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... is the entrywise ℓ1 norm. The ˆ Θ that solves (1) serves as an estimate of Σ−1 . This estimate will be positive definite for any λ > 0, and sparse when λ is sufficiently large, due to the ℓ1 penalty =-=[10]-=- in (1). We refer to (1) as the graphical lasso formulation. This formulation is convex, and efficient algorithms for solving it are available [6, 4, 5, 7, 11]. 22.2 The fused graphical lasso In rece...

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...e a GGM on the basis of n observations, X1, . . . , Xn ∈ R p , which are independent and identically distributed N(0, Σ). It is well known that this amounts to learning the sparsity structure of Σ −1 =-=[1, 2]-=-. When n > p, one can estimate Σ −1 by maximum likelihood, but when p > n this is not possible because the empirical covariance matrix is singular. Consequently, a number of authors [3, 4, 5, 6, 7, 8,...

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...e a GGM on the basis of n observations, X1, . . . , Xn ∈ R p , which are independent and identically distributed N(0, Σ). It is well known that this amounts to learning the sparsity structure of Σ −1 =-=[1, 2]-=-. When n > p, one can estimate Σ −1 by maximum likelihood, but when p > n this is not possible because the empirical covariance matrix is singular. Consequently, a number of authors [3, 4, 5, 6, 7, 8,...

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... ⎨ ⎩ L(Θ1 , Θ 2 ) − λ1∥Θ 1 ∥1 − λ1∥Θ 2 p∑ ∥1 − λ2 ∥Θ 1 j − Θ 2 ⎫ ⎬ j∥2 , (4) ⎭ maximize Θ 1 ∈S p ++ ,Θ2 ∈S p ++ where Θ k j is the jth column of the matrix Θk . This amounts to applying a group lasso =-=[15]-=- penalty to the columns of Θ 1 − Θ 2 . Since a group lasso penalty simultaneously shrinks all elements to which it is applied to zero, it appears that this will give the desired node perturbation stru...

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...t easy to compute and in addition the proximal operator to RCON is non-trivial to compute. In this section we present a fast and scalable alternating directions method of multipliers (ADMM) algorithm =-=[22]-=- to solve the problem (7). We first reformulate (7) by introducing new variables, so as to decouple some of the terms in the objective function that are difficult to optimize jointly. This will result...

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...ure of Σ −1 [1, 2]. When n > p, one can estimate Σ −1 by maximum likelihood, but when p > n this is not possible because the empirical covariance matrix is singular. Consequently, a number of authors =-=[3, 4, 5, 6, 7, 8, 9]-=- have considered maximizing the penalized log likelihood maximize Θ∈S p {log det Θ − trace(SΘ) − λ∥Θ∥1} , (1) ++ where S is the empirical covariance matrix based on the n observations, λ is a positive...

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...ure of Σ −1 [1, 2]. When n > p, one can estimate Σ −1 by maximum likelihood, but when p > n this is not possible because the empirical covariance matrix is singular. Consequently, a number of authors =-=[3, 4, 5, 6, 7, 8, 9]-=- have considered maximizing the penalized log likelihood maximize Θ∈S p {log det Θ − trace(SΘ) − λ∥Θ∥1} , (1) ++ where S is the empirical covariance matrix based on the n observations, λ is a positive...

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.... . . , ˆ ΘK that solve (2) serve as estimates for (Σ1) −1 , . . . , (ΣK) −1 . In particular, [13] considered the use of P (Θ 1 ij, . . . , Θ K ij ) = ∑ |Θ k ij − Θ k′ ij |, (3) a fused lasso penalty =-=[14]-=- on the differences between pairs of network edges. When λ1 is large, the network estimates will be sparse, and when λ2 is large, pairs of network estimates will have identical edges. We refer to (2) ...

