Complete Dictionary Recovery Using Nonconvex Optimization

Ju Sun, Qing Qu, John Wright
Proceedings of the 32nd International Conference on Machine Learning, PMLR 37:2351-2360, 2015.

Abstract

We consider the problem of recovering a complete (i.e., square and invertible) dictionary mb A_0, from mb Y = mb A_0 mb X_0 with mb Y ∈\mathbb R^n \times p. This recovery setting is central to the theoretical understanding of dictionary learning. We give the first efficient algorithm that provably recovers mb A_0 when mb X_0 has O(n) nonzeros per column, under suitable probability model for mb X_0. Prior results provide recovery guarantees when mb X_0 has only O(\sqrtn) nonzeros per column. Our algorithm is based on nonconvex optimization with a spherical constraint, and hence is naturally phrased in the language of manifold optimization. Our proofs give a geometric characterization of the high-dimensional objective landscape, which shows that with high probability there are no spurious local minima. Experiments with synthetic data corroborate our theory. Full version of this paper is available online: \urlhttp://arxiv.org/abs/1504.06785.

Cite this Paper

BibTeX
@InProceedings{pmlr-v37-sund15, title = {Complete Dictionary Recovery Using Nonconvex Optimization}, author = {Sun, Ju and Qu, Qing and Wright, John}, booktitle = {Proceedings of the 32nd International Conference on Machine Learning}, pages = {2351--2360}, year = {2015}, editor = {Bach, Francis and Blei, David}, volume = {37}, series = {Proceedings of Machine Learning Research}, address = {Lille, France}, month = {07--09 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v37/sund15.pdf}, url = {https://proceedings.mlr.press/v37/sund15.html}, abstract = {We consider the problem of recovering a complete (i.e., square and invertible) dictionary mb A_0, from mb Y = mb A_0 mb X_0 with mb Y ∈\mathbb R^n \times p. This recovery setting is central to the theoretical understanding of dictionary learning. We give the first efficient algorithm that provably recovers mb A_0 when mb X_0 has O(n) nonzeros per column, under suitable probability model for mb X_0. Prior results provide recovery guarantees when mb X_0 has only O(\sqrtn) nonzeros per column. Our algorithm is based on nonconvex optimization with a spherical constraint, and hence is naturally phrased in the language of manifold optimization. Our proofs give a geometric characterization of the high-dimensional objective landscape, which shows that with high probability there are no spurious local minima. Experiments with synthetic data corroborate our theory. Full version of this paper is available online: \urlhttp://arxiv.org/abs/1504.06785.} }
Endnote
%0 Conference Paper %T Complete Dictionary Recovery Using Nonconvex Optimization %A Ju Sun %A Qing Qu %A John Wright %B Proceedings of the 32nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2015 %E Francis Bach %E David Blei %F pmlr-v37-sund15 %I PMLR %P 2351--2360 %U https://proceedings.mlr.press/v37/sund15.html %V 37 %X We consider the problem of recovering a complete (i.e., square and invertible) dictionary mb A_0, from mb Y = mb A_0 mb X_0 with mb Y ∈\mathbb R^n \times p. This recovery setting is central to the theoretical understanding of dictionary learning. We give the first efficient algorithm that provably recovers mb A_0 when mb X_0 has O(n) nonzeros per column, under suitable probability model for mb X_0. Prior results provide recovery guarantees when mb X_0 has only O(\sqrtn) nonzeros per column. Our algorithm is based on nonconvex optimization with a spherical constraint, and hence is naturally phrased in the language of manifold optimization. Our proofs give a geometric characterization of the high-dimensional objective landscape, which shows that with high probability there are no spurious local minima. Experiments with synthetic data corroborate our theory. Full version of this paper is available online: \urlhttp://arxiv.org/abs/1504.06785.
RIS
TY - CPAPER TI - Complete Dictionary Recovery Using Nonconvex Optimization AU - Ju Sun AU - Qing Qu AU - John Wright BT - Proceedings of the 32nd International Conference on Machine Learning DA - 2015/06/01 ED - Francis Bach ED - David Blei ID - pmlr-v37-sund15 PB - PMLR DP - Proceedings of Machine Learning Research VL - 37 SP - 2351 EP - 2360 L1 - http://proceedings.mlr.press/v37/sund15.pdf UR - https://proceedings.mlr.press/v37/sund15.html AB - We consider the problem of recovering a complete (i.e., square and invertible) dictionary mb A_0, from mb Y = mb A_0 mb X_0 with mb Y ∈\mathbb R^n \times p. This recovery setting is central to the theoretical understanding of dictionary learning. We give the first efficient algorithm that provably recovers mb A_0 when mb X_0 has O(n) nonzeros per column, under suitable probability model for mb X_0. Prior results provide recovery guarantees when mb X_0 has only O(\sqrtn) nonzeros per column. Our algorithm is based on nonconvex optimization with a spherical constraint, and hence is naturally phrased in the language of manifold optimization. Our proofs give a geometric characterization of the high-dimensional objective landscape, which shows that with high probability there are no spurious local minima. Experiments with synthetic data corroborate our theory. Full version of this paper is available online: \urlhttp://arxiv.org/abs/1504.06785. ER -
APA
Sun, J., Qu, Q. & Wright, J.. (2015). Complete Dictionary Recovery Using Nonconvex Optimization. Proceedings of the 32nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 37:2351-2360 Available from https://proceedings.mlr.press/v37/sund15.html.

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