Many problems in AI are simplified by clever representations of sensory or symbolic input. How to discover such representations automatically, from large amounts of unlabeled data, remains a fundamental challenge. The goal of statistical methods for dimensionality reduction is to detect and discover low dimensional structure in high dimensional data. In this paper, we review a recently proposed algorithm - maximum variance unfolding - for learning faithful low dimensional representations of high dimensional data. The algorithm relies on modern tools in convex optimization that are proving increasingly useful in many areas of machine~learning.