One-Bit Compressed Sensing: Provable Support and Vector Recovery

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.

Authors

• Sivakanth Gopi
• Praneeth Netrapalli
• Prateek Jain 0002
• Aditya V. Nori

Keywords

No keywords are indexed for this paper.