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We propose a new procedure for estimating high dimensional Gaussian graphical models. Our approach is asymptotically tuning-free and non-asymptotically tuninginsensitive: it requires very few efforts to choose the tuning parameter in finite sample settings. Computationally, our procedure is significantly faster than existing methods due to its tuning-insensitive property. Theoretically, the obtained estimator is simultaneously minimax optimal for precision matrix estimation under different norms. Empirically, we illustrate the advantages of our method using thorough simulated and real examples. The R package bigmatrix implementing the proposed methods is available on the Comprehensive R Archive Network: http://cran.r-project.org/. IntroductionWe consider the problem of learning high dimensional Gaussian graphical models: let x 1 , . . . , x n be n data points from a d-dimensional random vector X = (X 1 , ..., X d ) T with X ∼ N d (0, Σ). We want to estimate an undirected graph denoted by G = (V, E), where V contains nodes corresponding to the d variables in X, and the edge set E describes the conditional independence relationships between X 1 , ..., X d . Let X \{j,k} := {X : = i, j}. We say the joint distribution of X is Markov to G if X j is independent of X k given X \{j,k} for all (j, k) / ∈ E. For Gaussian distributions, the graph G is known to be encoded by the precision matrix Θ := Σ −1 . More specifically, no edge connects X j and X k if and only if Θ jk = 0. The graph estimation problem is then reduced to the estimation of the precision matrix Θ. Such a problem is also called covariance selection (Dempster, 1972).

Recently, the alternating direction method of multipliers (ADMM) has received intensive attention from a broad spectrum of areas. The generalized ADMM (GADMM) proposed by Eckstein and Bertsekas is an efficient and simple acceleration scheme of ADMM. In this paper, we take a deeper look at the linearized version of GADMM where one of its subproblems is approximated by a linearization strategy. This linearized version is particularly efficient for a number of applications arising from different areas. Theoretically, we show the worst-case 𝒪(1/k) convergence rate measured by the iteration complexity (k represents the iteration counter) in both the ergodic and a nonergodic senses for the linearized version of GADMM. Numerically, we demonstrate the efficiency of this linearized version of GADMM by some rather new and core applications in statistical learning. Code packages in Matlab for these applications are also developed.

A popular mode of shape synthesis involves mixing and matching parts from different objects to form a coherent whole. The key challenge is to efficiently synthesize shape variations that are plausible, both locally and globally. A major obstacle is to assemble the objects with local consistency, i.e., all the connections between parts are valid with no dangling open connections. The combinatorial complexity of this problem limits existing methods in geometric and/or topological variations of the synthesized models. In this work, we introduce replaceable substructures as arrangements of parts that can be interchanged while ensuring boundary consistency. The consistency information is extracted from part labels and connections in the original source models. We present a polynomial time algorithm that discovers such substructures by working on a dual of the original shape graph that encodes inter-part connectivity. We demonstrate the algorithm on a range of test examples producing plausible shape variations, both from a geometric and from a topological viewpoint.