structured graph learning via laplacian spectral constraints...squirrel tiger lion cat dog wolf seal...

Structured Graph Learning Via LaplacianSpectral Constraints

Sandeep Kumar, Jiaxi Ying, José Vinícius de M. Cardoso,and Daniel P. Palomar

The Hong Kong University of Science and Technology (HKUST)

NeurIPS 2019, Vancouver, Canada

11 December 2019

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Outline

1 Graphical modeling

2 Probabilistic graphical model: GMRF

3 Structured graph learning (SGL): motivation, challenges and direction

4 Proposed framework for SGL via Laplacian spectral constraints

5 Algorithm: SGL via Laplacian spectral constraints

6 Experiments

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Outline






6 Experiments

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Graphical models

Representing knowledge through graphical models

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I Nodes correspond to the entities (variables).

I Edges encode the relationships between entities (dependencies betweenthe variables)

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Why do we need graphical models?

I Graphs are intuitive way of representing and visualising therelationships between entities.

I Graphs allow us to abstract out the conditional independencerelationships between the variables from the details of their parametricforms. Thus we can answer questions like: “Is x1 dependent on x6given that we know the value of x8?” just by looking at the graph.

I Graphs are widely used in a variety of applications in machine learning,graph CNN, graph signal processing, etc.

I Graphs offer a language through which different disciplines canseamlessly interact with each other.

I Graph-based approaches with big data and machine learning are drivingthe current research frontiers.

Graphical Models = Statistics × Graph Theory × Optimization × Engineering

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Why do we need graph learning?

Graphical models are about having a graph representation that can encoderelationships between entities.

In many cases, the relationships between entities are straightforward:I Are two people friends in a social network?I Are two researchers co-authors in a published paper?

In many other cases, relationships are not known and must be learned:I Does one gene regulate the expression of others?I Which drug alters the pharmacologic effect of another drug?

The choice of graph representation affects the subsequent analysis andeventually the performance of any graph-based algorithm.

The goal is to learn a graph representation of data with specific properties(e.g., structures).

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Schematic of graph learning

I Given a data matrix X ∈ Rn×p = [x1,x2, . . . ,xp], each columnxi ∈ Rn is assumed to reside on one of the p nodes and each of the nrows of X is a signal (or feature) on the same graph.

I The goal is to obtain a graph representation of the data.

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Graph is a simple mathematical structure of form G = (V, E), whereI V contains the set of nodes V = {1, 2, 3, . . . , p}, andI E = {(1, 2), (1, 3), . . . , (i, j), . . . , (p, p− 1)} contains the set of edges

between any pair of nodes (i, j).I Weights {w12, w13, . . . , wij , . . .} encode the relationships strength.

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Examples

Learning relational dependencies among entities benefits numerousapplication domains.

Figure 1: Financial Graph

Objective: To infer inter-dependencies offinancial companies.Input xi is the economic indices (stockprice, volume, etc.) of each entity.

Figure 2: Social Graph

Objective: To model behavioral similarity/influence between people.Input: Input xi is the individual onlineactivities (tagging, liking, purchase).

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Types of graphical models

I Models encoding direct dependencies: simple and intuitive.I Sample correlation based graph.

I Similarity function (e.g., Gaussian RBF) based graph.

I Models based on some assumption on the data: X ∼ F(G)I Statistical models: F represents a distribution by G (e.g., Markov model

and Bayesian model).

I Physically-inspired models: F represents generative model on G (e.g.,diffusion process on graphs).

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Outline






6 Experiments

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Gaussian Markov random field (GMRF)

A random vector x = (x1, x2, . . . , xp)> is called a GMRF with parameters

(0, Θ), if its density follows:

p(x) = (2π)(−p/2)(det(Θ))12 exp

(−1

2(x>Θx)

).

The nonzero pattern of Θ determines a conditional graph G = (V, E) :

Θij 6= 0 ⇐⇒ {i, j} ∈ E ∀ i 6= j

xi ⊥ xj |x/(xi, xj) ⇐⇒ Θij = 0

I For a Gaussian distributed data x ∼ N (0,Σ = Θ†) the graphlearning is simply an inverse covariance (precision) matrix estimationproblem [Lauritzen, 1996].

I If the rank(Θ) < p then x is called an improper GMRF (IGMRF)[Rue and Held, 2005].

I If Θij ≤ 0 ∀ i 6= j then x is called an attractive improper GMRF[Slawski and Hein, 2015].

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Historical timeline of Markov graphical models

Data X = {x(i) ∼ N (0,Σ = Θ†)}ni=1, S = 1n

∑ni=1(x(i))(x(i))>

I Covariance selection [Dempster, 1972]: graph from the elements of S−1

inverse sample covariance matrix. Not applicable when samplecovariance is not invertible!

