Maximum variance unfolding (MVU) is an effective heuristic for dimensionality reduction. It produces a low-dimensional representation of the data by maximizing the variance of their embeddings while preserving the local distances of the original data. We show that MVU also optimizes a statistical dependence measure which aims to retain the identity of individual observations under the distance-preserving constraints. This general view allows us to design "colored" variants of MVU, which produce low-dimensional representations for a given task, e.g. subject to class labels or other side information.