{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,17]],"date-time":"2026-07-17T08:04:42Z","timestamp":1784275482887,"version":"3.55.0"},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2026,7,17]],"date-time":"2026-07-17T00:00:00Z","timestamp":1784246400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,7,17]],"date-time":"2026-07-17T00:00:00Z","timestamp":1784246400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100002341","name":"Research Council of Finland","doi-asserted-by":"publisher","award":["Grant 326291"],"award-info":[{"award-number":["Grant 326291"]}],"id":[{"id":"10.13039\/501100002341","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Stat Comput"],"published-print":{"date-parts":[[2026,8]]},"abstract":"Abstract<\/jats:title>\n \n In high-dimensional regression, the choice of regularization penalty typically forces a rigid assumption upon the underlying signal structure, dichotomizing data into strictly sparse (Lasso,\n \n \n $$q=1$$<\/jats:tex-math>\n \n \n q<\/mml:mi>\n =<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n ) or entirely dense (Ridge,\n \n \n $$q=2$$<\/jats:tex-math>\n \n \n q<\/mml:mi>\n =<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n ) regimes. However, real-world data generating mechanisms frequently exist on a continuum between these extremes, requiring flexible geometries to handle varying degrees of sparsity and considerable multicollinearity. In this work, we propose a data-driven framework to learn the optimal regularization norm by elevating the\n \n \n $$L_q$$<\/jats:tex-math>\n \n \n L<\/mml:mi>\n q<\/mml:mi>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n exponent (\n \n \n $$q \\in (0, 2]$$<\/jats:tex-math>\n \n \n q<\/mml:mi>\n \u2208<\/mml:mo>\n (<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n 2<\/mml:mn>\n ]<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n ) from a discrete choice to a strictly continuous, learnable hyper-parameter. To overcome the computational bottleneck of evaluating non-convex and non-smooth penalty landscapes, we develop a universal proximal coordinate descent solver that utilizes a safeguarded jumping threshold operator and a novel empirical Karush-Kuhn-Tucker (KKT) verification strategy. This solver is coupled with a stochastic Tree-structured Parzen Estimator (TPE) utilizing randomized internal validation splits, enabling the rapid discovery of optimal penalty geometries without over-fitting. We evaluate the framework on simulated architectures, demonstrating its dynamic adaptivity to structural sparsity, collinearity, and varying signal-to-noise ratios. Applied to four high-dimensional genomic datasets (scaling up to\n \n \n $$P \\approx 50,000$$<\/jats:tex-math>\n \n \n P<\/mml:mi>\n \u2248<\/mml:mo>\n 50<\/mml:mn>\n ,<\/mml:mo>\n 000<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n features), our generalized adaptive bridge regression (GABR) framework successfully identifies optimal, off-grid grouping architectures (\n \n \n $$q \\approx 1.63$$<\/jats:tex-math>\n \n \n q<\/mml:mi>\n \u2248<\/mml:mo>\n 1.63<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n to 1.80), outperforming purely sparse and purely dense alternatives. These results demonstrate that the exact regression geometry can be efficiently learned from the data, enabling a unified approach to high-dimensional inference without the computational restrictions of exhaustive discrete grid searches.\n <\/jats:p>","DOI":"10.1007\/s11222-026-10938-1","type":"journal-article","created":{"date-parts":[[2026,7,17]],"date-time":"2026-07-17T07:14:54Z","timestamp":1784272494000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Generalized adaptive bridge regression: a unified framework for high-dimensional architecture discovery"],"prefix":"10.1007","volume":"36","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2390-6609","authenticated-orcid":false,"given":"Patrik","family":"Waldmann","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,7,17]]},"reference":[{"key":"10938_CR1","doi-asserted-by":"crossref","unstructured":"Attouch, H., Bolte, J., Svaiter, B.F.: Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods. Math. Program. 