Gaussian graphical models (GGM) have been widely used in a number of fields such as finance, bioinformatics, and image analysis [3, 25, 38, 49] . Graph learning under GGM can be formulated to estimate the precision matrix that captures the conditional dependency relations between random variables [10, 36] . In this paper, the goal is to learn a sparse graph under the Laplacian constrained GGM, where the precision matrix obeys the Laplacian structural constraints.
The general GGM has received broad interest in statistical machine learning, where the problem can be formulated as a sparse precision matrix estimation. The authors Banerjee et al. [3] , d'Aspremont et al. [9] , Yuan and Lin [70] proposed the 1 -norm penalized maximum likelihood estimation, also known as graphical lasso, to encourage sparsity in its entries. Numerous methods have been developed for solving this optimization problem. To solve the primal problem, first-order methods including Nesterovs smooth gradient method [9] , augmented Lagrangian method [54] , and second-order methods like inexact interior point method [39] , and Newton's method [11, 28, 48, 55, 64] have been proposed. To solve the dual problem, block coordinate ascent method [3, 22] , projected subgradient method [16] , and accelerated gradient descent method [43] have been explored. In addition, various extensions of graphical lasso and their theoretical properties have also been studied [27, 45, 50, 57, 68, 69] . However, those methods mentioned above focus on general graphical models and cannot be directly extended to Laplacian constrained GGM because of the multiple constraints on the precision matrix. Moreover, unlike the case of GGM, this paper shows that the 1 -norm is not effective in promoting sparsity in the penalized maximum likelihood estimation under Laplacian constrained GGM.
In recent years, Laplacian constrained GGM has received increasing attention in signal processing and machine learning over graphs [13, 33, 37, 47, 58] . Under Laplacian constrained GGM, graph Laplacian learning can be formulated as Laplacian structural precision matrix estimation. Unlike general GGM with a general positive definite precision matrix, the precision matrix in Laplacian constrained GGM enjoys the spectral property that its eigenvalues and eigenvectors can be interpreted as spectral frequencies and Fourier basis [58] , which is very useful in computing graph Fourier transform in graph signal processing [47, 58] , and graph convolutional networks [5, 46, 53] . To learn a graph Laplacian, the authors Dong et al. [12] , Egilmez et al. [17] , Gadde and Ortega [23] , Zhang and Florêncio [71] formulated the graph signals as random variables under the Laplacian constrained GGM. The learned graph under Laplacian constrained GGM favours smooth graph signal representations [12] , since the graph Laplacian quadratic term quantifies the smoothness of graph signals [30, 31] . The authors Kumar et al. [31] proposed to learn structured graphs under the Laplacian constrained GGM where a number of graph structures can be learned by imposing different Laplacian spectral constraints. A regularized Laplacian constrained GGM was proposed for graph structure discovery [33] and dimensionality reduction [37] , where the precision matrix is a Laplacian matrix plus a very small diagonal matrix. The underlying assumption in [33, 37] is that each data feature is independent and identically distributed sampled from a regularized Laplacian constrained GGM. However, sparse graph learning under Laplacian constrained GGM remains to be further explored. For example, how to effectively and efficiently learn a sparse graph and how to characterize the statistical error of the estimation under Laplacian constrained GGM are to be investigated.
Recent works [17, 31, 40, 74] introduced the 1 -norm penalized maximum likelihood estimation under Laplacian constrained GGM with the goal of learning a sparse graph. However, this paper will show that the 1 -norm is not effective in imposing a sparse solution in this problem. The contributions of this paper are summarized as follows.
• First, we find an unexpected behavior of the 1 -norm in Laplacian constrained GGM. More specifically, through empirical evidence, we observe that the number of nonzero graph weights grows as the regularization parameter increases. From a theoretical perspective, we prove that a large regularization parameter of the 1 -norm will surprisingly lead to a solution representing a fully connected graph.
• Second, we propose a nonconvex estimation method by solving a sequence of weighted 1 -norm penalized sub-problems. We present the theoretical guarantees on estimation error of the proposed estimator, in which the statistical error matches the minimax lower bound. To the best of our knowledge, this is the first work to analyze the non-asymptotic optimization performance guarantees on both optimization error and statistical error under Laplacian constrained GGM.
• Third, to solve each sub-problem efficiently, we simplify the Laplacian structural constraints and develop a projected gradient descent algorithm which enjoys a linear convergence rate.
• Finally, numerical experiments on both synthetic and real-world data sets from the recent COVID-19 pandemic and financial stock markets demonstrate the effectiveness of the proposed method in learning sparse and interpretable graphs.
The remainder of the paper is organized as follows. Problem formulation and related work are provided in Section 2. We present the proposed method and provide the theoretical results in Section 3. The experimental results are provided in Section 4. The proofs of the theorems and technical lemmas are in Section 5 and 6, respectively. We draw the conclusions in Section 7.
Notation Lower case bold letters denote vectors and upper case bold letters denote matrices. Both X ij and [X] ij denote the (i, j)-th entry of the matrix X. X denotes transpose of X.
[p] denotes the set {1, . . . , p}. Let supp + (x) = {i ∈ [p]|x i > 0} for any x ∈ R p . We use x ≥ 0 to denote that each element of x is non-negative. x denotes the least integer greater than or equal to x. The all-zero and all-one vectors or matrices are denoted by 0 and 1. x , X F and X 2 denote Euclidean norm, Frobenius norm and operator norm, respectively. Let λ max (X) denote the maximum eigenvalue of X. The inner product of two vectors is defined as x, y = i x i y i . Let x max = max i |x i | and x min = min i |x i |. For functions f (n) and g(n), we use f (n) g(n) if f (n) ≤ Cg(n) for some constant C ∈ (0, +∞). S p + and S p ++ denote the sets of positive semi-definite and positive definite matrices with size p × p, respectively. R + denotes the set of positive real numbers. E c denotes the complement of the set E.
We first present the definition of Laplacian constrained Gaussian Markov random fields and formulate the problem of learning a graph under the Laplacian constrained GGM. After that, we discuss relevant related work.
We define a weighted, undirected graph G = (V, E, W ), where V denotes the set of nodes, and the pair {i, j} ∈ E if and only if there is an edge between node i and node j. W ∈ R p×p is the weighted adjacency matrix with W ij denoting the graph weight between node i and node j. The graph Laplacian L ∈ R p×p , also known as combinatorial graph Laplacian, is defined as
where D is a diagonal matrix with D ii = p j=1 W ij . In this paper, we focus on the case of connected graphs, implying that there is only one graph component. From spectral graph theory [8] , the rank of the Laplacian matrix for a connected graph with p nodes is p − 1. Then, the set of Laplacian matrices for connected graphs can be formulated as
where 0 and 1 denote the constant zero and one vectors, respectively. Next, we will define Laplacian constrained Gaussian Markov random fields, and without loss of generality we assume that the random vector x has zero mean.
Definition 2.1. A zero-mean random vector x = [x 1 , . . . , x p ] ∈ V p−1 is called a Laplacian constrained Gaussian Markov Random Fields (L-GMRF) with parameters (0, Θ) with Θ ∈ S L , if and only if its density function q L :
where det denotes the pseudo determinant defined by the product of nonzero eigenvalues [26] , and
Note that we restrict x into a subspace because the L-GMRF does not have a density with respect to the p-dimensional Lebesgue measure. According to the disintegration theorem, we can construct a conditional probability measure defined on V p−1 and then the density of L-GMRF with this measure satisfies (3) . In this sense, the L-GMRF can be interpreted as a GMRF conditioned on the linear constraint 1 x = 0 and thus each observation x (k) of an L-GMRF also satisfies 1 x (k) = 0. For convenience, we still denote Θ in (3) as the precision matrix, though it formally does not exist [52] . Notice that the precision matrix in L-GMRF has sign constraints on the entries, which can also be found in Gaussian distributions with multivariate total positivity [18, 35, 59] , where the precision matrix is a symmetric M -matrix.
Sparse graph learning under the Laplacian constrained Gaussian graphical model can be formulated as the penalized maximum likelihood of the precision matrix with Laplacian structural constraints,
where S is the sample covariance matrix, h λ is a regularizer, depending on a regularization parameter λ ≥ 0, which serves to enforce sparsity, e.g., h λ (Θ ij ) = λ|Θ ij |. J = 1 p 1 p×p is a constant matrix with each element equal to 1 p . Note that we replace det (Θ) with det(Θ + J ) in (4) as done in [17] , which follows from the fact that the matrix J is rank one, and the nonzero eigenvalue of J is 1 whose eigenvector is orthogonal to the row and column spaces of Θ ∈ S L .
