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info.json
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{
"abstract": "This work studies the strong duality of non-convex matrix factorization problems: we show that under certain dual conditions, these problems and the 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 prove that under certain dual conditions, the optimal solution of the matrix factorization program is the same as that of 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 is 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. These are examples of efficiently recovering a hidden matrix given limited reliable observations. Our framework shows that exact recoverability and strong duality hold with nearly-optimal sample complexity for the two problems.",
"authors": [
"Maria-Florina Balcan",
"Yingyu Liang",
"Zhao Song",
"David P. Woodruff",
"Hongyang Zhang"
],
"emails": [
"ninamf@cs.cmu.edu",
"yliang@cs.wisc.edu",
"zhaos@g.harvard.edu",
"dwoodruf@cs.cmu.edu",
"hongyanz@cs.cmu.edu"
],
"id": "17-611",
"issue": 102,
"pages": [
1,
56
],
"title": "Non-Convex Matrix Completion and Related Problems via Strong Duality",
"volume": 20,
"year": 2019
}