Finite-Sample Linear Filter Optimization in Wireless Communications and Financial Systems

We study the problem of linear filter optimization with finite sample size, which has wide applications such as beamformer design in wireless communications and portfolio optimization in finance. Traditional methods in both fields are not robust against the imprecise channel vector and the noise covariance matrix (or the mean return and the covariance of assets in finance) due to finite sample size. We consider estimation errors both in the channel vector and the noise covariance matrix (or the mean return and the covariance) simultaneously. We resort to high-dimensional asymptotics to account for the fact that the observation dimension is of the same order of magnitude as the number of samples, and use the diagonal loading method (or the shrinkage estimator) to improve the robustness. The channel vector (or mean return) and the noise covariance matrix are estimated from the training data, and then corrected under several widely-used criteria. In an asymptotic setting where the number of samples is comparable to the observation dimension, we obtain linear filters that are as good as the optimal filters with a shrinkage structure and a perfect channel vector (or mean return) under different criteria. Monte Carlo simulations show the advantage of our linear filters in the finite sample size regime.