I Neighborhood regression [Meinshausen and Bühlmann, 2006]:

arg minβ1

|x(1) − β1X/x(1) |2 + α‖β1‖1

I `1-regularized MLE [Friedman et al., 2008, Banerjee et al., 2008]:

maximizeΘ�0

log det(Θ)− tr(ΘS)− α‖Θ‖1.

I Ising model: `1-regularized logistic regression [Ravikumar et al., 2010].I Attractive IGMRF [Slawski and Hein, 2015].I Laplacian structure in Θ [Lake and Tenenbaum, 2010].I `1-regularized MLE with Laplacian structure

[Egilmez et al., 2017, Zhao et al., 2019]Limitation: Existing methods are not suitable for learning graphs withspecific structures.

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Outline






6 Experiments

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Structured graphs

(i) Multi-componentgraph

(ii) Regular graph (iii) Modular graph

(iv) Bipartite graph (v) Grid graph (vi) Tree graph

Figure 3: Useful graph structures

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Structured graphs: importance

Useful structures:I Multi-component: graph for clustering, classification.I Bipartite: graph for matching and constructing two-channel filter

banks.I Multi-component bipartite: graph for co-clustering.I Tree: graphs for sampling algorithms.I Modular: graph for social network analysis.I Connected sparse: graph for graph signal processing applications.

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Structured graph learning: challenges

Structured graph learning from dataI involves both the estimation of structure (graph connectivity) and

parameters (graph weights),I parameter estimation is well explored (e.g., maximum likelihood),I but structure is a combinatorial property which makes structure

estimation very challenging.Structure learning is NP-hard for a general class of graphical models[Bogdanov et al., 2008].

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Structured graph learning: direction

State-of-the-art direction:I The effort has been on characterizing the families of structures for

which learning can be made feasible e.g., maximum weight spanningtree for tree structure [Chow and Liu, 1968] and local-separation andwalk summability for Erdos-Renyi graphs, power-law graphs, andsmall-world graphs [Anandkumar et al., 2012].

I Existing methods are restricted to some particular structures and it isdifficult to extend them to learn other useful structures, e.g.,multi-component, bipartite, etc.

I A recent method in [Hao et al., 2018], for learning multi-componentstructure follows a two-stage approach: non-optimal and not scalable tolarge-scale problems.

Proposed direction: Graph (structure) ⇐⇒ Graph matrix (spectrum)

I Spectral properties of a graph matrix is one such characterization[Chung, 1997] which is considered in the present work.

I Under this framework, structure learning of a large class of graphstructures can be expressed as an eigenvalue problem of the graphLaplacian matrix.

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Outline






6 Experiments

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Problem statement

To learn structured graphs via Laplacianspectral constraints

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Laplacian matrix

A set of p× p symmetric graph Laplacian matrices Θ:

SΘ ={

Θ|Θij = Θji ≤ 0 for i 6= j,Θii = −∑j 6=i

Θij

}.

Properties of Θ: Symmetric, diagonally dominant, positive semi-definite,and eigenvalues of Θ encodes the structural properties of many importantstructures.

Laplacian quadratic energy function:

tr(SΘ) =∑i,j

−Θij(xi − xj)2

I The above trace term is used to quantify smoothness of graph signals:a smaller tr(SΘ) indicating a smoother signal x.

I A graph learned by minimizing the trace term tends to put more weighton the relationship of xi, xj if they are similar, and vice versa.

I If the signals xi, xj are similar then the learned Laplacian weights |Θij |will be large, and vice versa.

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Motivating example: structure via Laplacian eigenvalues

Spectral graph theory: Graph (structure)⇐⇒ Graph Matrix (spectrum)

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0 10 20 30 40 50 60Number of nodes(eigenvalues)

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Eig

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of L

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rix

3 zero eigenavlues of 3 component graph Laplacian

A graph and its Laplacian matrix eigenvalues: k = 3 zero eigenvalues corresponding tok = 3 connected components.

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Proposed framework for structured graph learning

maximizeΘ

log gdet(Θ)− tr(ΘS)− αh(Θ),

subject to Θ ∈ SΘ, λ(T (Θ)) ∈ Sλ,

I gdet is the generalized determinant defined as the non-zeroeigenvalues product,

I SΘ encodes the typical constraints of a Laplacian matrix,I λ(T (Θ)) is the vector containing the eigenvalues of matrix T (Θ),I T (·) is the transformation matrix to consider the eigenvalues of

different graph matrices, andI Sλ allows to include spectral constraints in the eigenvalues.I Precisely Sλ will facilitate the process of incorporating the spectral

properties required for enforcing structure.

