Sparse Matrix Inversion with Scaled Lasso

Details Sparse Matrix Inversion with Scaled Lasso Sparse Matrix Inversion with Scaled Lasso

2012-02-13

arxiv.org/abs/1202.2723v1

Mathematics

Statistics Theory

Scientific paper

We propose a new method of learning a sparse nonnegative-definite target matrix. Our primary example of the target matrix is the inverse of a population covariance matrix or correlation matrix. The algorithm first estimates each column of the matrix by scaled Lasso, a joint estimation of regression coefficients and noise level, and then adjusts the matrix estimator to be symmetric. The procedure is efficient in the sense that the penalty level of the scaled Lasso for each column is completely determined by the data via convex minimization, without using cross-validation. We prove that this method guarantees the fastest proven rate of convergence in the spectrum norm under conditions of weaker form than those in the existing analyses of other $\ell_1$ algorithms, and has faster guaranteed rate of convergence when the ratio of the $\ell_1$ and spectrum norms of the target inverse matrix diverges to infinity. A simulation study also demonstrates the competitive performance of the proposed estimator.

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