Described is using semi-Riemannian geometry in supervised learning to learn a discriminant subspace for classification, e.g., labeled samples are used to learn the geometry of a semi-Riemannian submanifold. For a given sample, the K nearest classes of that sample are determined, along with the nearest samples that are in other classes, and the nearest samples in that sample's same class. The distances between these samples are computed, and used in computing a metric matrix. The metric matrix is used to compute a projection matrix that corresponds to the discriminant subspace. In online classification, as a new sample is received, it is projected into a feature space by use of the projection matrix and classified accordingly.