Gaussian Approximation of Quantization Error for Estimation from Compressed Data
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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.
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