The proposed formulation has converted the combinatorial structuralconstraints into analytical spectral constraints.

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Structures via Laplacian spectral constraints

T (Θ) = Θ

I Connected: Sλ = {λ1 = 0, c1 ≤ λ2 ≤ · · · ≤ λp ≤ c2}

I k−component: Sλ = {{λi = 0}ki=1, c1 ≤ λk+1 ≤ · · · ≤ λp ≤ c2}

I d−regular: Sλ = {{λi = 0}ki=1, c1 ≤ λk+1 ≤ · · · ≤ λp ≤ c2} andDiag(Θ) = dI

I Popular connected structures, e.g., Grid, Modular, and Erdos-Renyican also be learned under the connected spectral constraint.

Note: By properly specifying the transformation matrix T (·) in the proposedformulation, the spectral properties of other than graph Laplacian, e.g.,adjacency, normalized Laplacian, and signless Laplacian can also beutilized to learn more non-trivial structures (e.g., bipartite andmulti-component bipartite graph structures)[Van Mieghem, 2010, Kumar et al., 2019, Chung, 1997].

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Outline






6 Experiments

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Problem formulation for Laplacian spectral constraints

maximizeΘ,λ,U

log gdet(Θ)− tr(ΘS)− α ‖Θ‖1 ,

subject to Θ ∈ SΘ, Θ = UDiag(λ)UT , λ ∈ Sλ, UTU = I,

where λ = [λ1, λ2, . . . , λp] is the vector of eigenvalues and U is the matrixof eigenvectors.

The resulting formulation is still complicated and intractable:I Laplacian structural constraints,I non-convex constraints coupling Θ,U,λ, andI non-convex constraints on U.

In order to derive a feasible formulation:I we first introduce a linear operator L that transforms the Laplacian

structural constraints to simple algebraic constraints andI then relax the eigen-decomposition expression into the objective

function.

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Linear operator for Θ ∈ SΘ

SΘ ={

Θ|Θij = Θji ≤ 0 for i 6= j,Θii = −∑j 6=i

Θij

},

Θij = Θji ≤ 0 and Θ1 = 0 implying the target matrix is symmetric withdegrees of freedom of Θ equal to p(p− 1)/2.We define a linear operator L : w ∈ Rp(p−1)/2+ → Lw ∈ Rp×p, which maps aweight vector w to the Laplacian matrix:

[Lw]ij = [Lw]ji ≤ 0; i 6= j

[Lw]ii = −∑j 6=i

[Lw]ij

Example of Lw on w = [w1, w2, w3, w4, w5, w6]>:

Lw =

∑i=1,2,3 wi −w1 −w2 −w3

−w1

∑i=1,4,5 wi −w4 −w5

−w2 −w4

∑i=2,4,6 wi −w6

−w3 −w5 −w6

∑i=3,5,6 wi

.

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Problem reformulation

maximizeΘ,λ,U

log gdet(Θ)− tr(ΘS)− α ‖Θ‖1 ,

subject to Θ ∈ SΘ, Θ = UDiag(λ)UT , λ ∈ Sλ, UTU = I,

Using: i) Θ = Lw and ii) tr(ΘS)

+ αh(Θ) = tr(ΘK

), K = S + H and

H = α(2I− 11T ) the proposed problem formulation becomes:

⇓

maximizew,λ,U

log gdet(Diag(λ))− tr(KLw)− β

2‖Lw −UDiag(λ)UT ‖2F ,

subject to w ≥ 0, λ ∈ Sλ, UTU = I.

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SGL algorithm for k−component graph learning

I Variables: X = (w, λ, U)

I Spectral constraint: Sλ = {{λj = 0}kj=1, c1 ≤ λk+1 ≤ · · · ≤ λp ≤ c2}.I Positivity constraint: w ≥ 0

I Orthogonality constraint: UTU = Ip−k

We develop a block majorization-minimization (block-MM) type methodwhich updates each block sequentially while keeping the other blocksfixed [Sun et al., 2016, Razaviyayn et al., 2013].

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Update for w

Sub-problem for w:

minimizew≥0

tr (KLw) +β

2‖Lw −UDiag(λ)UT ‖2F .

minimizew≥0

f(w) =1

2‖Lw‖2F − cTw,

This problem is a convex quadratic program, but does not have aclosed-form solution due to the non-negativity constraint w ≥ 0.

We obtain a closed-form update by using the MM technique[Sun et al., 2016]

wt+1 =

(wt − 1

2p∇f(wt)

)+

,

where (a)+ = max(a, 0).

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Update for U

Sub-problem for U:

maximizeU

tr(UTLwUDiag(λ)

)subject to UTU = Ip−k.