The 1 -norm regularized maximum likelihood estimation under GGM has been extensively studied [22, 28, 45, 51, 57, 70] . To reduce estimation bias, nonconvex regularizers have been introduced in estimating a sparse precision matrix [4, 7, 34, 41, 56, 72] . Some popular nonconvex penalties include the smooth clipped absolute deviation (SCAD) [19] , minimax concave penalty (MCP) [72] , and capped 1 -penalty [73] . Several types of algorithms including local quadratic appoximation [19] , minimization-maximization [29, 61] , and local linear approximation [75] were proposed to solve the nonconvex optimization. Recently, the authors Fan et al. [20] , Loh and Wainwright [41] , Sun et al. [60] presented some theoretical analysis to characterize the nonconvex estimators with desired statistical guarantees. However, all of those methods cannot be directly extended to Laplacian constrained GGM because we aim to learn a precision matrix that must satisfy the constraints in (2), while the learned precision matrix under GGM is a general positive definite matrix.
Recent works [17, 31, 40, 74] proposed the 1 -norm penalized maximum likelihood estimation under Laplacian constrained GGM to learn a graph Laplacian. The authors Egilmez et al. [17] , Zhao et al. [74] designed a primal-dual algorithm that introduces additional variables to handle Laplacian structural constraints. The authors Liu et al. [40] proposed a block coordinate descent method to solve the optimization problem. The authors Kumar et al. [31] proposed a 1 -norm regularized maximum likelihood estimation method with Laplacian spectral constrains to learn structured graphs such as k-component graphs. More recently, the authors Kumar et al. [32] proposed a framework with re-weighted 1 -norm to learn structured graphs by imposing spectral constraints on graph matrices. But, the proposed algorithms in [32, 31] have to compute the eigenvalue decomposition in each iteration which is computationally expensive in the highdimensional regime. Note that all the methods mentioned above lack theoretical analysis on estimation error and algorithm convergence rate. In this paper, we will show that the 1 -norm is not effective in promoting sparsity in the Laplacian constrained GGM, and further propose a nonconvex estimator with theoretical guarantees on estimation error. The approach of theoretical analysis in this paper may be extended to characterize statistical properties of other estimators under the Laplacian constrained GGM.
In this section, we first present an unexpected behavior of the 1 -norm in learning a sparse graph under Laplacian constrained GGM. Then, we propose a nonconvex penalized maximum likelihood estimation method by solving a sequence of weighted 1 -norm sub-problems, and develop a projected gradient descent algorithm to solve each sub-problem. Finally, we present the theoretical results in the analysis of the estimation error and algorithm convergence of the proposed method.
Sparsity is often explored in high-dimensional Gaussian graphical models in order to reduce the number of samples required. The effectiveness of the 1 -norm regularized maximum likelihood estimation, also known as graphical lasso, has been widely demonstrated in a number of fields. One common rule of thumb for graphical lasso is that the estimated graph will get sparser when a larger regularization parameter is used. However, we find an unexpected behavior of the 1 -norm in Laplacian constrained GGM. We will prove that a large regularization parameter of the 1 -norm will lead to a solution that represents a fully connected graph instead of a sparse graph. The 1 -norm regularized maximum likelihood estimation under the Laplacian constrained Gaussian graphical model [17, 74] can be formulated as
where S is the sample covariance matrix with each sample independently sampled from L-GMRF, and λ is the regularization parameter. It is easy to check that (5) is a convex optimization problem.
Theorem 3.1. LetΘ ∈ R p×p be the global minimum of (5) with p > 3. Define s 1 = max k S kk and s 2 = min ij S ij . If the regularization parameter λ in (5) satisfies λ ∈ [(2 + 2 √ 2)(p + 1)(s 1 − s 2 ), +∞), then the estimated graph weightŴ ij = −Θ ij obeyŝ
Theorem 3.1 states that a large regularization parameter of the 1 -norm will force every graph weight to be strictly positive, thus the estimated graph is fully connected. This theoretical result is consistent with empirical observations depicted in Figure 1 , which shows that the number of positive edges learned by (5) grows along with the increase of λ, and finally the estimated graph in Figure 1 (d) is fully connected. Figure  2 illustrates the histograms of the nonzero graph weights (associated with the positive edges) learned in Figure 1 . Figure 2 (d) depicts the histogram for the graph in Figure 1 (d), where every weight is strictly positive. It is further observed in Figure 2 (d) that all the graph weights are very small. Therefore, a large regularization parameter will lead to a graph with every weight strictly positive and small. We can see that the histogram in Figure 2 (d) is significantly different from the true histogram in Figure 2 (a), implying that the estimated model fails to identify the true relationships among the data variables.
The unexpected behavior of the 1 -norm characterized in Theorem 3.1 is due to the Laplacian structural constraints in the optimization (5) . Because of the constrains Θ · 1 and Θ ij = Θ ji ≤ 0 for any i = j, the term tr (ΘS) + λ i>j |Θ ij | in (5) can be written as i>j (λ + [L * S] k )|Θ ij |, where L * is the adjoint of the linear operator L defined in (7) , and k = i − j + j−1 2 (2p − j). To intuitively understand the behavior of the 1 -norm with a large regularization parameter imposed in (5) , suppose that λ is sufficiently large such that (5) can be approximated well by
LetΘ be the optimal solution of (6). By calculation, we obtain thatW ij = −Θ ij = 2 pλ for any i = j. Notice that every estimated graph weightW ij is strictly positive, and a large λ will lead each weight to be very small. EveryW ij with i = j is equal with each other because the derivative of |Θ ij | is constant for Θ ij ≤ 0. In the general GGM, however, the estimated precision matrix only needs to be positive definite and there are no Laplacian structural constraints imposed. The resultant optimization problem for precision matrix estimation in general GGM cannot be approximated well by (6) . This is the reason why the 1 -norm behaves differently in the Laplacian constrained GGM and general GGM.
To solve the issue of the 1 -norm in Laplacian constrained GGM, we propose to introduce the nonconvex regularizer, and the effectiveness of the proposed method with the nonconvex regularizer has been demonstrated by numerical experiments in Section 4. In the next subsection, we will develop an efficient algorithm to solve the penalized maximum likelihood estimation with the nonconvex regularizer under Laplacian constrained GGM. 
Problem (4) is a constrained optimization problem with Θ ∈ S L including multiple constraints. We first simplify the Laplacian structural constraints in (2) as follows.
The constraints Θ ij = Θ ji and Θ · 1 = 0 in (2) are linear and there are only p(p−1) 2 free variables in Θ. Therefore, we use a linear operator defined in [32] that maps a vector x ∈ R p(p−1)/2 to a matrix Lx ∈ R p×p as below.
Definition 3.2. The linear operator L : R p(p−1)/2 → R p×p , x → Lx, is defined by
where k = i − j + j−1 2 (2p − j). A simple example is given below which illustrates the definition of the operator L. Let x ∈ R 3 . Then we have
The adjoint operator L * of L is defined so as to satisfy Lx, Y = x, L * Y , ∀x ∈ R p(p−1)/2 and Y ∈ R p×p . Definition 3.3. The adjoint operator L * :
where i, j ∈ [p] obeying k = i − j + j−1 2 (2p − j) and i > j. By introducing the linear operator L, we can simplify the definition of S L in (2) as below.
Theorem 3.4. The Laplacian set S L defined in (2) can be written as
where J = 1 p 1 p×p and x ≥ 0 means every entry of x is non-negative. As a result of Theorem 3.4, we introduce the linear operator L defined in (7) and reformulate the optimization (4) as
Notice that we remove the constraint (Lx + J ) ∈ S p ++ in (10) compared with the constraint set in (9), because any w in the feasible set of (10) must obey (Lx + J ) ∈ S p ++ as shown as follows. Let F (w) = − log det(Lw + J ) + tr (SLw) + i h λ (w i ). The feasible set of (10) is S w = {w |w ≥ 0, w ∈ dom(F )}, where dom(F ) denotes the domain of the function F . One can verify that
The set S w can be equivalently written as
which is due to the reason that Lw + J must be positive semi-definite, because Lw is positive semi-definite for any w ≥ 0 following from (39) , while the matrix J is rank one, and the nonzero eigenvalue of J is 1 whose eigenvector is orthogonal to the row and column spaces of Lw. The condition det(Lw + J ) ∈ R + in (11) implies that Lw + J is non-singular. The non-singularity and positive semi-definiteness of Lw + J together lead to (Lw + J ) ∈ S p ++ . To solve the problem (10), we follow the majorization-minimization framework [61] , which consists of two steps. In the majorization step, we design a majorized function f (w|ŵ (k−1) ) that locally approximates the objective function F (w) atŵ (k−1) satisfying f (w|ŵ (k−1) ) ≥ F (w) and f (ŵ (k−1) |ŵ (k−1) ) = F (ŵ (k−1) ). (13) Then in the minimization step, we minimize the majorized function f (w|ŵ (k−1) ). We assume h λ is concave (refer to Assumption 3.5 for more details about the choices of h λ ). Here we find the majorized function f (w|ŵ (k−1) ) by linearizing i h λ (w i ). Set f k (w) = f (w|ŵ (k−1) ) to simplify the notation and obtain
By minimizing f k (w), we establish a sequence {ŵ (k) } k≥1 bŷ
where z
≥ 0 by Assumption 3.5. Thus the problem (15) can be viewed as a weighted 1 -norm penalized maximum likelihood estimation under Laplacian constrained Gaussian graphical model. The final estimatorŵ is obtained by solving a sequence of optimization problems (15) . The iteration procedure is summarized in Algorithm 1.
