### Abstract

We propose a rank-k variant of the classical Frank-Wolfe algorithm to solve convex optimization over a trace-norm ball. Our algorithm replaces the top singular-vector computation (1-SVD) in Frank-Wolfe with a top-k singular-vector computation (k-SVD), which can be done by repeatedly applying 1-SVD k times. Alternatively, our algorithm can be viewed as a rank-k restricted version of projected gradient descent. We show that our algorithm has a linear convergence rate when the objective function is smooth and strongly convex, and the optimal solution has rank at most k. This improves the convergence rate and the total time complexity of the Frank-Wolfe method and its variants.

Original language | English (US) |
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Pages (from-to) | 6192-6201 |

Number of pages | 10 |

Journal | Advances in Neural Information Processing Systems |

Volume | 2017-December |

State | Published - Jan 1 2017 |

Event | 31st Annual Conference on Neural Information Processing Systems, NIPS 2017 - Long Beach, United States Duration: Dec 4 2017 → Dec 9 2017 |

### All Science Journal Classification (ASJC) codes

- Computer Networks and Communications
- Information Systems
- Signal Processing

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## Cite this

*Advances in Neural Information Processing Systems*,

*2017-December*, 6192-6201.