  In this paper, we study a remote source coding scenario in which binary phase
shift keying (BPSK) modulation sources are corrupted by additive white Gaussian
noise (AWGN). An intermediate node, such as a relay, receives these
observations and performs additional compression to balance complexity and
relevance. This problem can be further formulated as an information bottleneck
(IB) problem with Bernoulli sources and Gaussian mixture observations. However,
no closed-form solution exists for this IB problem. To address this challenge,
we propose a unified achievable scheme that employs three different
compression/quantization strategies for intermediate node processing by using
two-level quantization, multi-level deterministic quantization, and soft
quantization with the hyperbolic tangent ($\tanh$) function, respectively. In
addition, we extend our analysis to the vector mixture Gaussian observation
problem and explore its application in machine learning for binary
classification with information leakage. Numerical evaluations show that the
proposed scheme has a near-optimal performance over various signal-to-noise
ratios (SNRs), compared to the Blahut-Arimoto (BA) algorithm, and has better
performance than some existing numerical methods such as the information
dropout approach. Furthermore, experiments conducted on the realistic MNIST
dataset also validate the superior classification accuracy of our method
compared to the information dropout approach.
