We introduce a new perspective on spectral dimensionality reduction which views these methods as Gaussian random fields (GRFs). Our unifying perspective is based on the maximum entropy principle which is in turn inspired by maximum variance unfolding. The resulting probabilistic models are based on GRFs. The resulting model is a nonlinear generalization of principal component analysis. We show that parameter fitting in the locally linear embedding is approximate maximum likelihood in these models. We develop new algorithms that directly maximize the likelihood and show that these new algorithms are competitive with the leading spectral approaches on a robot navigation visualization and a human motion capture data set. Finally the maximum likelihood perspective allows us to introduce a new approach to dimensionality reduction based on L1 regularization of the GRF via the graphical lasso.