To solve the optimization problem in (15), we develop an algorithm based on projected gradient descent with backtracking line search. To obtainŵ (k) , the algorithm starts with w (k) 0 and then establishes the sequence {w (k) t } t≥0 by the projected gradient descent as below. In the t-th iteration, we update w
where P + (a) = max(a, 0) and ∇f k (w
t } t≥0 will converge toŵ (k) . Here we set w To establish the theoretical results in Section 3.3, the initial pointŵ (0) of Algorithm 1 is chosen such
, and s is the number of the nonzero graph weights in the true graph. In other words,ŵ (0) has no more than s positive elements. Through the analysis of Algorithm 2 in Section 3.3, we will see that the sequence {w (k) t } t≥0 can be guaranteed in the feasible set of (10), implying that the estimated graph must be connected (refer to the proof of Theorem 3.10 for more details). Though the proposed algorithm has two loops, we prove in Section 3.3 that for the outer loop, only 4 log(4αc −1 0 ) iterations in Algorithm 1 are needed to achieve the desired order of the estimation error, where α and c 0 are two constants, independent of the dimension size p and sample size n. For the inner loop, the projected gradient descent in Algorithm 2 enjoys a linear convergence rate.
Input: Sample covariance S, λ,ŵ (0) ; k ← 1; 1: while Stopping criteria not met do k ← k + 1; 5: end while Output:ŵ. t ← t + 1; 8: end while Output:ŵ (k) .
We first list the assumptions needed for establishing our theorems, and then present the theoretical analysis on estimation error and algorithm convergence of the proposed method.
We denote the true graph weights by w ∈ R p(p−1)/2 , which are non-negative, i.e., w ≥ 0. Let S = {i ∈ [p(p − 1)/2] |w i > 0} be the support set of w and s be the number of the nonzero weights, i.e., |S | = s. This paper focuses on learning connected graphs and thus w corresponds to the weights of a connected graph, implying that Lw ∈ S L in (2) and (9) . We impose some mild conditions on the sparsity-promoting function h λ in Assumption 3.5 and the true graph weights w in Assumption 3.6.
Assumption 3.5. The function h λ : R → R satisfies the following conditions:
The minimal nonzero graph weight satisfies min i∈S w i ≥ (c + γ)λ λ, where c and γ are defined in Assumption 3.5. There exists τ ≥ 1 such that
where λ 2 (Lw ) and λ max (Lw ) are the second smallest eigenvalue and maximum eigenvalue of Lw , respectively. Note that the smallest eigenvalue of Lw is 0.
Remark 3.7. In Assumption 3.5, the conditions on h λ (x) are mainly made over x ∈ [0, +∞) because of the nonnegativity constraint in the optimization (15) . In Assumption 3.5, the first two conditions are made to promote sparsity and unbiasedness [42] , and hold for a variety of nonconvex sparsity-promoting functions including the popular choices SCAD [19] and MCP [72] . The two conditions also ensure that the derivative of h λ (x) is not constant to avoid the issue in Theorem 3.1. In the third condition, we specify h λ (x) for x ≤ 0 only for theoretical analysis. The condition h λ (cλ) ≥ λ/2 can always hold by tuning parameters due to the conditions h λ (0) = λ and h λ (γλ) = 0. Therefore, it is flexible to design a sparsity-promoting function satisfying the conditions stated in 3.5. In Assumption 3.6, the conditions on the true graph weights w are mild. In our theorems, the regularization parameter λ is taken with the order log p/n that could be very small when the sample size n increases. The assumptions on the minimal magnitude of signals are often employed in the analysis of nonconvex optimization [20, 65] . The assumptions on the minimum nonzero eigenvalue are frequently made [7, 44, 65] to establish a local contraction region.
We characterize the estimation error of the proposed estimator. In the following theorem, the choice of the regularization parameter λ is set according to a user-defined parameter α > 2. A larger α yields a larger probability with which the claims hold, but also leads to a more stringent requirement on the number of samples.
Theorem 3.8. Under Assumptions 3.5 and 3.6, take the regularization parameter λ = 4αc −1 0 log p/n for some α > 2. If the sample size n is lower bounded by
then with probability at least 1 − 1/p α−2 , the sequenceŵ (k) returned by Algorithm 1 satisfies
Theorem 3.8 holds with overwhelming probability. According to Theorem 3.8, the estimation error between the estimated and true graph weights is bounded by two terms, i.e., the optimization error and statistical error. The optimization error, ( 3 2+ √ 2 ) k ŵ (0) − w , decays to zero at a linear rate with respect to the iteration number k. The statistical error depends on the sample size n, and a large sample size will lead to a small statistical error. We can see the statistical error is independent of k, and thus will not decrease during iterations in the algorithm. Therefore, the estimation error of the output of the algorithm is mainly from the statistical error. Corollary 3.9. Under the same assumptions and conditions as stated in Theorem 3.8, the sequenceŵ (k) returned by Algorithm 1 satisfies
, then the estimation error is dominated by the statistical error and we further obtain
max is a constant. Corollary 3.9 presents the estimation error between the estimated and true precision matrices in Laplacian constrained GGM. Similar to Theorem 3.8, the estimation error decays to zero at a linear rate. The order of statistical error is s log p/n, which matches the order of minimax lower bound [6, 51, 50] in Gaussian graphical model. Furthermore, the proposed estimation method can achieve the order of minimax lower bound by solving only 4 log(4αc −1 0 ) sub-problems. Notice that 4 log(4αc −1 0 ) is independent of the dimension size p and sample size n.
In this subsection, we will establish the convergence rate of Algorithm 2 in solving the sub-problem (15) . In the theorem statement, the parameter δ > 1 is a user-defined parameter. A larger δ leads to a faster convergence rate, but a larger number of samples are required.
Theorem 3.10. Under Assumptions 3.5 and 3.6, take the regularization parameter λ = 4αc −1 0 log p/n for some α > 2. If the sample size n satisfies
Theorem 3.10 shows that the designed projected gradient descent in Algorithm 2 enjoys a linear convergence rate in solving sub-problems (15) . Note that the claims in Theorem 3.10 may not hold for k = 1, because we only set a very mild condition on the initial pointŵ (0) that |supp + (ŵ (0) )| ≤ s. Thusŵ (0) may not be in the local region B(w ; 1
(1−δ −1 ) 2 according to Lemma 5.9. One may impose more conditions on initial point such thatŵ (0) ∈ B(w ; 1 √ 2pδτ ), then the claims in Theorem 3.10 also holds for k = 1. Here B(w ; r) = {w | w − w ≤ r}, and τ is defined in Assumption 3.6.
It is observed that the required sample size in Theorem 3.10 is larger than the one in Theorem 3.8. Using the sample size in Theorem 3.8, one can still establish a local contraction region
0 )} such that Algorithm 2 also has a linear convergence. However, the region
In this section, we conduct numerical experiments on both synthetic data and real-world data sets from the recent COVID-19 pandemic and financial stock markets to verify the performance of the proposed method in learning sparse and interpretable graphs. The compared methods include the state-of-the-art GLE-ADMM algorithm [74] and the baseline projected gradient descent with 1 -norm. All the experiments are conducted on a PC with a 2.8 GHz Inter Core i7 CPU and 16 GB RAM.
We use the relative error (RE) and F-score (FS) to evaluate the performance of the algorithms. Those performance measures are defined as
whereΘ = Lŵ and Θ = Lw denote the estimated and true precision matrices, respectively. The true positive number is denoted as tp, i.e., the case that there is an actual edge and the algorithm detects it, the false positive is denoted as fp, i.e., the case that there is no actual edge but algorithm detects one, and the false negative is denoted as fn, i.e., the case that the algorithm failed to detect an actual edge. The F-score takes values in [0, 1], where 1 indicates perfect structure recovery. For the proposed method, we test two nonconvex penalties, MCP and SCAD, defined respectively by
where we define h MCP,λ (x) and h SCAD,λ (x) only for x ≥ 0 because of the nonnegativity constraint in (15) . We set γ equal to 1.01 and 2.01 in h MCP,λ (x) and h SCAD,λ (x) in all the experiments, respectively. Note that the proposed method is not limited to using MCP and SCAD regularizers, and one can explore other sparsity-inducing functions [2] for interest.
