Mean-Squared Error Estimation for Linear Systems with Block Circulant Uncertainty

TLDR: A minimax mean-squared error (MSE) approach in which the linear estimator that minimizes the worst-case MSE over a BC structured uncertainty region is sought and it is shown that the minimax MSE estimator can be formulated as a solution to a semidefinite programming problem (SDP), which can be solved efficiently.

Abstract: We consider the problem of estimating a vector ${\bf x}$ in the linear model ${\bf A}{\bf x} \approx {\bf y}$, where ${\bf A}$ is a block circulant (BC) matrix with $N$ blocks and ${\bf x}$ is assumed to have a weighted norm bound. In the case where both ${\bf A}$ and ${\bf y}$ are subjected to noise, we propose a minimax mean-squared error (MSE) approach in which we seek the linear estimator that minimizes the worst-case MSE over a BC structured uncertainty region. For an arbitrary choice of weighting, we show that the minimax MSE estimator can be formulated as a solution to a semidefinite programming problem (SDP), which can be solved efficiently. For a Euclidean norm bound on ${\bf x}$, the SDP is reduced to a simple convex program with $N+1$ unknowns. Finally, we demonstrate through an image deblurring example the potential of the minimax MSE approach in comparison with other conventional methods.