On Distributed Sampling for Mismatched Estimation of Remote Sources

In this work, we study the problem of distributed sampling for the recovery of a remote source under information mismatch at the estimator. In particular, a centralized estimator seeks to estimate a remote Gaussian random signal, where unlike in the ‘classical’ estimation setup, we assume that the estimator has a fixed, unknown mismatch vis-à-vis source statistics, in particular, the source covariance matrix. Such a mismatched estimator deploys multiple samplers in the field, where each sampler observes an independently noise corrupted version of the remote source and then forwards its sampled version to the estimator. The estimator has a fixed limit on the number of samples it can concurrently process; given such a total sampling budget, it seeks to distribute these samples optimally among samplers so as to obtain a reasonably high fidelity sampled noisy observation of the remote source via the samplers. Using this sampled data, the mismatched estimator then outputs a source estimate which minimizes distortion (i.e., the overall mean squared error).Our principal goal in this work is to understand the distortion-versus-sampling rate trade-off for the mismatched Gaussian source estimation problem under general distributed configurations. In the high-rate sampling regime, where the estimator has a ‘large’ sampling budget and essentially every sampler can operate at ‘high’ sampling rate, we show the interesting result that for a wide range of parameters, the optimal distributed sampling strategy is a uniform sampling strategy but one which, interestingly, does not depend on the mismatch at the estimator. We also characterize the optimal distortion, which we show does indeed depend on the degree of mismatch. Our results also bring to the fore an interesting phenomenon where the optimal distortion behaves asymmetrically w.r.t. the nature of mismatch, i.e., even for identical mismatch magnitude, the distortion is significantly different depending on the sign of the mismatch.