For experiments on synthetic data, we consider two types of graphs: Barabasi-Albert graphs of degree one [1] , also known as tree graphs, and modular graphs. Estimating tree graphs is a crucial task that has an impact in the performance of many data intensive applications such as image segmentation [67] . Modular graph consists of a connected graph which contains clusters of nodes that share similar properties. The sparsity level in modular graphs are twofold: (1) intra-modular and (2) inter-modular. The first one refers to the number of connections among nodes that belong to the same module, whereas the second one refers to the number of connections among nodes in different modules. Modular graphs have been applied to a myriad of social network tasks such as community detection [21] . We generate the data matrix X ∈ R p×n with each column of X independently drawn from L-GMRF (See Definition 2.1) with Θ = Lw , where w contains the true weights from Barabasi-Albert graphs of degree one and modular graphs. The number of nodes in the Barabasi-Albert graph and modular graph is p = 50 and 100, respectively. The weights associated with positive edges in both Barabasi-Albert graph and modular graph are uniformly sampled from U (2, 5). The modular graph is generated randomly with probabilities of intra-module and inter-module connections 0.25 and 0.005, respectively. The sample covariance matrix is constructed by S = 1 n XX , where n is the number of samples. The curves in Figures 3, 4 , 5 and 6 are the results of an average of 100 Monte Carlo realizations and the shaded areas around the curves represent the one-standard deviation confidence interval. Figure 3 presents the results of learning Barabasi-Albert graphs by GLE-ADMM, projected gradient descent with 1 -norm, and the proposed method with the two regularizers. It is observed that the numbers of positive edges learned by GLE-ADMM and projected gradient descent with 1 -norm rise along with the increase of λ, implying that the learned graphs will get denser as λ increases. Figure 3 shows that both GLE-ADMM and projected gradient descent with 1 -norm achieve the best performance in terms of sparsity, relative error, and F-score when λ = 0, which defies the purpose of introducing the 1 -norm regularizer. In contrast, the proposed NGL-SCAD and NGL-MCP enhance sparsity when λ increases. It is observed that, with λ equal to 0.1 or 0.25, NGL-SCAD and NGL-MCP achieve the true number of positive edges, and an F-score of 1, implying that all the connections and disconnections between nodes in the true graph are correctly identified. Figure 4 shows that the proposed NGL-SCAD and NGL-MCP always outperform both GLE-ADMM and the baseline projected gradient descent in terms of sparsity, relative error, and F-score under different samples size ratios. Figure 5 presents the results of learning modular graphs by GLE-ADMM [74] , projected gradient descent with 1 -norm, and the proposed NGL-SCAD and NGL-MCP. Similar to the phenomenon in learning Barabasi-Albert graphs, it is observed that a larger regularization parameter λ for GLE-ADMM and projected gradient descent with 1 -norm will lead to a worse performance in terms of sparsity, relative error, and F-score. On the contrary, NGL-SCAD and NGL-MCP enhance the sparsity and improve the relative error and F-score as λ increases. Figure 6 shows that the proposed method always leads to a better performance in learning modular graphs in terms of sparsity, relative error, and F-score, than the compared methods under different sample size ratios. Figure 7 shows a sample result of learning a Barabasi-Albert graph via GLE-ADMM, NGL-SCAD and NGL-MCP. It is observed that the learned graphs via NGL-SCAD and NGL-MCP present the connection between any two nodes correctly, while there are many incorrect connections in the graph learned via GLE-ADMM. In addition, performance measures including sparsity, relative error, and F-score also indicate a better performance of the proposed method. 
In this section, we compare the proposed method with the benchmark GLE-ADMM [74] on three real-world data sets.
We first conduct numerical experiments on the COVID-19 data set 1 from 98 anonymous Chinese patients affected by the outbreak of COVID-19 on early February, 2020. The features include age (integer), gender (categorical), and location (categorical). The label is a binary variable representing the life status of patients, alive (green) or no longer alive (red). Our goal is to construct a graph from the data features. To this end, we first pre-process the feature matrix so as to transform the categorical features into numerical ones via one-hot-encoding. The pre-processed feature matrix X has the dimension 98 × 32, i.e., p = 98 and n = 32. We then compute the sample covariance matrix and learn the graphs. Figure 9 shows that the benchmark GLE-ADMM is unable to impose sparsity, diminishing interpretation capabilities of the graph severely. On the other hand, the proposed NGL-SCAD and NGL-MCP obtain sparse graphs with clearer connections. The learned graphs possibly provide guidance on priority setting in health care because green nodes (patients alive) that have stronger connections with red nodes (patients that passed away) may suffer a higher health risk. We next perform experiments on the COVID-19 data set 2 provided by the Israelite Hospital Albert Einstein in Brazil. The data set contains anonymized data from patients who had samples collected to perform the test for SARS-CoV-2. The features in the data set are mainly clinical coming from blood, urine, and saliva exams, e.g., hemoglobin level, platelets, red blood cells, etc. The original data set contains 108 features from 558 patients. Due to the high number of missing values, we do not consider features that were measured for at most 10 patients. In addition, a large number of patients had no record of any features. Finally, we end up with a data matrix of 182 patients with 57 features, i.e., p = 182 and n = 57. The remaining missing values were filled in with zeros. We then compare the proposed method with the GLE-ADMM method on this data set. It is observed from Figure 10 that the proposed NGL-SCAD and NGL-MCP output a more interpretable representation of the network, where blue and red nodes denote patients who tested negative and positive for SARS-CoV-2, respectively.
The third data set is from stocks composing the S&P 500 index. We select log-returns from 181 stocks from 4 sectors, namely: "Industrials", "Consumer Staples", "Energy", "Information Technology", during a period of 4 years from January 1st 2016 to May 20th 2020, with a total of 1101 observations. Then the data matrix X has a size of 181 × 1101. The graphs are learned on the basis of the sample correlation matrix. Figure 11 shows that the graphs learned via NGL-SCAD and NGL-MCP are able to vividly display the sectors modularity, whereas the graph learned via GLE-ADMM fails to do so. Note that the real-world data sets may not exactly follow the Laplacian constrained Gaussian graphical models. In this case, the log-determinant Bregman divergence regularized optimization with Laplacian structural constraints can be interpreted as learning a graph such that the features are smooth on the learned graph [12, 32] .
This section includes the proofs of Theorems 3.1, 3.4, 3.8 and 3.10, and Corollary 3.9. Before proving the theorems, we first present some technical lemmas.
Lemma 5.1. Let f (w) = − log det(Lw + J ), with w ∈ R p(p−1)/2 . Then for any x ∈ R p(p−1)/2 , we have
Lemma 5.2. For any given w ∈ R p(p−1)/2 satisfying (Lw + J ) ∈ S p ++ , there must exist an unique x ∈ R p(p−1)/2 such that
holds for any b = 0, where J = 1 p 1 p×p , in which 1 p×p ∈ R p×p with each element equal to 1.
and Ω t is an index set defined by
Furthermore, we have λ min (M ) = 2 and λ max (M ) = 2p.
Lemma 5.4. Take λ = 4αc −1 0 log p/n and suppose n ≥ 94αc −1 0 λ 2 max (Lw )s log p for some α > 2, where c 0 is a constant defined in Lemma 5.8. Let
where S is the support of w .
Lemma 5.5. Take λ = 4αc −1 0 log p/n and suppose n ≥ 94αc −1 0 λ 2 max (Lw )s log p for some α > 2, where c 0 is a constant defined in Lemma 5.8. Define the set E (k) by holds andŵ (0) satisfies |supp + (ŵ (0) )| ≤ s, then for any k ≥ 1,ŵ (k) defined in (15) obeys
and
Lemma 5.7. Take λ = 4αc −1 0 log p/n and suppose n ≥ 8α log p for some α > 2, where c 0 is a constant defined in Lemma 5.8. Then one has (17) and δ > 1. In other words, for any w 1 , w 2 ∈ B(w ; 1 √ 2pδτ ), we have
Lemma 5.10. Let f (w) = − log det(Lw + J ). Define a local region of w by
for any x ∈ R p(p−1)/2 with M defined in Lemma 5.3, and S w = {w |w ≥ 0, (Lw + J ) ∈ S p ++ }. Then for any w 1 , w 2 ∈ B M (w ; r), we have
Lemma 5.11.
