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Abstract:
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Estimation of covariance matrices has a number of applications (e.g., principal component analysis, discriminant analysis, and inferring independence relationships) and the sample covariance matrix performs poorly in high dimensions. Recent research mostly focused on regularization when variables have a natural ordering. When no such ordering exists, regularization must be invariant under variable permutations. We discuss several new permutation-invariant sparse estimators, with convergence rates that make explicit the trade-offs between dimension, sample size, and sparsity of the true model. We also discuss an alternative method for finding a "good" ordering of the variables based on the Isomap, a manifold projection method. Based on joint papers with P. Bickel, A. Rothman, A. Wagaman, and J. Zhu.
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