Phase Retrieval Meets Statistical Learning Theory: A Flexible Convex Relaxation

Sohail Bahmani, Justin Romberg
Proceedings of the 20th International Conference on Artificial Intelligence and Statistics, PMLR 54:252-260, 2017.

Abstract

We propose a flexible convex relaxation for the phase retrieval problem that operates in the natural domain of the signal. Therefore, we avoid the prohibitive computational cost associated with “lifting” and semidefinite programming (SDP) in methods such as PhaseLift and compete with recently developed non-convex techniques for phase retrieval. We relax the quadratic equations for phaseless measurements to inequality constraints each of which representing a symmetric “slab”. Through a simple convex program, our proposed estimator finds an extreme point of the intersection of these slabs that is best aligned with a given anchor vector. We characterize geometric conditions that certify success of the proposed estimator. Furthermore, using classic results in statistical learning theory, we show that for random measurements the geometric certificates hold with high probability at an optimal sample complexity. Phase transition of our estimator is evaluated through simulations. Our numerical experiments also suggest that the proposed method can solve phase retrieval problems with coded diffraction measurements as well.

Cite this Paper

BibTeX
@InProceedings{pmlr-v54-bahmani17a, title = {{Phase Retrieval Meets Statistical Learning Theory: A Flexible Convex Relaxation}}, author = {Bahmani, Sohail and Romberg, Justin}, booktitle = {Proceedings of the 20th International Conference on Artificial Intelligence and Statistics}, pages = {252--260}, year = {2017}, editor = {Singh, Aarti and Zhu, Jerry}, volume = {54}, series = {Proceedings of Machine Learning Research}, month = {20--22 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v54/bahmani17a/bahmani17a.pdf}, url = {https://proceedings.mlr.press/v54/bahmani17a.html}, abstract = {We propose a flexible convex relaxation for the phase retrieval problem that operates in the natural domain of the signal. Therefore, we avoid the prohibitive computational cost associated with “lifting” and semidefinite programming (SDP) in methods such as PhaseLift and compete with recently developed non-convex techniques for phase retrieval. We relax the quadratic equations for phaseless measurements to inequality constraints each of which representing a symmetric “slab”. Through a simple convex program, our proposed estimator finds an extreme point of the intersection of these slabs that is best aligned with a given anchor vector. We characterize geometric conditions that certify success of the proposed estimator. Furthermore, using classic results in statistical learning theory, we show that for random measurements the geometric certificates hold with high probability at an optimal sample complexity. Phase transition of our estimator is evaluated through simulations. Our numerical experiments also suggest that the proposed method can solve phase retrieval problems with coded diffraction measurements as well.} }
Endnote
%0 Conference Paper %T Phase Retrieval Meets Statistical Learning Theory: A Flexible Convex Relaxation %A Sohail Bahmani %A Justin Romberg %B Proceedings of the 20th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2017 %E Aarti Singh %E Jerry Zhu %F pmlr-v54-bahmani17a %I PMLR %P 252--260 %U https://proceedings.mlr.press/v54/bahmani17a.html %V 54 %X We propose a flexible convex relaxation for the phase retrieval problem that operates in the natural domain of the signal. Therefore, we avoid the prohibitive computational cost associated with “lifting” and semidefinite programming (SDP) in methods such as PhaseLift and compete with recently developed non-convex techniques for phase retrieval. We relax the quadratic equations for phaseless measurements to inequality constraints each of which representing a symmetric “slab”. Through a simple convex program, our proposed estimator finds an extreme point of the intersection of these slabs that is best aligned with a given anchor vector. We characterize geometric conditions that certify success of the proposed estimator. Furthermore, using classic results in statistical learning theory, we show that for random measurements the geometric certificates hold with high probability at an optimal sample complexity. Phase transition of our estimator is evaluated through simulations. Our numerical experiments also suggest that the proposed method can solve phase retrieval problems with coded diffraction measurements as well.
APA
Bahmani, S. & Romberg, J.. (2017). Phase Retrieval Meets Statistical Learning Theory: A Flexible Convex Relaxation. Proceedings of the 20th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 54:252-260 Available from https://proceedings.mlr.press/v54/bahmani17a.html.

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