[66] Suppose a positive matrix A ∈ R p×p is diagonally scaled such that A ii = 1, i = 1, . . . , p, and 0 < A ij < 1, i = j. Let y and x be the lower and upper bounds satisfying
and define s by
Then the inverse matrix of A exists and A is an inverse M -matrix if s −1 ≥ p − 2 with p > 3.
Lemma 5.12.
[63] (Sub-exponential tail bound) Suppose X is sub-exponential with parameters (υ, α). Then
t ∈ ( υ 2 α , +∞).
Proof. As a result of Theorem 3.4, the optimization (5) can be equivalently written as min w≥0 − log det(Lw + J ) + tr (LwS) + λ w 1 .
Due to the non-negativity constraint w ≥ 0, (21) can be further rewritten as min w≥0 − log det(Lw + J ) + L * S + λ1, w ,
where 1 = [1, . . . , 1] . We first prove that the optimization (22) has one global minimizer if λ > 0. Let f (w) = − log det(Lw + J ) + L * S + λ1, w . The feasible set of (22) is S w = {w ∈ R p(p−1)/2 |w ≥ 0, (Lw + J ) ∈ S p ++ }, which is the same with the feasible set of (10). For any w ∈ S w , the minimum eigenvalue of ∇ 2 f (w) can be lower bounded by
where the second equality is due to Lemma 5.1; the third equality follows from Lemma 5.3; the last equality follows from the property of Kronecker product that the eigenvalues of A ⊗ B are λ i µ j for i, j ∈ [p], where λ i and µ j are the eigenvalues of A ∈ R p×p and B ∈ R p×p , respectively, and λ min (M ) = 2 following from Lemma 5.3; the last inequality follows from the fact that w ∈ S w . Therefore, the optimization (22) is strictly convex, and thus (22) has at most one global minimizer. The existence of minimizers of (22) can be guaranteed by the coercivity of f (w). The function f (w) can be lower bounded by
where the second equality follows from (56) with b = 1; the forth equality follows from Lw · 1 = 0; the last inequality holds because w ≥ 0, and L * S ≥ 0, which follows from (92); the first inequality holds because the smallest eigenvalue λ 1 (Lw) = 0 and a 1 + a 2 + . . . + a n n ≥ n √ a 1 · a 2 · · · a n holds for any non-negative real numbers of a 1 , . . . , a n . A function g : Ω → R ∪ {+∞} is called coercive over Ω, if every sequence x k ∈ Ω with x k → +∞ obeys lim k→∞ g(x k ) = +∞, where Ω ⊂ R n . Let
A simple calculation yields lim z→+∞ h(z) = +∞ if λ > 0. For any sequence w k ∈ cl(S w ) with w k → +∞, where cl(S w ) is the closure of S w , one has
where the first inequality follows from (23) . Hence, f (w) is coercive over cl(S w ). Following from the Extreme Value Theorem [14] , if Ω ⊂ R n is non-empty and closed, and g : Ω → R ∪ {+∞} is lower semi-continuous and coercive, then the optimization min x∈Ω g(x) has at least one global minimizer. Therefore, by the coercivity of f (w), (22) has at least one global minimizer in cl(S w ).
Let Ω A = {w ∈ R p(p−1)/2 |w ≥ 0} and Ω B = {w ∈ R p(p−1)/2 |(Lw + J ) ∈ S p ++ }. Ω A is a closed set and Ω B is an open set. Then S w can be rewritten as S w = Ω A ∩ Ω B . Consider the set V := cl(S w ) \ S w , we have
where ∂Ω B is the boundary of Ω B . Notice that every matrix on the boundary of the set of positive definite matrices is positive semi-definite and has zero determinant. Hence, one has ∂Ω B = {w ∈ R p(p−1)/2 |(Lw + J ) ∈ S p + , det(Lw + J ) = 0}. As a result, for any w b ∈ cl(S w ) \ S w , f (w b ) = +∞. Therefore, (22) has at least one global minimizer and the minimizer must belong to the set S w . On the other hand, by the strict convexity of f (w), (22) has at most one global minimizer in S w . Totally, we conclude that (22) has an unique global minimizer in S w if λ > 0.
We prove the theorem through the KKT conditions. The Lagrangian of the optimization (22) is
where υ is a KKT multiplier. Let (ŵ,υ) be any pair of points that satisfies the KKT conditions. Then we have
w iυi = 0, for i = 1, . . . , p(p − 1)/2; (26) w ≥ 0,υ ≥ 0;
As we know, for any convex optimization with differentiable objective and constraint functions, any point that satisfies the KKT conditions (under Slater's constraint qualification) must be primal and dual optimal. Therefore,ŵ must obeyŵ = arg min w≥0 f (w). Note that the pair of points (ŵ,υ) that satisfies the KKT conditions is unique. To prove the optimal solutionŵ > 0 holds for (22), we can equivalently prove that the KKT conditions (25)- (27) hold for (ŵ > 0,υ = 0). It is further equivalent to prove that
holds forŵ > 0. Following from Lemma 5.2 with the fact thatŵ ∈ S w , there must exist an unique x such that
holds for any b = 0. Thus one has
where the first equality follows from (29) with b = 1; the second equality holds because J ∈ N (L * ) where N (L * ) is the null space of L * defined by N (L * ) := {X ∈ R p×p | L * X = 0}. Combining (28) and (30) yields
where L * L is invertible according to Lemma 5.3 . Recall that S is the sample covariance matrix defined by
where x (1) , . . . , x (n) are the samples independently drawn from L-GMRF in Definition 2.1. According to the density function of L-GMRF defined in (3), we get 1 x (k) = 0 for k = 1, . . . , n. Therefore, S is symmetric and obeys S · 1 = 0. It is easy to verify that S ∈ R(L), where R(L) is the range space of L defined by R(L) := {Ly | y ∈ R p(p−1)/2 }. Hence, there must exist a y ∈ R p(p−1)/2 such that S = Ly. Thus L * S = L * Ly. One further obtains
Then a simple calculation yields
Next, we construct a matrix X = Lx + aJ with a > 0 and have
where the first equality follows from (31); the second equality follows from (32); the third equality holds because L * L1 = 2p1 and then one has (L * L) −1 1 = 1 2p 1. LetX = D − 1 2 XD − 1 2 be the normalized matrix of X, where D is a diagonal matrix containing the diagonals of X. Notice that each diagonal element ofX is 1. Next, we will prove that, under some conditions,X is an inverse M -matrix, that is, (X) −1 is a M -matrix. We say A ∈ R p×p is an M -matrix if
where B ∈ R p×p is an element-wise non-negative matrix and s > ρ(B), the spectral radius of B. According to (33) , one has
Define S ij = max i =j S ij ,S ij = min i =j S ij , S kk = max k S kk andS kk = min k S kk . By the definition ofX, the lower bound y and upper bound x of the elements off the diagonal ofX can be obtained as below,
Define s by x 2 = sy + (1 − s)y 2 . According to Lemma 5.11 , if 0 < y ≤ x < 1 and s −1 ≥ p − 2 with p > 3, thenX is an inverse M -matrix. We can see, provided that y = 1 p+1 and the inequalities
hold, then one has
where the first inequality follows from x ≤ √ 2y. Therefore, if y = 1 p+1 and (37) holds, thenX is an inverse M -matrix.
Next, we will prove that if a = S kk − (p + 1)S ij + λ with λ ≥ 2(
and (37) holds. Substituting a = S kk − (p + 1)S ij + λ into (35) yields y = 1 p+1 . Then it is clear that y > 0 and √ 2y < 1. By comparing x and y defined in (35) and (36), respectively, one has y ≤ x. A simple algebra yields
where the first inequality follows from S ij ≤ S kk andS kk ≥S ij because S ∈ S p + and S · 1 = 0. It is easy to verify that λ ≥ 2( √ 2 + 1)(p + 1)( S kk −S ij ) is large enough to establish the second inequality. Therefore, all the inequalities in (37) hold.
Consequently, by Lemma 5.11, we conclude thatX is an inverse M -matrix when a = S kk − (p + 1)S ij + λ with λ ≥ 2( √ 2 + 1)(p + 1)( S kk −S ij ). Therefore,X −1 = D 1 2 X −1 D 1 2 is an M -matrix. Notice that the elements off the diagonal of an M -matrix are non-positive according to (34) . As a result, the elements off the diagonal of D 1 2 X −1 D 1 2 are non-positive, implying that the elements off the diagonal of X −1 are also non-positive, because X = Lx + aJ is positive definite and thus the diagonal elements of D are positive. The application of (29) with b = 1 a yields
One further obtains
. Therefore, we establisĥ
concluding that (28) holds forŵ > 0. Note that a = S kk − (p + 1)S ij + λ > 0 because S kk > 0, λ > 0, andS ij ≤ 0, whereS ij ≤ 0 following from S · 1 = 0 and each diagonal element S ii ≥ 0 since S is positive semi-definite.