137(1\u20132), 91\u2013129 (2013)","DOI":"10.1007\/s10107-011-0484-9"},{"key":"10938_CR2","doi-asserted-by":"crossref","unstructured":"Beck, A.: First-Order Methods in Optimization. MOS-SIAM Series on Optimization, vol. 25. Society for Industrial and Applied Mathematics, Philadelphia, PA (2017)","DOI":"10.1137\/1.9781611974997"},{"key":"10938_CR3","unstructured":"Bergstra, J., Bardenet, R., Bengio, Y., K\u00e9gl, B.: Algorithms for hyper-parameter optimization. In Advances in Neural Information Processing Systems, 24, 2546\u20132554 (2011). Curran Associates, Inc.,"},{"issue":"1","key":"10938_CR4","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1137\/141000671","volume":"59","author":"J Bezanson","year":"2017","unstructured":"Bezanson, J., Edelman, A., Karpinski, S., Shah, V.B.: Julia: A fresh approach to numerical computing. SIAM Rev. 59(1), 65\u201398 (2017)","journal-title":"SIAM Rev."},{"issue":"1\u20132","key":"10938_CR5","doi-asserted-by":"publisher","first-page":"459","DOI":"10.1007\/s10107-013-0701-9","volume":"146","author":"J Bolte","year":"2014","unstructured":"Bolte, J., Sabach, S., Teboulle, M.: Proximal alternating linearized minimization for nonconvex and nonsmooth problems. Math. Program. 146(1\u20132), 459\u2013494 (2014)","journal-title":"Math. Program."},{"key":"10938_CR6","unstructured":"Bouchard, K.E., Bujan, A.F., Roosta, S., et al.: Numerical characterization of support recovery in sparse regression with correlated design. Communications in Statistics-Simulation and Computation 51(10), 5922\u20135945 (2022)"},{"issue":"1","key":"10938_CR7","doi-asserted-by":"publisher","first-page":"232","DOI":"10.1214\/10-AOAS388","volume":"5","author":"P Breheny","year":"2011","unstructured":"Breheny, P., Huang, J.: Coordinate descent algorithms for nonconvex penalized regression, with applications to biological feature selection. The Annals of Applied Statistics 5(1), 232\u2013253 (2011)","journal-title":"The Annals of Applied Statistics"},{"key":"10938_CR8","doi-asserted-by":"crossref","unstructured":"Candes, E.J., Wakin, M.B., Boyd, S.P.: Enhancing sparsity by reweighted l1 minimization. J. Fourier Anal. Appl. 14(5), 877\u2013905 (2008)","DOI":"10.1007\/s00041-008-9045-x"},{"key":"10938_CR9","doi-asserted-by":"crossref","unstructured":"Cleveland, M.A., Hickey, J.M., Forni, S.: A common dataset for genomic analysis of livestock populations. G3: Genes|Genomes|Genetics 2(4), 429\u2013435 (2012)","DOI":"10.1534\/g3.111.001453"},{"issue":"456","key":"10938_CR10","doi-asserted-by":"publisher","first-page":"1348","DOI":"10.1198\/016214501753382273","volume":"96","author":"J Fan","year":"2001","unstructured":"Fan, J., Li, R.: Variable selection via nonconcave penalized likelihood and its oracle properties. J. Am. Stat. Assoc. 96(456), 1348\u20131360 (2001)","journal-title":"J. Am. Stat. Assoc."},{"key":"10938_CR11","doi-asserted-by":"crossref","unstructured":"Frank, L.E., Friedman, J.H.: A statistical view of some chemometrics regression tools. Technometrics 35(2), 109\u2013135 (1993)","DOI":"10.1080\/00401706.1993.10485033"},{"issue":"1","key":"10938_CR12","doi-asserted-by":"publisher","first-page":"1","DOI":"10.18637\/jss.v033.i01","volume":"33","author":"J Friedman","year":"2010","unstructured":"Friedman, J., Hastie, T., Tibshirani, R.: Regularization paths for generalized linear models via coordinate descent. J. Stat. Softw. 33(1), 1 (2010)","journal-title":"J. Stat. Softw."