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One-Bit Compressed Sensing: Provable Support and Vector Recovery

Sivakant Gopi, Praneeth Netrapalli, Prateek Jain, Aditya Nori
JMLR W&CP 28 (3) : 154–162, 2013


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 . We study this problem in two settings: a) support recovery: recover \(\supp(\x^*)\), b) approximate vector recovery: recover a unit vector \(\hx\) s.t. \(|| \hat{x}-\x^*/||\x^*|| ||_2\leq \epsilon\). 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.

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