Proof. Let x ∈ R p(p−1)/2 . According to the definition of L in (7), Lx must obey [Lx] ij = [Lx] ji , for any i = j and (Lx) · 1 = 0. Next, we will show that Lx is positive semi-definite for any x ≥ 0 by the Gershgorin circle theorem [62] . Given a matrix X ∈ R p×p with entries X ij . Let R i (X) = j =i |X ij | be the sum of the absolute values of the non-diagonal entries in the i-th row. Then a Gershgorin disc is the disc D(X ii , R i (X)) centered at X ii on the complex plane with radius R i (X). Gershgorin circle theorem [24] shows that each eigenvalue of X lies within at least one of the Gershgorin discs. For any x ≥ 0, R i (Lx) = [Lx] ii holds for each i because (Lx) · 1 = 0 and [Lx] ij ≤ 0 for any i = j. For any given eigenvalue λ of Lx, by Gershgorin circle theorem, there must exist one Gershgorin disc D([Lx] ii , R i (Lx)) such that
indicating that λ ≥ 0. Note that the eigenvalues of Lx are real since Lx is symmetric. Therefore, one has
Finally, we will prove that rank(Lx) = p − 1 ⇔ (Lx + J ) ∈ S p ++ , for any x ≥ 0. On one hand, if rank(Lx) = p − 1, then Lx + J admits the eigenvalue decomposition U ΛU , where U = [U s 1 √ p 1] and Λ is a diagonal matrix with the diagonal elements [λ 2 , . . . , λ p , 1]. Here λ p i=2 are the nonzero eigenvalues of Lx and U s is a p × (p − 1) matrix whose columns are the corresponding eigenvectors of Lx. Note that the nonzero eigenvalue λ p i=2 > 0 because Lx ∈ S p + . Therefore, one has (Lx + J ) ∈ S p ++ . On the other hand, if (Lx + J ) ∈ S p ++ , then rank(Lx) ≥ rank(Lx + J ) − rank(J ) = p − 1 because Lx + J is full rank and J is rank one. Furthermore, rank(Lx) ≤ p − 1 because (Lx) · 1 = 0. Therefore, we conclude that rank(Lx) = p − 1, completing the proof.
Proof. We first prove Theorem 3.8. Take the regularization parameter λ = 4αc −1 0 log p/n for some α > 2, and the sample size
where c 0 is a constant defined in Lemma 5.8. Notice that the sample size n in (40) 
Taking
Under the event J , L * (Lw + J ) −1 − S S can be bounded by
Therefore, under the event J , which holds with probability at least 1 − 1/p α−2 , one has
Next, we prove Corollary 3.9. Under the event J , one applies Lemma 5.6 and obtains
holds for any k ≥ 1.
Similarly, according to (42) , one obtains,
Alternative to (41) , one obtains
and correspondingly establishes
To apply Lemma 5.4, we first check the necessary conditions of the lemma. Let z (0) satisfy z Assumption 3.5 . According to (20) ,
where the first inequality holds becauseŵ (0) i < b for any i ∈ S (1) c , and h λ is non-increasing according to Assumption 3.5; the second inequality directly follows from Assumption 3.5. Hence one has
One also obtains |E (1) | < 2s by Lemma 5.5, and S ⊆ E (1) by the definition of E (1) . Therefore, one can apply Lemma 5.4 with E = E (1) and z = z (0) and obtains
3 , a simple algebra yields
Taking
, and combining (44) and (45) together, we can conclude that
completing the proof.
Proof. Take the regularization parameter λ = 4αc −1 0 log p/n for some α > 2, and the sample size
where c 0 is a constant defined in Lemma 5.8. The sample size n in (46) Recall that the initial pointŵ (0) of Algorithm 1 satisfies |supp + (ŵ (0) )| ≤ s. Define a local region around w
where τ is defined in (17) and δ > 1, B(w ; r) = {w ∈ R p(p−1)/2 | w − w ≤ r} and S w = {w ∈ R p(p−1)/2 |w ≥ 0, (Lw+J ) ∈ S p ++ }. Notice that B(w ; 1 √ 2pδτ ) must be nonempty because w ∈ S w according to Theorem 3.4 together with the fact that w contains the weights from a connected graph. According to Lemma 5.9, − log det(Lw + J ) is µ-strongly convex and L-smooth in B(w ; 1 √ 2pδτ ) with µ = 2 (1+δ −1 ) 2 τ 2 and L = 2pτ 2 (1−δ −1 ) 2 . Therefore, f k (w) = − log det(Lw + J ) + tr (SLw) + z (k−1) , w as defined in (14) is also µ-strongly convex and L-smooth in B(w ; 1 √ 2pδτ ), for any k ≥ 1. We define another local region
Next, we will prove that under the event J ,
holds for any k ≥ 1, whereŵ (k) is defined in (15) . Before applying Lemma 5.4, we first check the necessary conditions of the lemma. Let z (k−1) satisfy z Assumption 3.5 . According to the definition of E (k) in (20) , one has S ⊆ E (k) . For any i ∈ S (k) c , one further has
where the first inequality holds becauseŵ (k−1) i < b for any i ∈ S (k) c , and h λ is non-increasing by Assumption 3.5; the second inequality follows from Assumption 3.5. Hence one obtains
Under the event J , |E (k) | ≤ 2s holds for any k ≥ 1 following from Lemma 5.5. Therefore, all the conditions in Lemma 5.4 are satisfied with E = E (k) and z = z (k−1) , and one has
where the first inequality follows from (91) and the third inequality is established by plugging λ = 4αc −1 0 log p/n with n ≥ 840αc −1 0 (δτ 2 +1) 4 δ 2 τ 2 sp log p. The (47) indicates that
holds for any k ≥ 1. For any w ∈ B(ŵ (k) ; 2
Therefore, f k (w) is µ-strongly convex and L-smooth in B(ŵ (k) ; 2 3 √ 2pδτ ), for any k ≥ 1. Then we will establish that the whole sequence {w (k) t } t≥0 } returned from Algorithm 2 is within the region B(ŵ (k) ; 2 3 √ 2pδτ ) for any k ≥ 2. We first prove that w (k) t ∈ S w for any k ≥ 2 and t ≥ 0. Any w (k) t returned from Algorithm 2 must obey w (k) t ≥ 0 because of the projection P + . Notice that Lw + J must be positive semi-definite for any w ≥ 0, because Lw is positive semi-definite for any w ≥ 0 according to (39) , and J has an unique nonzero eigenvalue 1 with the eigenvector orthogonal to the row and column spaces of Lw. Therefore, any w (k) t / ∈ S w will lead Lw + J to be singular and positive semi-definite, and thus the objective function f k (w (k) t ) will go to infinity. On the other hand, one has
where the first inequality is established by the backtracking exit inequality in Algorithm 2; the second inequality is established by
t−1 ). The equality follows from the updating rule w
t−1 ) and the inequality holds because of the projection theorem together with the fact that w (k) t−1 ≥ 0. More specifically, let S ⊆ R n be a closed and convex set. Then P S (x) is the projection of x ∈ R n on S, if and only if one can establish
The (50) indicates that the objective function f k (w (k) t ) is not increasing with the increase of t, and consequently one has f k (w
t ) is also upper bounded and thus w (k) t ∈ S w . Notice that Algorithm 2 takes the initial point w (k) 0 =ŵ (k−1) , which is the minimizer of the optimization (15) . Thus, − log det(Lŵ (k−1) + J ) + tr SLŵ (k−1) is upper bounded.
is also upper bounded following from Assumption 3.5. Therefore, f k (w (k) 0 ) is upper bounded for any k ≥ 2. Then, we can conclude that
holds for any k ≥ 2 and t ≥ 0. We will prove by induction that the sequence {w
2pδτ ) for any k ≥ 2. For any given k ≥ 2, when t = 0, one has
where the second inequality follows from (48) . Together with (52), we conclude that w
On the other hand, one has
where the first inequality follows from the backtracking exit inequality, and f k (w) is µ-strongly convex in the region B(ŵ (k) ; 2
). The last equality is obtained by plugging 1 η w
. The second inequality follows from
t−1 ) and w (k) t = P + (w). The second equality follows from the updating rule w
t−1 ) and the inequality is established by the projection theorem as shown in (51),
Substituting (55) into (53) yields
). Together with (52), we conclude that w
, completing the induction. Therefore, the whole sequence {w
Finally, we will establish the lower bound of the step size η for each iteration.