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...ure of Σ −1 [1, 2]. When n > p, one can estimate Σ −1 by maximum likelihood, but when p > n this is not possible because the empirical covariance matrix is singular. Consequently, a number of authors =-=[3, 4, 5, 6, 7, 8, 9]-=- have considered maximizing the penalized log likelihood maximize Θ∈S p {log det Θ − trace(SΘ) − λ∥Θ∥1} , (1) ++ where S is the empirical covariance matrix based on the n observations, λ is a positive...

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...t of Θ 1 − Θ 2 is in both the ith (row) and jth (column) groups. In the presence of overlapping groups, the group lasso penalty yields estimates whose support is the complement of the union of groups =-=[16, 17]-=-. Figure 2 shows a simple example of (Σ 1 ) −1 −(Σ 2 ) −1 in the case of node perturbation, as well as the estimate obtained using (4). The figure reveals that (4) cannot be used to detect node pertur...

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...imal operator corresponding to the ℓ1/ℓq norm. For q = 1, 2, ∞, Tq takes a simple form, which we omit here due to space constraints. A description of these operators can also be found in Section 5 of =-=[25]-=-. Algorithm 1 can be interpreted as an approximate dual gradient ascent method. The approximation is due to the fact that the gradient of the dual to the augmented Lagrangian in each iteration is comp...

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... when λ is sufficiently large, due to the ℓ1 penalty [10] in (1). We refer to (1) as the graphical lasso formulation. This formulation is convex, and efficient algorithms for solving it are available =-=[6, 4, 5, 7, 11]-=-. 22.2 The fused graphical lasso In recent literature, convex formulations have been proposed for extending the graphical lasso (1) to the setting in which one has access to a number of observations ...

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...through more than two such groups. Although investigation of the convergence properties of ADMM algorithms for an arbitrary number of groups is an ongoing research area in the optimization literature =-=[23, 24]-=- and specific convergence results for our algorithm are not known, we empirically observe very good convergence behavior. Further study of this issue is a direction for future work. We initialize the ...

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...ure of Σ −1 [1, 2]. When n > p, one can estimate Σ −1 by maximum likelihood, but when p > n this is not possible because the empirical covariance matrix is singular. Consequently, a number of authors =-=[3, 4, 5, 6, 7, 8, 9]-=- have considered maximizing the penalized log likelihood maximize Θ∈S p {log det Θ − trace(SΘ) − λ∥Θ∥1} , (1) ++ where S is the empirical covariance matrix based on the n observations, λ is a positive...

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...nt to which PNJGL with q = 2 outperforms others is more apparent when n is small. 5.2 Inferring biological networks We applied the PNJGL method to a recently-published cancer gene expression data set =-=[26]-=-, with mRNA expression measurements for 11,861 genes in 220 patients with glioblastoma multiforme (GBM), a brain cancer. Each patient has one of four distinct clinical subtypes: Proneural, Neural, Cla...

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... However, such a general approach does not fully exploit the structure of the problem and will not scale well to large-scale instances. Other algorithms proposed for overlapping group lasso penalties =-=[19, 20, 21]-=- do not apply to our setting since the PNJGL formulation has a combination of Gaussian log-likelihood loss (instead of squared error loss) and the RCON penalty along with a positivedefinite constraint...

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...ure of Σ −1 [1, 2]. When n > p, one can estimate Σ −1 by maximum likelihood, but when p > n this is not possible because the empirical covariance matrix is singular. Consequently, a number of authors =-=[3, 4, 5, 6, 7, 8, 9]-=- have considered maximizing the penalized log likelihood maximize Θ∈S p {log det Θ − trace(SΘ) − λ∥Θ∥1} , (1) ++ where S is the empirical covariance matrix based on the n observations, λ is a positive...

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...ber of observations from K distinct conditions. The goal of the formulations is to estimate a graphical model for each condition under the assumption that the K networks share certain characteristics =-=[12, 13]-=-. Suppose that Xk 1 , . . . , Xk nk ∈ Rp are independent and identically distributed from a N(0, Σk ) distribution, for k = 1, . . . , K. Letting Sk denote the empirical covariance matrix for the kth ...

