Algebraic Reconstruction Bounds and Explicit Inversion for Phase Retrieval at the Identifiability Threshold

Franz Király, Martin Ehler
Proceedings of the Seventeenth International Conference on Artificial Intelligence and Statistics, PMLR 33:503-511, 2014.

Abstract

We study phase retrieval from magnitude measurements of an unknown signal as an algebraic estimation problem. Indeed, phase retrieval from rank-one and more general linear measurements can be treated in an algebraic way. It is verified that a certain number of generic rank-one or generic linear measurements are sufficient to enable signal reconstruction for generic signals, and slightly more generic measurements yield reconstructability for all signals. Our results solve few open problems stated in the recent literature. Furthermore, we show how the algebraic estimation problem can be solved by a closed-form algebraic estimation technique, termed ideal regression, providing non-asymptotic success guarantees.

Cite this Paper

BibTeX
@InProceedings{pmlr-v33-kiraly14, title = {{Algebraic Reconstruction Bounds and Explicit Inversion for Phase Retrieval at the Identifiability Threshold}}, author = {Király, Franz and Ehler, Martin}, booktitle = {Proceedings of the Seventeenth International Conference on Artificial Intelligence and Statistics}, pages = {503--511}, year = {2014}, editor = {Kaski, Samuel and Corander, Jukka}, volume = {33}, series = {Proceedings of Machine Learning Research}, address = {Reykjavik, Iceland}, month = {22--25 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v33/kiraly14.pdf}, url = {https://proceedings.mlr.press/v33/kiraly14.html}, abstract = {We study phase retrieval from magnitude measurements of an unknown signal as an algebraic estimation problem. Indeed, phase retrieval from rank-one and more general linear measurements can be treated in an algebraic way. It is verified that a certain number of generic rank-one or generic linear measurements are sufficient to enable signal reconstruction for generic signals, and slightly more generic measurements yield reconstructability for all signals. Our results solve few open problems stated in the recent literature. Furthermore, we show how the algebraic estimation problem can be solved by a closed-form algebraic estimation technique, termed ideal regression, providing non-asymptotic success guarantees.} }
Endnote
%0 Conference Paper %T Algebraic Reconstruction Bounds and Explicit Inversion for Phase Retrieval at the Identifiability Threshold %A Franz Király %A Martin Ehler %B Proceedings of the Seventeenth International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2014 %E Samuel Kaski %E Jukka Corander %F pmlr-v33-kiraly14 %I PMLR %P 503--511 %U https://proceedings.mlr.press/v33/kiraly14.html %V 33 %X We study phase retrieval from magnitude measurements of an unknown signal as an algebraic estimation problem. Indeed, phase retrieval from rank-one and more general linear measurements can be treated in an algebraic way. It is verified that a certain number of generic rank-one or generic linear measurements are sufficient to enable signal reconstruction for generic signals, and slightly more generic measurements yield reconstructability for all signals. Our results solve few open problems stated in the recent literature. Furthermore, we show how the algebraic estimation problem can be solved by a closed-form algebraic estimation technique, termed ideal regression, providing non-asymptotic success guarantees.
RIS
TY - CPAPER TI - Algebraic Reconstruction Bounds and Explicit Inversion for Phase Retrieval at the Identifiability Threshold AU - Franz Király AU - Martin Ehler BT - Proceedings of the Seventeenth International Conference on Artificial Intelligence and Statistics DA - 2014/04/02 ED - Samuel Kaski ED - Jukka Corander ID - pmlr-v33-kiraly14 PB - PMLR DP - Proceedings of Machine Learning Research VL - 33 SP - 503 EP - 511 L1 - http://proceedings.mlr.press/v33/kiraly14.pdf UR - https://proceedings.mlr.press/v33/kiraly14.html AB - We study phase retrieval from magnitude measurements of an unknown signal as an algebraic estimation problem. Indeed, phase retrieval from rank-one and more general linear measurements can be treated in an algebraic way. It is verified that a certain number of generic rank-one or generic linear measurements are sufficient to enable signal reconstruction for generic signals, and slightly more generic measurements yield reconstructability for all signals. Our results solve few open problems stated in the recent literature. Furthermore, we show how the algebraic estimation problem can be solved by a closed-form algebraic estimation technique, termed ideal regression, providing non-asymptotic success guarantees. ER -
APA
Király, F. & Ehler, M.. (2014). Algebraic Reconstruction Bounds and Explicit Inversion for Phase Retrieval at the Identifiability Threshold. Proceedings of the Seventeenth International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 33:503-511 Available from https://proceedings.mlr.press/v33/kiraly14.html.

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