NBIHT: An Efficient Algorithm for 1-Bit Compressed Sensing With Optimal Error Decay Rate

Michael P. Friedlander, Halyun Jeong, Yaniv Plan, Özgür Yilmaz

Abstract: The <i>Binary Iterative Hard Thresholding</i> (BIHT) algorithm is a popular reconstruction method for one-bit compressed sensing due to its simplicity and fast empirical convergence. Despite considerable research on this algorithm, a theoretical understanding of the corresponding approximation error and convergence rate still remains an open problem. This paper shows that the normalized version of BIHT (NBIHT) achieves an approximation error rate optimal up to logarithmic factors. More precisely, using <inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> one-bit measurements of an <inline-formula> <tex-math notation="LaTeX">$s$ </tex-math></inline-formula>-sparse vector <inline-formula> <tex-math notation="LaTeX">$x$ </tex-math></inline-formula>, we prove that the approximation error of NBIHT is of order <inline-formula> <tex-math notation="LaTeX">$O \left ({\frac{1 }{ m }}\right)$ </tex-math></inline-formula> up to logarithmic factors, which matches the information-theoretic lower bound <inline-formula> <tex-math notation="LaTeX">$\Omega \left ({\frac{1 }{ m }}\right)$ </tex-math></inline-formula> proved by Jacques, Laska, Boufounos, and Baraniuk in 2013. To our knowledge, this is the first theoretical analysis of a BIHT-type algorithm that explains the optimal rate of error decay empirically observed in the literature. This also makes NBIHT the first provable computationally-efficient one-bit compressed sensing algorithm that breaks the inverse square-root error decay rate <inline-formula> <tex-math notation="LaTeX">$O \left ({\frac{1 }{ m^{1/2} }}\right)\vphantom {{\left ({\frac{1 }{ m^{1/2} }}\right)}^{'}}$ </tex-math></inline-formula>.