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...ber of observations from K distinct conditions. The goal of the formulations is to estimate a graphical model for each condition under the assumption that the K networks share certain characteristics =-=[12, 13]-=-. Suppose that Xk 1 , . . . , Xk nk ∈ Rp are independent and identically distributed from a N(0, Σk ) distribution, for k = 1, . . . , K. Letting Sk denote the empirical covariance matrix for the kth ...

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...through more than two such groups. Although investigation of the convergence properties of ADMM algorithms for an arbitrary number of groups is an ongoing research area in the optimization literature =-=[23, 24]-=- and specific convergence results for our algorithm are not known, we empirically observe very good convergence behavior. Further study of this issue is a direction for future work. We initialize the ...

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...ure of Σ −1 [1, 2]. When n > p, one can estimate Σ −1 by maximum likelihood, but when p > n this is not possible because the empirical covariance matrix is singular. Consequently, a number of authors =-=[3, 4, 5, 6, 7, 8, 9]-=- have considered maximizing the penalized log likelihood maximize Θ∈S p {log det Θ − trace(SΘ) − λ∥Θ∥1} , (1) ++ where S is the empirical covariance matrix based on the n observations, λ is a positive...

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...t of Θ 1 − Θ 2 is in both the ith (row) and jth (column) groups. In the presence of overlapping groups, the group lasso penalty yields estimates whose support is the complement of the union of groups =-=[16, 17]-=-. Figure 2 shows a simple example of (Σ 1 ) −1 −(Σ 2 ) −1 in the case of node perturbation, as well as the estimate obtained using (4). The figure reveals that (4) cannot be used to detect node pertur...

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... However, such a general approach does not fully exploit the structure of the problem and will not scale well to large-scale instances. Other algorithms proposed for overlapping group lasso penalties =-=[19, 20, 21]-=- do not apply to our setting since the PNJGL formulation has a combination of Gaussian log-likelihood loss (instead of squared error loss) and the RCON penalty along with a positivedefinite constraint...

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... However, such a general approach does not fully exploit the structure of the problem and will not scale well to large-scale instances. Other algorithms proposed for overlapping group lasso penalties =-=[19, 20, 21]-=- do not apply to our setting since the PNJGL formulation has a combination of Gaussian log-likelihood loss (instead of squared error loss) and the RCON penalty along with a positivedefinite constraint...

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...ure of Σ −1 [1, 2]. When n > p, one can estimate Σ −1 by maximum likelihood, but when p > n this is not possible because the empirical covariance matrix is singular. Consequently, a number of authors =-=[3, 4, 5, 6, 7, 8, 9]-=- have considered maximizing the penalized log likelihood maximize Θ∈S p {log det Θ − trace(SΘ) − λ∥Θ∥1} , (1) ++ where S is the empirical covariance matrix based on the n observations, λ is a positive...

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...rselves to the case of K = 2 in this paper. 44 An ADMM algorithm for the PNJGL formulation The PNJGL optimization problem (7) is convex, and so can be directly solved in the modeling environment cvx =-=[18]-=-, which calls conic interior-point solvers such as SeDuMi or SDPT3. However, such a general approach does not fully exploit the structure of the problem and will not scale well to large-scale instance...

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... was not identified as a frequently mutated gene in GBM [26]. However, there is recent evidence that CXCL13 plays a critical role in driving cancerous pathways in breast, prostate, and ovarian tissue =-=[27, 28]-=-. Our results suggest the possibility of a previously unknown role of CXCL13 in brain cancer. 6 Discussion and future work We have proposed the perturbed-node joint graphical lasso, a new approach for...

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... was not identified as a frequently mutated gene in GBM [26]. However, there is recent evidence that CXCL13 plays a critical role in driving cancerous pathways in breast, prostate, and ovarian tissue =-=[27, 28]-=-. Our results suggest the possibility of a previously unknown role of CXCL13 in brain cancer. 6 Discussion and future work We have proposed the perturbed-node joint graphical lasso, a new approach for...