The Minimum Volume Covering Ellipsoid Estimation in Kernel-Defined Feature Spaces

Abstract

Minimum volume covering ellipsoid estimation is important in areas such as systems identification, control, video tracking, sensor management, and novelty detection. It is well known that finding the minimum volume covering ellipsoid (MVCE) reduces to a convex optimisation problem. We propose a regularised version of the MVCE problem, and derive its dual formulation. This makes it possible to apply the MVCE problem in kernel-defined feature spaces. The solution is generally sparse, in the sense that the solution depends on a limited set of points. We argue that the MVCE is a valuable alternative to the minimum volume enclosing hypersphere for novelty detection. It is clearly a less conservative method. Besides this, we can show using statistical learning theory that the probability of a typical point being misidentified as a novelty is generally small. We illustrate our results on real data.