Thus any η ∈ [0, 1 L ] must satisfy the backtracking exit condition and the backtracking line search must terminate when η ≤ β L with β ∈ (0, 1), implying that η ≥ β L . Therefore, we establish
where ρ = 1 − β(1−δ −1 ) 2 pτ 4 (1+δ −1 ) 2 < 1, completing the proof.
This section contains the proofs of the technical lemmas used in Section 5.
Proof. The gradient of f (w) is ∇f (w) = −L * (Lw + J ) −1 and its Hessian matrix is ∇ 2 f (w) with the k-th column being
and zeros for the other elements, in which i, j ∈ Z + obeying k = i − j + j−1 2 (2p − j) and i > j. Therefore, ∇ 2 f (w) can be written as
where B k = (Lw + J ) −1 A k (Lw + J ) −1 , for k = 1, 2, . . . , p(p − 1)/2. Then one has
where the forth equality follows from the definition of L in (7), and the last equality follows from the property of Kronecker product that vec(ABC) = C ⊗ A vec(B).
Proof. Let X = Lw + bJ with any b = 0 and any given w ∈ R p(p−1)/2 obeying (Lw + J ) ∈ S p ++ . It is easy to verify that the column spaces as well as row spaces of Lw and J are orthogonal with each other. Hence X admits the eigenvalue decomposition
where Lw = U ΛU and bJ = buu with u = 1 √ p 1 p , in which 1 p ∈ R p with each element equal to 1. Notice that Λ is non-singular. X −1 admits the eigenvalue decomposition
It is easy to check that U Λ −1 U is symmetric and U Λ −1 U · 1 p = 0. Therefore, there must exist a x such that Lx = U Λ −1 U . One further obtains
Hence such x is fixed and unique for a given w. Note that x is independent of b, and thus Lx + 1 b J = (Lw + bJ ) −1 holds for any b = 0, completing the proof.
Proof. Define an index set Ω t by
According to the definition of L in (7), for any x ∈ R p(p−1)/2 , one obtains that Lx ∈ R p×p obeys
where k = i − j + j−1 2 (2p − j). By the definition of L * in (8), one further obtains that L * Lx ∈ R p(p−1)/2 satisfies
where i, j ∈ [p] satisfying k = i − j + j−1 2 (2p − j) and i > j. The last equality holds because
where the last equality follows from [Lx] ij = −x k according to (58) . According to (59) , we conclude that there exists a matrix M ∈ R
such that L * Lx = M x for any x ∈ R p(p−1)/2 , and M obeys
where i, j ∈ [p] satisfying k = i−j+ j−1 2 (2p−j) and i > j. Note that we use the fact that {Ω i \k}∩{Ω j \k} = ∅ with k = i − j + j−1 2 (2p − j). Finally, we will compute the minimum and maximum eigenvalues of M . To compute the minimum eigenvalue of M , one has x M x
with equality when each element of x is equal with each other, and thus λ max (M ) = 2p. The last second equality is obtained by plugging |Ω t | = p − 1 with t ∈ [p], which is easy to verify according to the definition of Ω t in (57); the last equality follows from p t=1 i∈Ωt x 2 i = 2 p(p−1)/2 k=1
x 2 k , because for any k ∈ [p(p − 1)/2], k ∈ Ω t only holds with t = {i, j}, where i, j ∈ [p] obeying k = i − j + j−1 2 (2p − j) and i > j.
Proof. Take λ = 4αc −1 0 log p/n and n ≥ 94αc −1 0 λ 2 max (Lw )s log p. Define a local region
It is easy to check that w ∈ B M (w ; λ max (Lw )).
Recall thatŵ minimizes the optimization min w≥0 − log det(Lw + J ) + tr (LwS) + z w,
where 0 ≤ z i ≤ λ for i ∈ [p(p − 1)/2]. We can see the optimization problems (10) and (61) have the same feasible set. Therefore, S w is also the feasible set of (61) and thusŵ ∈ S w must hold. Next, we will prove thatŵ ∈ B M (w ; λ max (Lw )). We first construct an intermediate estimator,
where t is taken such that w t − w M = λ max (Lw ) if ŵ − w M ≥ λ max (Lw ), and t = 1 otherwise. Hence w t − w M ≤ λ max (Lw ) always holds and t ∈ [0, 1]. One further has w t ∈ S w because both w ,ŵ ∈ S w and S w is a convex set as shown in (98). Therefore, we conclude that w t ∈ B M (w ; λ max (Lw )). Applying Lemma 5.10 with w 1 = w t , w 2 = w * and r = λ max (Lw ) yields
Let q(a) = − log det L w + a(ŵ − w ) + J + a L * (Lw + J ) −1 ,ŵ − w and a ∈ [0, 1]. One has
and
where w a = w + a(ŵ − w ), A = (Lw a + J ) −1 and B = (Lŵ − Lw ). Note that A is symmetric and positive definite because w t ∈ S w and B is symmetric. Let C = AB. According to Theorem 1 in Drazin and Haynsworth [15] , all the eigenvalues of a matrix X ∈ R p×p are real if there exists a symmetric and positive definite matrix Y ∈ R p×p such that XY are symmetric. It is easy to check that the matrix CA is symmetric with A symmetric and positive definite, and thus all the eigenvalues of C are real. Suppose λ 1 , . . . , λ p are the eigenvalues of C. Then the eigenvalues of CC are λ 2 1 , . . . , λ 2 p . Therefore, q (a) = p i=1 λ 2 i ≥ 0, implying that q (a) is non-decreasing with the increase of a. Then one obtains
where the first inequality holds because q (a) is non-decreasing and t ≤ 1, and the second inequality follows from (63) . The Lagrangian of the optimization (61) is
where υ is a KKT multiplier. Let (ŵ,υ) be the primal and dual optimal point. Then (ŵ,υ) must satisfy the KKT conditions as below
w iυi = 0, for i = 1, . . . , p(p − 1)/2; (67) w ≥ 0,υ ≥ 0;
According to (66) , one has
Substituting (69) into (65) yields
Next we will bound term I, II and III, respectively. The term I can be directly bounded by
where the equality follows from (67) and the inequality follows fromυ ≥ 0 in (68) and w ≥ 0.
For term II, we separate the support of z into two parts, S and its complementary set {S } c , where S is the support of w with |S | ≤ s. Take a set E satisfying S ⊆ E and |E| ≤ 2s. A simple algebra yields
where the first inequality follows from CauchySchwarz inequality and the second inequality follows from z ≥ 0,ŵ ≥ 0 and E c ⊆ {S } c . For term III, we separate the support of L * (Lw + J ) −1 − S into parts, E and E c . Then one has
Substituting (71), (72) and (73) into (70) yields
Notice that the inequality
holds becauseŵ ≥ 0, z E c min ≥ λ/2 and
where the last inequality follows from the conditions in Lemma 5.4. Combining (74) and (75) together yields
where the last inequality follows from
On the other hand, one has
Combining (76) and (77) together yields
Recall that z max ≤ λ and |S | ≤ s. Thus one has
One also has
Substituting (79) and (80) into (78) yields
which implies that t = 1 in (62), i.e., w t =ŵ. The last inequality is established by plugging λ = 4αc −1 0 log p/n with n ≥ 94αc −1 0 λ 2 max (Lw )s log p. Therefore, we conclude that
where the first inequality is established by (78) with t = 1, and the second inequality is established by plugging (79) and (80).
Proof.
Recall that
We prove |E (k) | ≤ 2s holds by induction. For k = 1, ∀i / ∈ supp + (ŵ (0) ), i.e., w (1) . In other words, S (1) ⊆ supp + (ŵ (0) ). Then one has
Therefore, |E (k) | ≤ 2s holds for k = 1. Assume |E (k−1) | ≤ 2s holds for some k ≥ 2. We separate the set E (k) into two parts, S and S (k) \S . For any i ∈ S (k) \S , one hasŵ (k−1) i ≥ b, and further obtains
Collecting the indices i ∈ S together and applying (86) with w =ŵ (k−1) yields
where h λ (ŵ
is a constant vector. Combining (85) and (87) together yields
By separating the set E (k) into two parts, S and S (k) \S , one has
We will show that the term L * (Lw + J ) −1 − S S (k) \S can be bounded in terms of ŵ (k−1) − w .
where the last second equality follows from (82). Thus one has
Substituting (89) into (88) yields
where the equality is established by plugging b = (2+ √ 2)λ 2 max (Lw )λ and following from h λ (w S −b) = 0 because w S min − b ≥ γλ and h λ (x) = 0 for any x ≥ γλ following from Assumption 3.6; the last inequality follows from
Similarly, one also obtains
Proof. We apply Lemma 5.8 with t = λ/2 and union sum bound, then get
for any λ ≤ 2t 0 , where t 0 = L * (Lw + J ) −1 max and c 0 = 1/ 8 L * (Lw + J ) −1 2 max . Take λ = 4αc −1 0 log p/n for some α > 2. To guarantee λ ≤ 2t 0 , one takes n ≥ 8α log p. By calculation, we establish
completing the proof.
