{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,23]],"date-time":"2026-04-23T11:11:27Z","timestamp":1776942687783,"version":"3.51.4"},"reference-count":36,"publisher":"Wiley","license":[{"start":{"date-parts":[[2016,1,1]],"date-time":"2016-01-01T00:00:00Z","timestamp":1451606400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computational and Mathematical Methods in Medicine"],"published-print":{"date-parts":[[2016]]},"abstract":"<jats:p>Variable selections for regression with high-dimensional big data have found many applications in bioinformatics and computational biology. One appealing approach is the<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mrow><mml:msub><mml:mrow><mml:mi>L<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn fontstyle=\"italic\">0<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>regularized regression which penalizes the number of nonzero features in the model directly. However, it is well known that<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mrow><mml:msub><mml:mrow><mml:mi>L<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn fontstyle=\"italic\">0<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>optimization is NP-hard and computationally challenging. In this paper, we propose efficient EM (<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mrow><mml:msub><mml:mrow><mml:mi>L<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn fontstyle=\"italic\">0<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>EM) and dual<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mrow><mml:msub><mml:mrow><mml:mi>L<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn fontstyle=\"italic\">0<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>EM (D<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mrow><mml:msub><mml:mrow><mml:mi>L<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn fontstyle=\"italic\">0<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>EM) algorithms that directly approximate the<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:mrow><mml:msub><mml:mrow><mml:mi>L<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn fontstyle=\"italic\">0<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>optimization problem. While<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\"><mml:mrow><mml:msub><mml:mrow><mml:mi>L<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn fontstyle=\"italic\">0<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>EM is efficient with large sample size, D<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\"><mml:mrow><mml:msub><mml:mrow><mml:mi>L<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn fontstyle=\"italic\">0<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>EM is efficient with high-dimensional (<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M10\"><mml:mi>n<\/mml:mi><mml:mo>\u226a<\/mml:mo><mml:mi>m<\/mml:mi><\/mml:math>) data. They also provide a natural solution to all<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M11\"><mml:mrow><mml:msub><mml:mrow><mml:mi>L<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>p<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>\u2009\u2009<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M12\"><mml:mi>p<\/mml:mi><mml:mo>\u2208<\/mml:mo><mml:mo stretchy=\"false\">[<\/mml:mo><mml:mn fontstyle=\"italic\">0,2<\/mml:mn><mml:mo stretchy=\"false\">]<\/mml:mo><\/mml:math>problems, including lasso with<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M13\"><mml:mi>p<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn fontstyle=\"italic\">1<\/mml:mn><\/mml:math>and elastic net with<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M14\"><mml:mi>p<\/mml:mi><mml:mo>\u2208<\/mml:mo><mml:mo stretchy=\"false\">[<\/mml:mo><mml:mn fontstyle=\"italic\">1,2<\/mml:mn><mml:mo stretchy=\"false\">]<\/mml:mo><\/mml:math>. The regularized parameter<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M15\"><mml:mrow><mml:mi>\u03bb<\/mml:mi><\/mml:mrow><\/mml:math>can be determined through cross validation or AIC and BIC. We demonstrate our methods through simulation and high-dimensional genomic data. The results indicate that<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M16\"><mml:mrow><mml:msub><mml:mrow><mml:mi>L<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn fontstyle=\"italic\">0<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>has better performance than lasso, SCAD, and MC+, and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M17\"><mml:mrow><mml:msub><mml:mrow><mml:mi>L<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn fontstyle=\"italic\">0<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>with AIC or BIC has similar performance as computationally intensive cross validation. The proposed algorithms are efficient in identifying the nonzero variables with less bias and constructing biologically important networks with high-dimensional big data.<\/jats:p>","DOI":"10.1155\/2016\/3456153","type":"journal-article","created":{"date-parts":[[2016,10,24]],"date-time":"2016-10-24T21:01:05Z","timestamp":1477342865000},"page":"1-11","source":"Crossref","is-referenced-by-count":26,"title":["Efficient Regularized Regression with<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mrow><mml:msub><mml:mrow><mml:mi>L<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn fontstyle=\"italic\">0<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>Penalty for Variable Selection and Network Construction"],"prefix":"10.1155","volume":"2016","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1535-4322","authenticated-orcid":true,"given":"Zhenqiu","family":"Liu","sequence":"first","affiliation":[{"name":"Samuel Oschin Comprehensive Cancer Institute, Cedars-Sinai 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