Recovery Guarantee and Reconstruction Algorithms for 1-bit Compressive Sens- Ing

Compressive sensing is an emerging method for signal acquisition in which the number of samples ensuring exact reconstruction of the signal to be acquired is far less than the one in the conventional Nyquist sampling approach. In compressive sensing, the signal is acquired by means of few linear non-adaptive measurements, and then reconstructed by finding the sparsest solution via an l1-minimization. In the classic compressive sensing setup, each measurement outcome is described by a real value. In practice, for further processing and storage purposes, often the real-valued measurements need to be converted to finiteprecision numbers. 1-bit compressive sensing refers to the extreme case where the quantizer is a simple sign comparator and each measurement is represented using one bit only, i.e., +1 or −1. Several algorithms have been introduced in the literature for solving efficiently the reconstruction problem in the 1-bit compressive sensing setting, e.g., renormalized fixed point iteration (RFPI) and binary iterative hard thresholding (BIHT). However, these algorithms can not reconstruct the signal accurately when there is noise, i.e., bit flips, in the binary measurements. Adaptive outlier pursuit (AOP) is an algorithm which reconstructs the signal robustly against bit flips in the binary measurements. AOP requires the sparsity level of the signal to be reconstructed as an input. In many practical cases, however, the sparsity level of the signal is unknown and time variant. In this thesis, we address reconstruction problem in 1-bit compressive sensing. We introduce a new algorithm for 1-bit compressive sensing which reconstructs the signal robustly from the noisy binary measurements. This new reconstruction algorithm does not require the sparsity level of the signal as an input. Therefore, our algorithm can be applied in the most practical scenarios in which the sparsity level of the signal is unknown.