Gaussian Approximation of Quantization Error for Estimation from Compressed Data

We consider the statistical connection between the quantized representation of a high dimensional signal X using a random spherical code and the observation of X under an additive white Gaussian noise (AWGN). We show that given X, the conditional Wasserstein distance between its bitrate-R quantized version and its observation under AWGN of signal-to-noise ratio 2
^2R
- 1 is sub-linear in the problem dimension. We then utilize this fact to connect the mean squared error (MSE) attained by an estimator based on an AWGN-corrupted version of X to the MSE attained by the same estimator when fed with its bitrate-R quantized version.