{"id":164990,"date":"2013-06-01T00:00:00","date_gmt":"2013-06-01T00:00:00","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/one-bit-compressed-sensing-provable-support-and-vector-recovery\/"},"modified":"2018-10-16T21:22:54","modified_gmt":"2018-10-17T04:22:54","slug":"one-bit-compressed-sensing-provable-support-and-vector-recovery","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/one-bit-compressed-sensing-provable-support-and-vector-recovery\/","title":{"rendered":"One-bit Compressed Sensing: Provable Support and Vector Recovery"},"content":{"rendered":"<div class=\"asset-content\">\n<p>In this paper, we study the problem of onebit compressed sensing (1-bit CS), where the goal is to design a measurement matrix A and a recovery algorithm such that a k-sparse unit vector x\u2217 can be efficiently recovered from the sign of its linear measurements, i.e., b = sign(Ax\u2217). This is an important problem for signal acquisition and has several learning applications as well, e.g., multi-label classification (Hsu et al., 2009). We study this problem in two settings: a) support recovery: recover the support of x\u2217, b) approximate vector recovery: recover a unit vector hat x such that ||hat x &#8211; x*||\u2264 \u03b5. For support recovery, we propose two novel and efficient solutions based on two combinatorial structures: union free families 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 all the signals. For approximate recovery, we propose the first method to recover a sparse vector using a near optimal number of measurements. We also empirically validate our algorithms and demonstrate that our algorithms recover the true signal using fewer measurements than the existing methods.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>In this paper, we study the problem of onebit compressed sensing (1-bit CS), where the goal is to design a measurement matrix A and a recovery algorithm such that a k-sparse unit vector x\u2217 can be efficiently recovered from the sign of its linear measurements, i.e., b = sign(Ax\u2217). This is an important problem for [&hellip;]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","footnotes":""},"msr-content-type":[3],"msr-research-highlight":[],"research-area":[13556],"msr-publication-type":[193716],"msr-product-type":[],"msr-focus-area":[],"msr-platform":[],"msr-download-source":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-164990","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-artificial-intelligence","msr-locale-en_us"],"msr_publishername":"Journal of Machine Learning Research","msr_edition":"International Conference on Machine Learning 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