Directed Principal Component Analysis

We consider a problem involving estimation of a high-dimensional covariance matrix that is the sum of a diagonal matrix and a low-rank matrix, and making a decision based on the resulting estimate. Such problems arise, for example, in portfolio management, where a common approach employs principal component analysis (PCA) to estimate factors used in constructing the low-rank term of the covariance matrix. The decision problem is typically treated separately, with the estimated covariance matrix taken to be an input to an optimization problem. We propose directed PCA , an efficient algorithm that takes the decision objective into account when estimating the covariance matrix. Directed PCA effectively adjusts factors that would be produced by PCA so that they better guide the specific decision at hand. We demonstrate through computational studies that directed PCA yields significant benefit, and we prove theoretical results establishing that the degree of improvement over conventional PCA can be arbitrarily large.