6.8 Proof of Lemma 5.8
Proof. The L-GMRF is a constrained GMRF model with 1 x = 0 and we can follow the method called conditioning by Kriging [52] to sample L-GMRF. More specifically, to sample x for L-GMRF with precision matrix Lw ,we could first sample from a unconstrained GMRFx ∼ N (0, (Lw + J ) −1 ) and then correct for the constraint 1 x = 0 by
For any i ∈ [p(p − 1)/2], one has
where the indices a and b obey i = a − b + (b − 1)(2p − b)/2 and a > b. LetΣ = (Lw + J ) −1 and thusx ∼ N (0,Σ). We first introduce two auxiliary random variables Y k,i :=x 
We can see Y k,i ∼ N (0, σ 2 i ) because of the fact that any linear combination of p components inx has a univariate normal distribution. The variance of Y k,i is
Therefore Z k,i /σ 2 i ∼ χ 2 (1) and E[Z k,i /σ 2 i ] = 1. We say a random variable X is sub-exponential if there are non-negative parameters (υ, α) such that E exp λ(X − E[X]) ≤ exp( υ 2 λ 2 2 ), for all |λ| < 1 α .
(95)
6.9 Proof of Lemma 5.9
Proof. Recall that B(w ; r) = {w ∈ R p(p−1)/2 | w − w ≤ r}, and S w = {w |w ≥ 0, (Lw + J ) ∈ S p ++ }. It is easy to verify that the set B(w ; r) is convex. For any x 1 , x 2 ∈ S w , define x t = tx 1 + (1 − t)x 2 , t ∈ [0, 1]. It is clear that x t ≥ 0. Since S p ++ is a convex cone, one has Lx t + J = t(Lx 1 + J ) + (1 − t)(Lx 2 + J ) ∈ S p ++ ,
indicating that x t ∈ S w and thus the set S w is convex. Hence B(w ; r) is a convex set. For any w 1 , w 2 ∈ B(w ; 1 √ 2pδτ ) with δ > 1, by Mean Value Theorem, we have
where w t = tw 2 + (1 − t)w 1 with t ∈ [0, 1]. Then one obtains λ min ∇ 2 f (w t ) = inf
x =1
x ∇ 2 f (w t )x = inf
x =1
(vec(Lx)) (Lw t + J ) −1 ⊗ (Lw t + J ) −1 vec(Lx)
≥ inf
x =1
(vec(Lx)) (Lw t + J ) −1 ⊗ (Lw t + J ) −1 vec(Lx) (vec(Lx)) vec(Lx)
(vec(Lx)) (Lw t ) † + J ⊗ (Lw t ) † + J vec(Lx) (vec(Lx)) vec(Lx)
· inf
x =1
x M x = 2 inf
x =1
(vec(Lx)) (Lw t ) † ⊗ (Lw t ) † vec(Lx) (vec(Lx)) vec(Lx)
where λ 2 (Lw t ) † denotes the second smallest eigenvalue of (Lw t ) † and (Lw t ) † is the pseudo inverse of Lw t . The second equality is according to Lemma 5.1; the third equality is established by (Lw t +J ) −1 = (Lw t ) † +J because the row spaces as well as column spaces of Lw t and J are orthogonal with each other, and M is defined in Lemma 5.3 with λ min (M ) = 2; the last equality follows from vec(Lx) (Lw t ) † ⊗J vec(Lx) = 0, vec(Lx) J ⊗ (Lw t ) † vec(Lx) = 0, and vec(Lx) (J ⊗ J )vec(Lx) = 0 which are easy to verify; the last inequality holds because of the property of Kronecker product that the eigenvalues of A⊗B are λ i µ j with the corresponding eigenvector a i ⊗ b j , where λ 1 , . . . , λ p are the eigenvalues of A ∈ R p×p with the corresponding eigenvectors a 1 , . . . , a p , and µ 1 , . . . , µ p are the eigenvalues of B ∈ R p×p with the corresponding eigenvectors b 1 , . . . , b p . Notice that there is one and only one zero eigenvalue for Lw t because w t ∈ S w . Assume λ 1 , . . . , λ p are the eigenvalues of (Lw t ) † with the corresponding eigenvectors a 1 , . . . , a p . Without loss of generality, let λ 1 = 0 and then a 1 = 1 √ p 1. By calculation, one obtains (vec(Lx)) vec(a i ⊗ a 1 ) = 0, and (vec(Lx)) vec(a 1 ⊗ a i ) = 0, for any i = 1, . . . , p and any x ∈ R p(p−1)/2 , indicating that vec(Lx) is orthogonal to all the eigenvectors of (Lw t ) † ⊗ (Lw t ) † corresponding to zero eigenvalues. The smallest nonzero eigenvalue of (Lw t ) † ⊗ (Lw t ) † is λ 2 2 (Lw t ) † , establishing (100). One further has
where the second inequality follows from Lx 2 ≤ Lx F ≤ (x M x) 1 2 ≤ √ 2p x according to Lemma 5.3; the third inequality holds because both w 1 , w 2 ∈ B(w ; 1 √ 2pδτ ), and Lw 2 ≤ τ following from Assumption 3.6; the last inequality follows from 1 δτ ≤ δ −1 τ , where τ > 1. Substituting (101) into (100) yields λ min ∇ 2 g(w t ) ≥ 2
Combining (99) and (102) together yields f (w 2 ) ≥ f (w 1 ) + ∇f (w 1 ), w 2 − w 1 + 1
Next, we will bound λ max (∇ 2 f (w t )). Similarly, one has λ max ∇ 2 f (w t ) = sup
x =1
x ∇ 2 f (w t )x = sup
x =1
(vec(Lx)) (Lw t + J ) −1 ⊗ (Lw t + J ) −1 vec(Lx)
≤ sup
x =1
(vec(Lx)) (Lw t ) † + J ⊗ (Lw t ) † + J vec(Lx) (vec(Lx)) vec(Lx)
· sup
x =1
(vec(Lx)) (Lw t ) † ⊗ (Lw t ) † vec(Lx) (vec(Lx)) vec(Lx) · λ max (M )
where the last equality follows from Lemma 5.3. One further obtains,
where λ 2 (Lw t ) denotes the second smallest eigenvalue of Lw t , i.e., the minimum nonzero eigenvalue, which could be bounded by
where the first equality holds because Lw t has only one zero eigenvalue and its eigenvector is 1 √ p 1, and the eigvectors of Lw t associated with different eigenvalues are orthogonal with each other; the last inequality follows from Assumption 3.6. Substituting (104) and (105) into (103) yields
Combining (99) and (106) ) is a convex set because both B M (w ; r) and S w are convex. It is easy to check that S w is convex (See (98) for more details). For any w 1 , w 2 ∈ B M (w ; r), by Mean Value Theorem, one obtains f (w 2 ) = f (w 1 ) + ∇f (w 1 ), w 2 − w 1 + 1 2
where w t = tw 2 + (1 − t)w 1 with t ∈ [0, 1]. For any nonzero x ∈ R p(p−1)/2 , one has
x ∇ 2 f (w t )x = (vec(Lx)) (Lw t + J ) 
where λ 2 (Lw t ) † denotes the second smallest eigenvalue of (Lw t ) † , and (Lw t ) † is the pseudo inverse of Lw t . The first equality follows from Lemma 5.1 and the last inequality is according to (100). Notice that (108) also holds with x = 0. One further obtains
where the second inequality is established by Lx 2 ≤ Lx F and the fact that both w 1 , w 2 ∈ B M (w ; r). Substituting (109) into (108) yields
Combining (107) and (110) yields
and
Combining (111) and (112), we establish
completing the proof.
In this paper, we have considered learning a sparse graph under the Laplacian constrained GGM. First we have proved that a large regularization parameter of the 1 -norm leads to a fully connected graph. Then we have proposed a nonconvex penalized maximum likelihood estimation method with theoretical guarantees on the estimation error, by solving a sequence of weighted 1 -norm regularized sub-problems. We have proved that the statistical error of the proposed estimator matches the minimax lower bound. A projected gradient descent algorithm has designed to solve each sub-problem which enjoys a linear convergence rate. Numerical results have demonstrated the effectiveness of the proposed method. It is interesting to extend the proposed method to learn other structured graphs like k-component graphs, and characterize the statistical properties. We leave this for the future research.