One-Bit Compressed Sensing: Provable Support and Vector Recovery

Sivakant Gopi, Praneeth Netrapalli, Prateek Jain, Aditya Nori
Proceedings of the 30th International Conference on Machine Learning, PMLR 28(3):154-162, 2013.

Abstract

In this paper, we study the problem of one-bit compressed sensing (1-bit CS), where the goal is to design a measurement matrix A and a recovery algorithm s.t. a k-sparse vector \x^* can be efficiently recovered back from signed linear measurements, i.e., b=\sign(A\x^*). This is an important problem in the signal acquisition area and has several learning applications as well, e.g., multi-label classification \citeHsuKLZ10. We study this problem in two settings: a) support recovery: recover \supp(\x^*), b) approximate vector recovery: recover a unit vector \hx s.t. || \hatx-\x^*/||\x^*|| ||_2≤ε. For support recovery, we propose two novel and efficient solutions based on two combinatorial structures: union free family of sets and expanders. In contrast to existing methods for support recovery, our methods are universal i.e. a single measurement matrix A can recover almost all the signals. For approximate recovery, we propose the first method to recover sparse vector using a near optimal number of measurements. We also empirically demonstrate effectiveness of our algorithms; we show that our algorithms are able to recover signals with smaller number of measurements than several existing methods.

Cite this Paper

BibTeX
@InProceedings{pmlr-v28-gopi13, title = {One-Bit Compressed Sensing: Provable Support and Vector Recovery}, author = {Gopi, Sivakant and Netrapalli, Praneeth and Jain, Prateek and Nori, Aditya}, booktitle = {Proceedings of the 30th International Conference on Machine Learning}, pages = {154--162}, year = {2013}, editor = {Dasgupta, Sanjoy and McAllester, David}, volume = {28}, number = {3}, series = {Proceedings of Machine Learning Research}, address = {Atlanta, Georgia, USA}, month = {17--19 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v28/gopi13.pdf}, url = {https://proceedings.mlr.press/v28/gopi13.html}, abstract = {In this paper, we study the problem of one-bit compressed sensing (1-bit CS), where the goal is to design a measurement matrix A and a recovery algorithm s.t. a k-sparse vector \x^* can be efficiently recovered back from signed linear measurements, i.e., b=\sign(A\x^*). This is an important problem in the signal acquisition area and has several learning applications as well, e.g., multi-label classification \citeHsuKLZ10. We study this problem in two settings: a) support recovery: recover \supp(\x^*), b) approximate vector recovery: recover a unit vector \hx s.t. || \hatx-\x^*/||\x^*|| ||_2≤ε. For support recovery, we propose two novel and efficient solutions based on two combinatorial structures: union free family of sets and expanders. In contrast to existing methods for support recovery, our methods are universal i.e. a single measurement matrix A can recover almost all the signals. For approximate recovery, we propose the first method to recover sparse vector using a near optimal number of measurements. We also empirically demonstrate effectiveness of our algorithms; we show that our algorithms are able to recover signals with smaller number of measurements than several existing methods. } }
Endnote
%0 Conference Paper %T One-Bit Compressed Sensing: Provable Support and Vector Recovery %A Sivakant Gopi %A Praneeth Netrapalli %A Prateek Jain %A Aditya Nori %B Proceedings of the 30th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2013 %E Sanjoy Dasgupta %E David McAllester %F pmlr-v28-gopi13 %I PMLR %P 154--162 %U https://proceedings.mlr.press/v28/gopi13.html %V 28 %N 3 %X In this paper, we study the problem of one-bit compressed sensing (1-bit CS), where the goal is to design a measurement matrix A and a recovery algorithm s.t. a k-sparse vector \x^* can be efficiently recovered back from signed linear measurements, i.e., b=\sign(A\x^*). This is an important problem in the signal acquisition area and has several learning applications as well, e.g., multi-label classification \citeHsuKLZ10. We study this problem in two settings: a) support recovery: recover \supp(\x^*), b) approximate vector recovery: recover a unit vector \hx s.t. || \hatx-\x^*/||\x^*|| ||_2≤ε. For support recovery, we propose two novel and efficient solutions based on two combinatorial structures: union free family of sets and expanders. In contrast to existing methods for support recovery, our methods are universal i.e. a single measurement matrix A can recover almost all the signals. For approximate recovery, we propose the first method to recover sparse vector using a near optimal number of measurements. We also empirically demonstrate effectiveness of our algorithms; we show that our algorithms are able to recover signals with smaller number of measurements than several existing methods.
RIS
TY - CPAPER TI - One-Bit Compressed Sensing: Provable Support and Vector Recovery AU - Sivakant Gopi AU - Praneeth Netrapalli AU - Prateek Jain AU - Aditya Nori BT - Proceedings of the 30th International Conference on Machine Learning DA - 2013/05/26 ED - Sanjoy Dasgupta ED - David McAllester ID - pmlr-v28-gopi13 PB - PMLR DP - Proceedings of Machine Learning Research VL - 28 IS - 3 SP - 154 EP - 162 L1 - http://proceedings.mlr.press/v28/gopi13.pdf UR - https://proceedings.mlr.press/v28/gopi13.html AB - In this paper, we study the problem of one-bit compressed sensing (1-bit CS), where the goal is to design a measurement matrix A and a recovery algorithm s.t. a k-sparse vector \x^* can be efficiently recovered back from signed linear measurements, i.e., b=\sign(A\x^*). This is an important problem in the signal acquisition area and has several learning applications as well, e.g., multi-label classification \citeHsuKLZ10. We study this problem in two settings: a) support recovery: recover \supp(\x^*), b) approximate vector recovery: recover a unit vector \hx s.t. || \hatx-\x^*/||\x^*|| ||_2≤ε. For support recovery, we propose two novel and efficient solutions based on two combinatorial structures: union free family of sets and expanders. In contrast to existing methods for support recovery, our methods are universal i.e. a single measurement matrix A can recover almost all the signals. For approximate recovery, we propose the first method to recover sparse vector using a near optimal number of measurements. We also empirically demonstrate effectiveness of our algorithms; we show that our algorithms are able to recover signals with smaller number of measurements than several existing methods. ER -
APA
Gopi, S., Netrapalli, P., Jain, P. & Nori, A.. (2013). One-Bit Compressed Sensing: Provable Support and Vector Recovery. Proceedings of the 30th International Conference on Machine Learning, in Proceedings of Machine Learning Research 28(3):154-162 Available from https://proceedings.mlr.press/v28/gopi13.html.

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