Learning classifier decision boundaries and an informative two-dimensional linear subspace jointly through variational level set methods and optimization on the Stiefel manifold. The two classes of measurements, the black + markers and the magenta x markers are eight-dimensional. Only the first two dimensions are informative for classification: the points are separable by an ellipse in those two dimensions. The other six dimensions are independent Gaussian noise. Starting from a random initialization for a two-dimensional subspace, and a poor initialization for a decision boundary, the method converges to the plane of the first two dimensions and an elliptical decision boundary that separates the two classes. The blue line is the decision boundary. The green polygon outlines the projection of an eight-dimensional hypercube onto two dimensions using the current dimensionality reduction mapping. When aligned to two dimensions, the polygon is a square. Visit http://lids.mit.edu/research/project-highlights/classification-and-dimensionality.html for more information.