},{"key":"10938_CR13","volume-title":"and Martin Wainwright","author":"T Hastie","year":"2015","unstructured":"Hastie, T., Tibshirani, R.: and Martin Wainwright. The Lasso and Generalizations. CRC Press, Statistical Learning with Sparsity (2015)"},{"key":"10938_CR14","doi-asserted-by":"crossref","unstructured":"Hoerl, A.E., Kennard, R.W.: Ridge regression: Biased estimation for nonorthogonal problems. Technometrics 12(1), 55\u201367 (1970)","DOI":"10.1080\/00401706.1970.10488634"},{"issue":"2","key":"10938_CR15","doi-asserted-by":"publisher","first-page":"587","DOI":"10.1214\/009053607000000875","volume":"36","author":"J Huang","year":"2008","unstructured":"Huang, J., Horowitz, J.L., Ma, S.: Properties of bridge estimators in sparse high-dimensional regression models. Ann. Stat. 36(2), 587\u2013613 (2008)","journal-title":"Ann. Stat."},{"key":"10938_CR16","doi-asserted-by":"publisher","first-page":"29","DOI":"10.5109\/1495409","volume":"44","author":"S Kawano","year":"2012","unstructured":"Kawano, S.: Adaptive bridge regression modeling and selection of the tuning parameters. Bull. Inform. Cybernet. 44, 29\u201339 (2012)","journal-title":"Bull. Inform. Cybernet."},{"key":"10938_CR17","doi-asserted-by":"crossref","unstructured":"Kawano, S.: Selection of tuning parameters in bridge regression models via Bayesian information criterion. Stat. Pap. 55(4), 1207\u20131223 (2014)","DOI":"10.1007\/s00362-013-0561-7"},{"key":"10938_CR18","doi-asserted-by":"crossref","unstructured":"Lange, K., Won, J.-H., Landeros, A., Zhou, H.: Nonconvex optimization via MM algorithms: Convergence theory, (2021). arXiv:2106.02805 arXiv preprint","DOI":"10.1002\/9781118445112.stat08295"},{"issue":"6","key":"10938_CR19","doi-asserted-by":"crossref","first-page":"1489","DOI":"10.1109\/TSP.2014.2302740","volume":"62","author":"G Marjanovic","year":"2014","unstructured":"Marjanovic, G., Solo, V.: On $$l_q$$ optimization and matrix completion. IEEE Trans. Signal Process. 62(6), 1489\u20131499 (2014)","journal-title":"IEEE Trans. Signal Process."},{"issue":"495","key":"10938_CR20","doi-asserted-by":"publisher","first-page":"1125","DOI":"10.1198\/jasa.2011.tm09738","volume":"106","author":"R Mazumder","year":"2011","unstructured":"Mazumder, R., Friedman, J.H., Hastie, T.: SparseNet: Coordinate descent with nonconvex penalties. J. Am. Stat. Assoc. 106(495), 1125\u20131138 (2011)","journal-title":"J. Am. Stat. Assoc."},{"issue":"7","key":"10938_CR21","doi-asserted-by":"publisher","first-page":"685","DOI":"10.1080\/02664760500079464","volume":"32","author":"S Nadarajah","year":"2005","unstructured":"Nadarajah, S.: A generalized normal distribution. J. Appl. Stat. 32(7), 685\u2013694 (2005)","journal-title":"J. Appl. Stat."},{"key":"10938_CR22","unstructured":"Nocedal, J., Wright, S.J.: Numerical optimization. Springer Science & Business Media, New York, NY, 2nd edition, (2006)"},{"key":"10938_CR23","doi-asserted-by":"crossref","unstructured":"Parikh, N., Boyd, S.: Proximal algorithms. Foundations and Trends\u00ae in Optimization 1(3), 127\u2013239 (2014)","DOI":"10.1561\/2400000003"},{"key":"10938_CR24","doi-asserted-by":"crossref","unstructured":"P\u00e9rez, P., de los Campos, G.: Genome-wide regression and prediction with the BGLR statistical package. Genetics 198(2), 483\u2013495 (2014)","DOI":"10.1534\/genetics.114.164442"},{"key":"10938_CR25","doi-asserted-by":"crossref","unstructured":"Polson, N.G., Scott, J.G., Willard, B.T.: Proximal algorithms in statistics and machine learning. Stat. Sci. 30(4), 559\u2013581 (2015)","DOI":"10.1214\/15-STS530"},{"key":"10938_CR26","unstructured":"Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical recipes 3rd edition: The art of scientific computing. Cambridge University Press, Cambridge, UK, (2007)"},{"key":"10938_CR27","unstructured":"Shi, H.