615-3375","1615-3383"],"issn-type":[{"value":"1615-3375","type":"print"},{"value":"1615-3383","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,1,27]]},"assertion":[{"value":"2 December 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 September 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 October 2024","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 January 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}