MVEE

MVEE stands for minimum volume enclosing ellipsoid. It is the smallest ellipsoid that contains a given set of points in real space and is a common tool in data analysis and geometry. For a finite set of points x_i in R^d, the MVEE finds a center c in R^d and a positive definite matrix A in R^{d×d} such that the ellipsoid E = { x : (x − c)^T A (x − c) ≤ 1 } contains all the points and its volume is minimized. The volume of such an ellipsoid is proportional to det(A)^{-1/2}, so the optimization seeks to maximize det(A) subject to the containment constraints.

The problem is convex and can be formulated as a semidefinite program. A widely used practical approach is Khachiyan's algorithm, an iterative method that updates a weight vector associated with the points and reconstructs the center and shape of the ellipsoid from these weights. Other solution methods include interior-point techniques and fast approximate algorithms, especially for high-dimensional or large-point sets.

Applications of the MVEE include robust data normalization, outlier detection, pattern recognition, and preprocessing in machine learning to improve conditioning or reduce dimensionality. It also serves in computer graphics, computer vision, and geographic information systems as a bounding ellipsoid for point clouds or convex hulls. Variants and extensions address robustness to outliers, weighting schemes, and computational efficiency in large-scale problems.