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Non-negative matrix factorization (NMF) is a highly celebrated algorithm for matrix decomposition that guarantees non-negative factors. The underlying optimization problem is computationally intractable, yet in practice, gradient-descent-based methods often find good solutions. In this paper, we revisit the NMF optimization problem and analyze its loss landscape in non-worst-case settings. It has recently been observed that gradients in deep networks tend to point towards the final minimizer throughout the optimization procedure. We show that a similar property holds (with high probability) for NMF, provably in a non-worst case model with a planted solution, and empirically across an extensive suite of real-world NMF problems. Our analysis predicts that this property becomes more likely with growing number of parameters, and experiments suggest that a similar trend might also hold for deep neural networks---turning increasing dataset sizes and model sizes into a blessing from an optimization perspective.more » « less
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Bjorck, Johan ; Kabra, Anmol ; Weinberger, Kilian Q. ; Gomes, Carla ( , Proceedings of the AAAI Conference on Artificial Intelligence)
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Suram, Santosh K. ; Xue, Yexiang ; Bai, Junwen ; Le Bras, Ronan ; Rappazzo, Brendan ; Bernstein, Richard ; Bjorck, Johan ; Zhou, Lan ; van Dover, R. Bruce ; Gomes, Carla P. ; et al ( , ACS Combinatorial Science)