Computing minimum-volume enclosing ellipsoids

For a given multidimensional data set (point cloud), we investigate methods for computing the minimum-volume enclosing ellipsoid (MVEE), which provides an efficient representation of the data that is useful in many applications, including data analysis, optimal design, and computational geometry. Contrary to conventional wisdom, we demonstrate that careful exploitation of problem structure can enable high-order (Newton and Newton-like) methods with superlinear convergence rates to scale to very large MVEE problems. We also introduce a hybrid method that combines the benefits of both high-order and low-order methods, along with new initialization schemes that further enhance performance. Observing that computational cost depends significantly on the particular distribution of the data, we demonstrate that kurtosis serves as an excellent indicator of problem difficulty and provides useful guidance in choosing an appropriate solution algorithm and initialization.