2012/12/12 by Tommaso Cai, Weidong Liu, Cai, T. Tony +4 · 2 citations
Computer Science · Engineering · #62G09 (Secondary) #62H12 (Primary) 62F12 #Control Systems and Identification #Direction-of-Arrival Estimation Techniques #FOS: Computer and information sciences #FOS: Mathematics #Methodology (stat.ME) #Sparse and Compressive Sensing Techniques #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1212.2882
openalex publication_date 2012/12/12 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28
Precision matrix is of significant importance in a wide range of applications\nin multivariate analysis. This paper considers adaptive minimax estimation of\nsparse precision matrices in the high dimensional setting. Optimal rates of\nconvergence are established for a range of matrix norm losses. A fully data\ndriven estimator based on adaptive constrained \ℓ1 minimization is\nproposed and its rate of convergence is obtained over a collection of parameter\nspaces. The estimator, called ACLIME, is easy to implement and performs well\nnumerically.\n A major step in establishing the minimax rate of convergence is the\nderivation of a rate-sharp lower bound. A "two-directional" lower bound\ntechnique is applied to obtain the minimax lower bound. The upper and lower\nbounds together yield the optimal rates ofconvergence for sparse precision\nmatrix estimation and show that the ACLIME estimator is adaptively minimax rate\noptimal for a collection of parameter spaces and a range of matrix norm losses\nsimultaneously.\n