Sparse Precision Matrix Estimation with Calibration
We propose a semiparametric method for estimating sparse precision matrix of high dimensional elliptical distribution. The proposed method calibrates regularizations when estimating each column of the precision matrix. Thus it not only is asymptotically tuning free, but also achieves an improved finite sample performance. Theoretically, we prove that the proposed method achieves the parametric rates of convergence in both parameter estimation and model selection. We present numerical results on both simulated and real datasets to support our theory and illustrate the effectiveness of the proposed estimator.
Summary of the paper This paper presents a 
      semi-parametric tuning-free procedure for estimating sparse concentration 
      matrices. This method is applicable to the elliptical distribution family, 
      while most of its competitors only apply to the sub-Gaussian distribution 
      family. The procedure, called ALICE, learns the precision matrix column by 
      column in a similar fashion than the CLIME (Cai et al, 2011), yet with 
      important modifications: a first step is designed to learn the correlation 
      matrix and the associated variances/standard deviations by means of the 
      Kendall's Tau statistic as proposed in Liu et al, 2012. Then, the standard 
      deviations are estimated through a recent proposal of Catoni (2012). In 
      the second step, the inverse correlation is recovered by plugin-in the 
      correlation estimated in the first step in a convex program similar to the 
      CLIME, yet with a modification that allows for some calibration between 
      the columns. This leads to the tuning-free property of the ALICE. Finally, 
      the third step recovers the inverse covariance matrix after a simple 
      rescaling of the inverse correlation matrix. A dual inexact iterative 
      projection algorithm is given to (approximately) solve the convex program 
      in step 2. Theoretical guarantees, in both terms of estimation consistency 
      and selection consistency are provided, equivalent to those proposed for 
      the CLIME estimator. A simulation study on synthetic and breast cancer 
      data illustrates the good performance of the method compared to the 
      state-of-the-art methods. Comments The introductory part 
      in this work gives a concise and appropriate bibliography. Motivations for 
      semi-parametric estimators and limitations of the state-of-the art methods 
      are clearly introduced. Contributions of the current proposal (ALICE) are 
      clearly stated. When the method (part 3) is presented though, some 
      additional justification and explanation could have helped the reading and 
      the general understanding: why choosing Kendall's Tau + Catoni's M 
      estimator? Any other possibilities? It is also not very clear what is due 
      to the authors: using Kendall's Tau for correlation estimation in the 
      elliptical distribution framework is apparently due to a previous work 
      (Liu et al, 2012b): what of Catoni's M estimator? Is this already used in 
      such a context? The convex program solved looks like a 
      modification of the CLIME criterion. Some more connexion at this stage of 
      the paper (part 3.2) with the CLIME would have been appropriate. As a 
      matter of fact, it would have been easier to judge the originality of the 
      algorithm that follows, whose presentation is somewhat cumbersome. The 
      algorithm gives an approximation of the target estimator, but no 
      quantification are made on this approximation. Providing a global 
      complexity of the procedure would have been a good idea, too. The 
      same kind of remarks go for the theoretical part: though quite nice, the 
      properties derived by the authors are close to the properties existing for 
      the CLIME; and it is hard to evaluate how straightforward it can be 
      deduced from the work of Cai et al. Regarding the numerical 
      experiments, the good behavior of ALICE on non strictly Gaussian data is 
      nicely illustrated. Still, I do not understand why the proposal of Liu et 
      al., 2012b, is not included in the analysis. The same for the GLASSO with 
      a prior transformation of the sample covariance matrix, as implemented in 
      the huge package with the non paranormal transformation. It is a 
      pity that no illustration is made of the tuning-free property of ALICE, 
      since the calibration parameters are tuned with cross-validation in the 
      numerical experiments.