This work studies the strong duality of non-convex matrix factorization problems: we show that under certain dual conditions, these problems and its dual have the same optimum. This has been well understood for convex optimization, but little was known for non-convex problems. We propose a novel analytical framework and show that under certain dual conditions, the optimal solution of the matrix factorization program is the same as its bi-dual and thus the global optimality of the non-convex program can be achieved by solving its bi-dual which is convex. These dual conditions are satisfied by a wide class of matrix factorization problems, although matrix factorization problems are hard to solve in full generality. This analytical framework may be of independent interest to non-convex optimization more broadly. We apply our framework to two prototypical matrix factorization problems: matrix completion and robust Principal Component Analysis (PCA). These are examples of efficiently recovering a hidden matrix given limited reliable observations of it. Our framework shows that exact recoverability and strong duality hold with nearly-optimal sample complexity guarantees for matrix completion and robust PCA.
@InProceedings{balcan_et_al:LIPIcs.ITCS.2018.5, author = {Balcan, Maria-Florina and Liang, Yingyu and Woodruff, David P. and Zhang, Hongyang}, title = {{Matrix Completion and Related Problems via Strong Duality}}, booktitle = {9th Innovations in Theoretical Computer Science Conference (ITCS 2018)}, pages = {5:1--5:22}, 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.2018.5}, URN = {urn:nbn:de:0030-drops-83583}, doi = {10.4230/LIPIcs.ITCS.2018.5}, annote = {Keywords: Non-Convex Optimization, Strong Duality, Matrix Completion, Robust PCA, Sample Complexity} }
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