  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 or correlation matrix. The algorithm first estimates each column of
the target matrix by the scaled Lasso and then adjusts the matrix estimator to
be symmetric. The penalty level of the scaled Lasso for each column is
completely determined by data via convex minimization, without using
cross-validation.
  We prove that this scaled Lasso 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 $\ell_1$ regularized 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
demonstrates the computational feasibility and superb performance of the
proposed method.
  Our analysis also provides new performance bounds for the Lasso and scaled
Lasso to guarantee higher concentration of the error at a smaller threshold
level than previous analyses, and to allow the use of the union bound in
column-by-column applications of the scaled Lasso without an adjustment of the
penalty level. In addition, the least squares estimation after the scaled Lasso
selection is considered and proven to guarantee performance bounds similar to
that of the scaled Lasso.
