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Matrix completion has no spurious local minimum

Publication ,  Conference
Ge, R; Lee, JD; Ma, T
Published in: Advances in Neural Information Processing Systems
January 1, 2016

Matrix completion is a basic machine learning problem that has wide applications, especially in collaborative filtering and recommender systems. Simple non-convex optimization algorithms are popular and effective in practice. Despite recent progress in proving various non-convex algorithms converge from a good initial point, it remains unclear why random or arbitrary initialization suffices in practice. We prove that the commonly used non-convex objective function for positive semidefinite matrix completion has no spurious local minima - all local minima must also be global. Therefore, many popular optimization algorithms such as (stochastic) gradient descent can provably solve positive semidefinite matrix completion with arbitrary initialization in polynomial time. The result can be generalized to the setting when the observed entries contain noise. We believe that our main proof strategy can be useful for understanding geometric properties of other statistical problems involving partial or noisy observations.

Duke Scholars

Published In

Advances in Neural Information Processing Systems

ISSN

1049-5258

Publication Date

January 1, 2016

Start / End Page

2981 / 2989

Related Subject Headings

  • 4611 Machine learning
  • 1702 Cognitive Sciences
  • 1701 Psychology
 

Citation

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MLA
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Ge, R., Lee, J. D., & Ma, T. (2016). Matrix completion has no spurious local minimum. In Advances in Neural Information Processing Systems (pp. 2981–2989).
Ge, R., J. D. Lee, and T. Ma. “Matrix completion has no spurious local minimum.” In Advances in Neural Information Processing Systems, 2981–89, 2016.
Ge R, Lee JD, Ma T. Matrix completion has no spurious local minimum. In: Advances in Neural Information Processing Systems. 2016. p. 2981–9.
Ge, R., et al. “Matrix completion has no spurious local minimum.” Advances in Neural Information Processing Systems, 2016, pp. 2981–89.
Ge R, Lee JD, Ma T. Matrix completion has no spurious local minimum. Advances in Neural Information Processing Systems. 2016. p. 2981–2989.

Published In

Advances in Neural Information Processing Systems

ISSN

1049-5258

Publication Date

January 1, 2016

Start / End Page

2981 / 2989

Related Subject Headings

  • 4611 Machine learning
  • 1702 Cognitive Sciences
  • 1701 Psychology