  Within the realm of rapidly advancing wireless sensor networks (WSNs),
distributed detection assumes a significant role in various practical
applications. However, critical challenge lies in maintaining robust detection
performance while operating within the constraints of limited bandwidth and
energy resources. This paper introduces a novel approach that combines
model-driven deep learning (DL) with binary quantization to strike a balance
between communication overhead and detection performance in WSNs. We begin by
establishing the lower bound of detection error probability for distributed
detection using the maximum a posteriori (MAP) criterion. Furthermore, we prove
the global optimality of employing identical local quantizers across sensors,
thereby maximizing the corresponding Chernoff information. Subsequently, the
paper derives the minimum MAP detection error probability (MAPDEP) by
inplementing identical binary probabilistic quantizers across the sensors.
Moreover, the paper establishes the equivalence between utilizing all quantized
data and their average as input to the detector at the fusion center (FC). In
particular, we derive the Kullback-Leibler (KL) divergence, which measures the
difference between the true posterior probability and output of the proposed
detector. Leveraging the MAPDEP and KL divergence as loss functions, the paper
proposes model-driven DL method to separately train the probability controller
module in the quantizer and the detector module at the FC. Numerical results
validate the convergence and effectiveness of the proposed method, which
achieves near-optimal performance with reduced complexity for Gaussian
hypothesis testing.
