We consider the problems of sparse regression and column subset selection under L1 error. For both problems, we show that in the non-negative setting it is possible to obtain tight and efficient approximations, without any additional structural assumptions (such as restricted isometry, incoherence, expansion, etc.). For sparse regression, given a matrix A and a vector b with non-negative entries, we give an efficient algorithm to output a vector x of sparsity O(k), for which |Ax - b|_1 is comparable to the smallest error possible using non-negative k-sparse x. We then use this technique to obtain our main result: an efficient algorithm for column subset selection under L1 error for non-negative matrices.
@InProceedings{bhaskara_et_al:LIPIcs.ITCS.2018.7, author = {Bhaskara, Aditya and Lattanzi, Silvio}, title = {{Non-Negative Sparse Regression and Column Subset Selection with L1 Error}}, booktitle = {9th Innovations in Theoretical Computer Science Conference (ITCS 2018)}, pages = {7:1--7:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-060-6}, ISSN = {1868-8969}, year = {2018}, volume = {94}, editor = {Karlin, Anna R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.7}, URN = {urn:nbn:de:0030-drops-83548}, doi = {10.4230/LIPIcs.ITCS.2018.7}, annote = {Keywords: Sparse regression, L1 error optimization, Column subset selection} }
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