-J., Tu, S., Xu, Y., Yin, W.: A primer on coordinate descent algorithms, (2016). arXiv:1610.00040 arXiv preprint"},{"key":"10938_CR28","doi-asserted-by":"crossref","unstructured":"Sz\u00e9kely, G.J., Rizzo, M.L., Bakirov, N.K.: Measuring and testing dependence by correlation of distances. Ann. Stat. 35(6), 2769\u20132794 (2007)","DOI":"10.1214\/009053607000000505"},{"issue":"1","key":"10938_CR29","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1111\/j.2517-6161.1996.tb02080.x","volume":"58","author":"R Tibshirani","year":"1996","unstructured":"Tibshirani, R.: Regression shrinkage and selection via the lasso. J. Roy. Stat. Soc.: Ser. B (Methodol.) 58(1), 267\u2013288 (1996)","journal-title":"J. Roy. Stat. Soc.: Ser. B (Methodol.)"},{"issue":"3","key":"10938_CR30","doi-asserted-by":"publisher","first-page":"475","DOI":"10.1023\/A:1017501703105","volume":"109","author":"P Tseng","year":"2001","unstructured":"Tseng, P.: Convergence of a block coordinate descent method for nondifferentiable minimization. J. Optim. Theory Appl. 109(3), 475\u2013494 (2001)","journal-title":"J. Optim. Theory Appl."},{"key":"10938_CR31","doi-asserted-by":"crossref","unstructured":"Wainwright, M.J.: High-Dimensional Statistics: A Non-Asymptotic Viewpoint. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, (2019)","DOI":"10.1017\/9781108627771"},{"key":"10938_CR32","doi-asserted-by":"crossref","unstructured":"Wright, S.J.: Coordinate descent algorithms. Math. Program. 151(1), 3\u201334 (2015)","DOI":"10.1007\/s10107-015-0892-3"},{"issue":"23","key":"10938_CR33","first-page":"6306","volume":"64","author":"J Zeng","year":"2016","unstructured":"Zeng, J., Lin, S., Zongben, X.: Sparse regularization: Convergence of iterative jumping thresholding algorithm. IEEE Trans. Signal Process. 64(23), 6306\u20136319 (2016)","journal-title":"IEEE Trans. Signal Process."},{"issue":"2","key":"10938_CR34","doi-asserted-by":"publisher","first-page":"894","DOI":"10.1214\/09-AOS729","volume":"38","author":"C-H Zhang","year":"2010","unstructured":"Zhang, C.-H.: Nearly unbiased variable selection under minimax concave penalty. Ann. Stat. 38(2), 894\u2013942 (2010)","journal-title":"Ann. Stat."},{"key":"10938_CR35","first-page":"1081","volume":"11","author":"T Zhang","year":"2010","unstructured":"Zhang, T.: Analysis of multi-stage convex relaxation for sparse regularization. J. Mach. Learn. Res. 11, 1081\u20131107 (2010)","journal-title":"J. Mach. Learn. Res."},{"issue":"2","key":"10938_CR36","doi-asserted-by":"publisher","first-page":"301","DOI":"10.1111\/j.1467-9868.2005.00503.x","volume":"67","author":"H Zou","year":"2005","unstructured":"Zou, H., Hastie, T.: Regularization and variable selection via the elastic net. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 67(2), 301\u2013320 (2005)","journal-title":"Journal of the Royal Statistical Society: Series B (Statistical Methodology)"}],"container-title":["Statistics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-026-10938-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11222-026-10938-1","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-026-10938-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,17]],"date-time":"2026-07-17T07:14:59Z","timestamp":1784272499000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11222-026-10938-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,7,17]]},"references-count":36,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2026,8]]}},"alternative-id":["10938"],"URL":"https:\/\/doi.org\/10.1007\/s11222-026-10938-1","relation":{},"ISSN":["0960-3174","1573-1375"],"issn-type":[{"value":"0960-3174","type":"print"},{"value":"1573-1375","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,7,17]]},"assertion":[{"value":"22 March 2026","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 July 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 July 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no competing interests.","order":1,"name":"Ethics","label":"Competing interests","group":{"name":"EthicsHeading","label":"Declarations"}}],"article-number":"181"}}