  We introduce Cayley transform ellipsoid fitting (CTEF), an algorithm that
uses the Cayley transform to fit ellipsoids to noisy data in any dimension.
Unlike many ellipsoid fitting methods, CTEF is ellipsoid specific, meaning it
always returns elliptic solutions, and can fit arbitrary ellipsoids. It also
significantly outperforms other fitting methods when data are not uniformly
distributed over the surface of an ellipsoid. Inspired by growing calls for
interpretable and reproducible methods in machine learning, we apply CTEF to
dimension reduction, data visualization, and clustering in the context of cell
cycle and circadian rhythm data and several classical toy examples. Since CTEF
captures global curvature, it extracts nonlinear features in data that other
machine learning methods fail to identify. For example, on the clustering
examples CTEF outperforms 10 popular algorithms.
