Gaussian Approximation of Quantization Error for Estimation from Compressed Data

Abstract

We consider the distributional connection between the lossy compressed 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 the Wasserstein distance between a bitrate-R compressed version of X and its observation under an AWGN-channel of signal-to-noise ratio 2 2R −1 is sub-linear in the problem dimension. We utilize this fact to connect the risk of an estimator based on an AWGN-corrupted version of X to the risk attained by the same estimator when fed with its bitrate-R quantized version. We demonstrate the usefulness of this connection by deriving various novel results for inference problems under compression constraints, including noisy source coding and limited-bitrate parameter estimation.

Tracking an Auto-Regressive Process with Limited Communication per Unit Time

“…Note that after the first round of quantization, the error vector need not be Gaussian, and the analysis in [ 39 ] can only be applied after showing a closeness of the error vector distribution to Gaussian in the Wasserstein distance of order 2. While the original proof [ 39 ] overlooks this technical point, this gap can be filled using a recent result from [ 40 ] if spherical codes are used. However, we follow an alternative approach and show a direct analysis using vector quantizers.…”