Maximin Correlation

This paper is concerned with the problem of finding a vector from a given set in a Euclidean space that maximizes the worst-case correlation with vectors in this set, where ‘worst’ means smallest. This problem, called the maximin correlation problem (MCP), comes up in several disciplines including pattern classification, portfolio selection, statistics, and signal processing. With a general infinite set, the associated MCP is a semi-infinite program and so difficult to solve exactly. As an important tractable case, we show that the MCP with an ellipsoid or, more generally, the union of finitely many ellipsoids can be solved efficiently using an iterative method which alternates between optimization and worst-case correlation analysis. The optimization step is to solve an MCP with a finite set of sampled points from the set, and the worst-case correlation analysis step is to find a point in the set that minimizes the correlation with the point found at the preceding optimization step. The optimization step and the worst-case correlation analysis step can be reformulated as a second-order cone program (SOCP) and a semidefinite program (SDP), respectively, each of which can be solved with great efficiency using interior-point methods. Combined with a technique that approximates a general set with the union of finitely many ellipsoids, the iterative method can approximately solve the MCP with a general non-ellipsoidal set.