Sparse permutation invariant covariance estimation

The paper proposes a method for constructing a sparse estimator for the inverse covariance (concentration) matrix in high-dimensional settings. The estimator uses a penalized normal likelihood approach and forces sparsity by using a lasso-type penalty. We establish a rate of convergence in the Frobenius norm as both data dimension $p$ and sample size $n$ are allowed to grow, and show that the rate depends explicitly on how sparse the true concentration matrix is. We also show that a correlation-based version of the method exhibits better rates in the operator norm. We also derive a fast iterative algorithm for computing the estimator, which relies on the popular Cholesky decomposition of the inverse but produces a permutation-invariant estimator. The method is compared to other estimators on simulated data and on a real data example of tumor tissue classification using gene expression data.Comment: Published in at http://dx.doi.org/10.1214/08-EJS176 the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org

Brain covariance selection: better individual functional connectivity models using population prior

Spontaneous brain activity, as observed in functional neuroimaging, has been shown to display reproducible structure that expresses brain architecture and carries markers of brain pathologies. An important view of modern neuroscience is that such large-scale structure of coherent activity reflects modularity properties of brain connectivity graphs. However, to date, there has been no demonstration that the limited and noisy data available in spontaneous activity observations could be used to learn full-brain probabilistic models that generalize to new data. Learning such models entails two main challenges: i) modeling full brain connectivity is a difficult estimation problem that faces the curse of dimensionality and ii) variability between subjects, coupled with the variability of functional signals between experimental runs, makes the use of multiple datasets challenging. We describe subject-level brain functional connectivity structure as a multivariate Gaussian process and introduce a new strategy to estimate it from group data, by imposing a common structure on the graphical model in the population. We show that individual models learned from functional Magnetic Resonance Imaging (fMRI) data using this population prior generalize better to unseen data than models based on alternative regularization schemes. To our knowledge, this is the first report of a cross-validated model of spontaneous brain activity. Finally, we use the estimated graphical model to explore the large-scale characteristics of functional architecture and show for the first time that known cognitive networks appear as the integrated communities of functional connectivity graph.Comment: in Advances in Neural Information Processing Systems, Vancouver : Canada (2010

Randomized Dimension Reduction on Massive Data

Scalability of statistical estimators is of increasing importance in modern applications and dimension reduction is often used to extract relevant information from data. A variety of popular dimension reduction approaches can be framed as symmetric generalized eigendecomposition problems. In this paper we outline how taking into account the low rank structure assumption implicit in these dimension reduction approaches provides both computational and statistical advantages. We adapt recent randomized low-rank approximation algorithms to provide efficient solutions to three dimension reduction methods: Principal Component Analysis (PCA), Sliced Inverse Regression (SIR), and Localized Sliced Inverse Regression (LSIR). A key observation in this paper is that randomization serves a dual role, improving both computational and statistical performance. This point is highlighted in our experiments on real and simulated data.Comment: 31 pages, 6 figures, Key Words:dimension reduction, generalized eigendecompositon, low-rank, supervised, inverse regression, random projections, randomized algorithms, Krylov subspace method