  One-bit compressive sensing (CS) is an advanced version of sparse recovery in
which the sparse signal of interest can be recovered from extremely quantized
measurements. Namely, only the sign of each measurement is available to us. In
many applications, the ground-truth signal is not sparse itself, but can be
represented in a redundant dictionary. A strong line of research has addressed
conventional CS in this signal model including its extension to one-bit
measurements. However, one-bit CS suffers from the extremely large number of
required measurements to achieve a predefined reconstruction error level. A
common alternative to resolve this issue is to exploit adaptive schemes.
Adaptive sampling acts on the acquired samples to trace the signal in an
efficient way. In this work, we utilize an adaptive sampling strategy to
recover dictionary-sparse signals from binary measurements. For this task, a
multi-dimensional threshold is proposed to incorporate the previous signal
estimates into the current sampling procedure. This strategy substantially
reduces the required number of measurements for exact recovery. Our proof
approach is based on the recent tools in high dimensional geometry in
particular random hyperplane tessellation and Gaussian width. We show through
rigorous and numerical analysis that the proposed algorithm considerably
outperforms state of the art approaches. Further, our algorithm reaches an
exponential error decay in terms of the number of quantized measurements.
