Minimum Distortion Quantization with Specified Output Distribution

The paper presents a method for designing optimal quantizers of real-valued random variables that not only minimize the minimum mean squared error (MMSE) for estimating the original variable but also conform to a specified output distribution. The quantizer is defined as \(X = \sigma(F_{\sigma^{-1}(X)}^{-1}(F_W(W)))\), where \(\sigma\) is the optimal permutation for minimizing MMSE, and the findings leverage the concept of majorization for proof of optimality. This work is significant for practitioners as it enables the design of quantizers with controlled output entropy and tailored distributions, which can enhance performance in communication systems and data anonymization tasks.