Monday, June 08, 2015

Mismatched Estimation in Large Linear Systems

From Dror's Information Theoretic Results in Compressed Sensing research page:

Mismatched estimation in compressed sensing: Many scientific and engineering problems can be approximated as linear systems, where the input signal is modeled as a vector, and each observation is a linear combination of the entries in the input vector corrupted by white Gaussian noise. In joint work with Yanting Ma and Ahmad Beirami, the input signal is modeled as a realization of a vector of independent and identically distributed random variables. The goal is to estimate the input signal such that the mean square error (MSE), which is the Euclidean distance between the estimated signal and the true input signal averaged over all possible realizations of the input and the observation, is minimized. It is well-known that the best possible MSE, minimum mean square error (MMSE), can be achieved by computing conditional expectation, which is the mean or average value of the input given the observation vector, where the true distribution of the input is used. However, the true distribution is usually not known exactly in practice, and so conditional expectation is computed with a postulated distribution that differs from the true distribution; we call this procedure mismatched estimation, and it yields an MSE that is higher than the MMSE. We are interested in characterizing the excess MSE (EMSE) above the MMSE due to mismatched estimation in large linear systems, where the length of the input and the number of observations grow to infinity together, and their ratio is fixed......"
The rest on this page and the attendant paper is: Mismatched Estimation in Large Linear Systems by Yanting Ma, Dror Baron, Ahmad Beirami

We study the excess mean square error (EMSE) above the minimum mean square error (MMSE) in large linear systems where the posterior mean estimator (PME) is evaluated with a postulated prior that differs from the true prior of the input signal. We focus on large linear systems where the measurements are acquired via an independent and identically distributed random matrix, and are corrupted by additive white Gaussian noise (AWGN). The relationship between the EMSE in large linear systems and EMSE in scalar channels is derived, and closed form approximations are provided. Our analysis is based on the decoupling principle, which links scalar channels to large linear system analyses. Numerical examples demonstrate that our closed form approximations are accurate.

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