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Covariance Matrix Estimation via Convex Optimization

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Xi Luo, The Wharton School

What
  • Colloquium
When Mon, Mar 14, 2011
from 04:00 PM to 05:00 PM
Where Ritter 142
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Covariance Matrix Estimation via Convex Optimization: Theory, Methods, Algorithms and Applications

 In this talk, I will focus on estimating covariance and inverse covariance matrices in the high dimensional (large p, small n) setting.

Relational data have gained much attention recently.  Social network, gene network, computer network and voting blocks are among many motivating examples.  We take a graphical model approach, and we aim to estimate an s-sparse inverse covariance matrix based on a sample of n iid p-variate random variables.  We propose a constrained L1 minimization method, and the resulting estimator enjoys a number of desirable properties.  It is shown that the convergence rate is s(log p/n)1/2 under the spectral norm, for either exponential-type or polynomial-type distributions.  In addition, graphical model selection is presented.  The procedure can be easily implemented by linear programming. The numerical merits are illustrated using simulated data and a breast cancer data set.  In particular, it is applied to classification of cancer patients.

 

In the second part of the talk, I will consider the problem of estimating the covariance matrix.  Factor models and random effect models have been shown to provide good approximations in modeling multivariate observations in many settings.  These models motivate us to consider a general framework of covariance structures, which contains sparse and low rank components.  We propose a convex optimization criterion, and the resulting estimator is shown to recover exactly the rank and support of the low rank and sparse components respectively.  The convergence rates are also presented.  To solve the optimization problem, we propose an iterative algorithm with low memory cost, and it converges to the optimal with order K-2 for any finite K iterations.  Numerical performance is demonstrated using simulated data and stock portfolio selection on S&P 100.

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