%0 Journal Article %1 bogdan2015slope %A Bogdan, Małgorzata %A van den Berg, Ewout %A Sabatti, Chiara %A Su, Weijie %A Candès, Emmanuel J. %D 2015 %J The Annals of Applied Statistics %K owl slope structured_sparsity %N 3 %P 1103--1140 %R 10.1214/15-aoas842 %T SLOPE - Adaptive variable selection via convex optimization %U http://dx.doi.org/10.1214/15-aoas842 %V 9 %X We introduce a new estimator for the vector of coefficients \$\beta\$ in the linear model \$y=X\beta+z\$, where \$X\$ has dimensions \$np\$ with \$p\$ possibly larger than \$n\$. SLOPE, short for Sorted L-One Penalized Estimation, is the solution to \\min\_bınR^p12\Vert y-Xb\Vert \_\ell\_2^2+łambda\_1b\_(1)+łambda\_2\vert b\vert\_(2)+\cdots+łambda\_pb\vert\_(p),\ where \$łambda\_1\gełambda\_2\ge\cdots\gełambda\_p\ge0\$ and \$\vert b\vert\_(1)\geb\vert\_(2)\ge\cdots\geb\vert\_(p)\$ are the decreasing absolute values of the entries of \$b\$. This is a convex program and we demonstrate a solution algorithm whose computational complexity is roughly comparable to that of classical \$\ell\_1\$ procedures such as the Lasso. Here, the regularizer is a sorted \$\ell\_1\$ norm, which penalizes the regression coefficients according to their rank: the higher the rank - that is, stronger the signal - the larger the penalty. This is similar to the Benjamini and Hochberg J. Roy. Statist. Soc. Ser. B 57 (1995) 289-300 procedure (BH) which compares more significant \$p\$-values with more stringent thresholds. One notable choice of the sequence \$\łambda\_i\\$ is given by the BH critical values \$łambda\_BH(i)=z(1-iq/2p)\$, where \$qın(0,1)\$ and \$z(\alpha)\$ is the quantile of a standard normal distribution. SLOPE aims to provide finite sample guarantees on the selected model; of special interest is the false discovery rate (FDR), defined as the expected proportion of irrelevant regressors among all selected predictors. Under orthogonal designs, SLOPE with \$łambda\_BH\$ provably controls FDR at level \$q\$. Moreover, it also appears to have appreciable inferential properties under more general designs \$X\$ while having substantial power, as demonstrated in a series of experiments running on both simulated and real data.

@article{bogdan2015slope, abstract = {We introduce a new estimator for the vector of coefficients \$\beta\$ in the linear model \$y=X\beta+z\$, where \$X\$ has dimensions \$n\times p\$ with \$p\$ possibly larger than \$n\$. SLOPE, short for Sorted L-One Penalized Estimation, is the solution to \[\min\_{b\in\mathbb{R}^p}\frac{1}{2}\Vert y-Xb\Vert \_{\ell\_2}^2+\lambda\_1\vert b\vert \_{(1)}+\lambda\_2\vert b\vert\_{(2)}+\cdots+\lambda\_p\vert b\vert\_{(p)},\] where \$\lambda\_1\ge\lambda\_2\ge\cdots\ge\lambda\_p\ge0\$ and \$\vert b\vert\_{(1)}\ge\vert b\vert\_{(2)}\ge\cdots\ge\vert b\vert\_{(p)}\$ are the decreasing absolute values of the entries of \$b\$. This is a convex program and we demonstrate a solution algorithm whose computational complexity is roughly comparable to that of classical \$\ell\_1\$ procedures such as the Lasso. Here, the regularizer is a sorted \$\ell\_1\$ norm, which penalizes the regression coefficients according to their rank: the higher the rank - that is, stronger the signal - the larger the penalty. This is similar to the Benjamini and Hochberg [J. Roy. Statist. Soc. Ser. B 57 (1995) 289-300] procedure (BH) which compares more significant \$p\$-values with more stringent thresholds. One notable choice of the sequence \$\{\lambda\_i\}\$ is given by the BH critical values \$\lambda\_{\mathrm {BH}}(i)=z(1-i\cdot q/2p)\$, where \$q\in(0,1)\$ and \$z(\alpha)\$ is the quantile of a standard normal distribution. SLOPE aims to provide finite sample guarantees on the selected model; of special interest is the false discovery rate (FDR), defined as the expected proportion of irrelevant regressors among all selected predictors. Under orthogonal designs, SLOPE with \$\lambda\_{\mathrm{BH}}\$ provably controls FDR at level \$q\$. Moreover, it also appears to have appreciable inferential properties under more general designs \$X\$ while having substantial power, as demonstrated in a series of experiments running on both simulated and real data.}, added-at = {2018-12-07T09:10:16.000+0100}, archiveprefix = {arXiv}, author = {Bogdan, Ma{\l}gorzata and van den Berg, Ewout and Sabatti, Chiara and Su, Weijie and Cand\`{e}s, Emmanuel J.}, biburl = {https://www.bibsonomy.org/bibtex/235d7300f44975f4a72574281fdc92401/jpvaldes}, citeulike-article-id = {14027610}, citeulike-linkout-0 = {http://arxiv.org/abs/1407.3824}, citeulike-linkout-1 = {http://arxiv.org/pdf/1407.3824}, citeulike-linkout-2 = {http://dx.doi.org/10.1214/15-aoas842}, citeulike-linkout-3 = {http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4689150/}, citeulike-linkout-4 = {http://view.ncbi.nlm.nih.gov/pubmed/26709357}, citeulike-linkout-5 = {http://www.hubmed.org/display.cgi?uids=26709357}, day = 4, doi = {10.1214/15-aoas842}, eprint = {1407.3824}, interhash = {9fc31921245a756aa002c9b1316295d7}, intrahash = {35d7300f44975f4a72574281fdc92401}, issn = {1932-6157}, journal = {The Annals of Applied Statistics}, keywords = {owl slope structured_sparsity}, month = nov, number = 3, pages = {1103--1140}, pmcid = {PMC4689150}, pmid = {26709357}, posted-at = {2017-11-22 12:42:27}, priority = {2}, timestamp = {2018-12-07T09:40:35.000+0100}, title = {{SLOPE - Adaptive variable selection via convex optimization}}, url = {http://dx.doi.org/10.1214/15-aoas842}, volume = 9, year = 2015 }