Towards Optimal Sparse Inverse Covariance Selection through Non-Convex Optimization

We study the problem of reconstructing the graph of a sparse Gaussian Graphical Model from independent observations, which is equivalent to finding non-zero elements of an inverse covariance matrix. For a model of size $p$ and maximum degree $d$, information theoretic lower bounds established in prior works require that the number of samples needed for recovering the graph perfectly is at least $d \log p/\kappa^2$, where $\kappa$ is the minimum normalized non-zero entry of the inverse covariance matrix. Existing algorithms require additional assumptions to guarantee perfect graph reconstruction, and consequently, their sample complexity is dependent on parameters that are not present in the information theoretic lower bound. We propose an estimator, called SLICE, that consists of a cardinality constrained least-squares regression followed by a thresholding procedure. Without any additional assumptions we show that SLICE attains a sample complexity of $\frac{64}{\kappa^4}d \log p$, which differs from the lower bound by only a factor proportional to $1/\kappa^2$ and depends only on parameters present in the lower bound.