  The one-bit compressed sensing framework aims to reconstruct a sparse signal
by only using the sign information of its linear measurements. To compensate
for the loss of scale information, past studies in the area have proposed
recovering the signal by imposing an additional constraint on the L2-norm of
the signal. Recently, an alternative strategy that captures scale information
by introducing a threshold parameter to the quantization process was advanced.
In this paper, we analyze the typical behavior of the thresholding 1-bit
compressed sensing utilizing the replica method of statistical mechanics, so as
to gain an insight for properly setting the threshold value. Our result shows
that, fixing the threshold at a constant value yields better performance than
varying it randomly when the constant is optimally tuned, statistically.
Unfortunately, the optimal threshold value depends on the statistical
properties of the target signal, which may not be known in advance. In order to
handle this inconvenience, we develop a heuristic that adaptively tunes the
threshold parameter based on the frequency of positive (or negative) values in
the binary outputs. Numerical experiments show that the heuristic exhibits
satisfactory performance while incurring low computational cost.
