A new geometrically-motivated algorithm for nonnegative
matrix factorization is developed and applied to the discovery
of latent “topics” for text and image “document” corpora. The algorithm is
based on robustly finding and clustering extreme-points of empirical
cross-document word frequencies that correspond to novel “words” unique to each
topic. In contrast to related approaches that are based on solving non-convex
optimization problems using sub-optimal approximations, locally-optimal methods,
or heuristics, the new algorithm is convex, has polynomial complexity, and has
competitive qualitative and quantitative performance compared to the current
state-of-the-art approaches on synthetic and real-world datasets.