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# Efficient Algorithms for Robust One-bit Compressive Sensing

*Proceedings of the 31st International Conference on Machine Learning*, PMLR 32(2):820-828, 2014.

#### Abstract

While the conventional compressive sensing assumes measurements of infinite precision, one-bit compressive sensing considers an extreme setting where each measurement is quantized to just a single bit. In this paper, we study the vector recovery problem from noisy one-bit measurements, and develop two novel algorithms with formal theoretical guarantees. First, we propose a passive algorithm, which is very efficient in the sense it only needs to solve a convex optimization problem that has a closed-form solution. Despite the apparent simplicity, our theoretical analysis reveals that the proposed algorithm can recover both the exactly sparse and the approximately sparse vectors. In particular, for a sparse vector with s nonzero elements, the sample complexity is O(s \log n/ε^2), where n is the dimensionality and εis the recovery error. This result improves significantly over the previously best known sample complexity in the noisy setting, which is O(s \log n/ε^4). Second, in the case that the noise model is known, we develop an adaptive algorithm based on the principle of active learning. The key idea is to solicit the sign information only when it cannot be inferred from the current estimator. Compared with the passive algorithm, the adaptive one has a lower sample complexity if a high-precision solution is desired.