Adaptive Metric Dimensionality Reduction

Linear classifiers play a central role in supervised learning, with a rich and elegant theory. This setting assumes data is represented as points in a Hilbert space, either explicitly as feature vectors or implicitly via a kernel. A significant strength of the Hilbert-space model is its inner-product structure, which has been exploited statistically and algorithmically by sophisticated techniques from geometric and functional analysis, placing the celebrated hyperplane methods on a solid foundation. However, the success of the Hilbert-space model obscures its limitations -- perhaps the most significant of which is that it cannot represent many norms and distance functions that arise naturally in applications.