Sampling and Distortion Tradeoffs for Indirect Source Retrieval

Consider a continuous signal that cannot be observed directly. Instead, one has access to multiple corrupted versions of the signal. The available corrupted signals are correlated because they carry information about the common remote signal. The goal is to reconstruct the original signal from the data collected from its corrupted versions. Known as the indirect or remote reconstruction problem, it has been mainly studied in the literature from an information theoretic perspective. A variant of this problem for a class of Gaussian signals, known as the “Gaussian CEO problem,” has received particular attention; for example, it has been shown that the problem of recovering the remote signal is equivalent with the problem of recovering the set of corrupted signals (separation principle). The information theoretic formulation of the remote reconstruction problem assumes that the corrupted signals are uniformly sampled and the focus is on optimal compression of the samples. On the other hand, in this paper, we revisit this problem from a sampling perspective. More specifically, assuming restrictions on the sampling rate from each corrupted signal, we look at the problem of finding the best sampling locations for each signal to minimize the total reconstruction distortion of the remote signal. In finding the sampling locations, one can take advantage of the correlation among the corrupted signals. The statistical model of the original signal and its corrupted versions adopted in this paper are similar to the one considered for the Gaussian CEO problem; i.e., we restrict to a class of Gaussian signals. Our main contribution is a fundamental lower bound on the reconstruction distortion for any arbitrary nonuniform sampling strategy. This lower bound is valid for any sampling rate. Furthermore, it is tight and matches the optimal reconstruction distortion in low and high sampling rates. Moreover, it is shown that in the low sampling rate region, it is optimal to use a certain nonuniform sampling scheme on all the signals. On the other hand, in the high sampling rate region, it is optimal to uniformly sample all the signals. We also consider the problem of finding the optimal sampling locations to recover the set of corrupted signals, rather than the remote signal. Unlike the information theoretic formulation of the problem in which these two problems were equivalent, we show that they are not equivalent in our setting. Finally, another contribution of this paper is a new reverse majorization inequality that might be of independent interest.