  The Binary Iterative Hard Thresholding (BIHT) algorithm is a popular
reconstruction method for one-bit compressed sensing due to its simplicity and
fast empirical convergence. There have been several works about BIHT but a
theoretical understanding of the corresponding approximation error and
convergence rate still remains open.
  This paper shows that the normalized version of BIHT (NBHIT) achieves an
approximation error rate optimal up to logarithmic factors. More precisely,
using $m$ one-bit measurements of an $s$-sparse vector $x$, we prove that the
approximation error of NBIHT is of order $O \left(1 \over m \right)$ up to
logarithmic factors, which matches the information-theoretic lower bound
$\Omega \left(1 \over m \right)$ 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 $O \left(1 \over m^{1/2} \right)$.
