ABSTRACT:   Covariance estimation (especially in high dimensions) has become an important issue in statistics. Improving estimation of covariance matrices in high dimensions is usually either based on a known ordering of variables or ignores the ordering entirely. This talk discusses methodologies for estimating covariances and proposes a method for discovering meaningful orderings of variables based on their correlations using the Isomap, a non-linear dimension reduction technique designed for manifold embeddings. These orderings are then used to construct a sparse covariance estimator, which is block-diagonal and/or banded.  We show the advantages of this new methodology compared to other estimators on both simulated data and a data set on sources of protein consumption, where the variables (food types) have a structure that cannot be easily described a priori.