Abstract
The positive-definiteness and sparsity are the most important property of high-dimensional precision matrices. To better achieve those property, this paper uses a sparse lasso penalized D-trace loss under the positive-definiteness constraint to estimate high-dimensional precision matrices. This paper derives an efficient accelerated gradient method to solve the challenging optimization problem and establish its converges rate as . The numerical simulations illustrated our method have competitive advantage than other methods.
Highlights
In the past twenty years, the most popular direction of statistics is highdimensional data
Estimation of high-dimensional precision matrix is increasingly becoming a crucial question in many field
To gain a better estimator for high-dimensional precision matrix and achieve the more optimal convergence rate, this paper mainly propose an effective algorithm, an accelerated gradient method ([10]), with fast global convergence rates to solve problem (1)
Summary
In the past twenty years, the most popular direction of statistics is highdimensional data. Zhang et al [9] consider a constrained convex optimization framework for high-dimensional precision matrix They used lasso penalized D-trace loss replace traditional lasso function, and enforced the positive-definite constraint {Θ ≥ ε I} for some arbitrarily small ε > 0. To gain a better estimator for high-dimensional precision matrix and achieve the more optimal convergence rate, this paper mainly propose an effective algorithm, an accelerated gradient method ([10]), with fast global convergence rates to solve problem (1). This method mainly basis on the Nesterov's method for accelerating the gradient method ([11] [12]), showing that by exploiting the special structure of the trace norm, the classical gradient method for smooth problems can be adapted to solve the trace regularized nonsmooth problems. The proof of this theorem is easy by applying the soft-thresholding method
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