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Title: Towards a sparse, scalable, and stably positive definite (inverse) covariance estimator Authors:  Joong-Ho Won - Seoul National University (Korea, South) [presenting]
Abstract: High dimensional covariance and inverse covariance matrix estimation is notoriously difficult as the traditional estimate is not even positive definite. An important line of research in this regard is to shrink the extreme spectrum of the covariance matrix estimators. A separate line of research has considered sparse inverse covariance estimation which in turn gives rise to graphical models. In practice, however, a sparse covariance or inverse covariance matrix which is simultaneously well-conditioned and at the same time computationally tractable is desired. We consider imposing a condition number constraint to various types of losses used in covariance and inverse covariance matrix estimation. When the loss function can be decomposed as a sum of an orthogonally invariant function of the estimate and its inner product with a function of the sample covariance matrix, we show that a complete solution path can be obtained, involving a series of ordinary differential equations. An important finding is that the proximal operator for the condition number constraint, which turns out to be very useful in regularizing loss functions that are not orthogonally invariant and may yield non-positive-definite estimates, can be efficiently computed by this path algorithm.