This sub- problem is an optimization on the orthogonal Stiefel manifold[Absil et al., 2009, Benidis et al., 2016]. From the KKT optimalityconditions the solution is given by

Ut+1 = eigenvectors(Lwt+1)[k + 1 : p],

that is, the p− k principal eigenvectors of the matrix Lwt+1 in increasingorder of eigenvalue magnitude.

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Update for λ

Sub-problem for λ:

minimizeλ∈Sλ

− log det(λ) +β

2‖UT (Lw)U− Diag(λ)‖2F .

minimizec1≤λk+1≤···≤λp≤c2

−p−k∑i=1

log λk+i +β

2‖λ− d‖2,

The sub-problem is popularly known as a regularized isotonic regressionproblem. This is a convex optimization problem and the solution can beobtained from the KKT optimality conditions. We develop an efficientalgorithm with a fast convergence to the global optimum in a maximum ofp− k iterations [Kumar et al., 2019].

Sandeep Kumar, Jiaxi Ying, José Vinícius de M. Cardoso, and Daniel P. Palomar,“A Unified Framework for Structured Graph Learning via Spectral Constraints.” arXivpreprint arXiv:1904.09792 (2019).

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Proposed SGL algorithm summary

maximizew,λ,U

log gdet(Diag(λ))− tr(KLw)− β

2‖Lw −UDiag(λ)UT ‖2F ,

subject to w ≥ 0, λ ∈ Sλ, UTU = Ip−k.

Proposed algorithm:

1: Input: SCM S, k, c1, c2, β2: Output: Lw3: t← 04: while stopping criterion is not met do

5: wt+1 =(wt − 1

2p∇f(wt))+

6: Ut+1 ← eigenvectors(Lwt+1), suitably ordered.

7: Update λt+1 (via isotonic regression method with maxm iter p− k).

8: t← t+ 19: end while

10: return w(t+1)

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Convergence and the computational complexity

The worst-case computational complexity of the proposed algorithm isO(p3).

Theorem: The limit point (w?,U?,λ?) generated by this algorithmconverges to the set of KKT points of the optimization problem.

Sandeep Kumar, Jiaxi Ying, José Vinícius de M. Cardoso, and Daniel P. Palomar,“Structured graph learning via Laplacian spectral constraints,” in Advances in NeuralInformation Processing Systems (NeurIPS), 2019.

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Outline






6 Experiments

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Synthetic experiment setup

I Generate a graph with desired structure.I Sample weights for the graph edges.I Obtain true Laplacian Θtrue.I Sample data X = {x(i) ∈ Rp ∼ N (0,Σ = Θ†true)}ni=1.I S = 1

n

∑ni=1(x(i))(x(i))>.

I Use S and some prior spectral information, if available.I Performance metric

Relative Error =

∥∥∥Θ̂?−Θtrue

∥∥∥F

‖Θtrue‖F, F-Score =

2tp2tp + fp + fn

I Where Θ̂?is the final estimation result the algorithm and Θtrue is the

true reference graph Laplacian matrix, and tp, fp, fn correspond to truepositives, false positives, and false negatives, respectively.

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Grid graph

(i) True (ii) [Egilmez et al., 2017] (iii) SGL with k = 1

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Noisy multi-component graph

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(vii) True graph (viii) Noisy graph (ix) Learned

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Model mismatch

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Popular multi-component structures

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Real data: cancer dataset [Weinstein et al., 2013]

(xxii) CLR (Nie et al., 2016) (xxiii) SGL with k = 5

Clustering accuracy (ACC): CLR = 0.9862 and SGL = 0.99875.

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Animal dataset [Osherson et al., 1991]

ElephantRhino

Horse

CowCamel

Giraffe

Chimp

Gorilla

Mouse

Squirrel

TigerLion

CatDog

Wolf

Seal

Dolphin

Robin

Eagle

Chicken

Salmon

Trout

Bee

Iguana

Alligator

Butterfly

Ant

Finch

Penguin

Cockroach

Whale

Ostrich

Deer

(xxiv) GGL [Egilmez et al., 2017]

Elephant

Rhino

HorseCow

Camel

Giraffe

Chimp

Gorilla

Mouse

Squirrel

Tiger

Lion

Cat

Dog Wolf

Seal

Dolphin

Robin

Eagle

Chicken

Salmon

TroutBee

Iguana

Alligator

Butterfly

Ant

Finch

Penguin

Cockroach

Whale

Ostrich

Deer

(xxv) GLasso [Friedman et al., 2008]

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Animal dataset contd...

Elephant

Rhino

Horse

Cow

Camel

Giraffe

ChimpGorilla

Mouse

Squirrel

Tiger

LionCat

DogWolf

Seal Dolphin

Robin

EagleChicken

Salmon

Trout

Bee

IguanaAlligator

Butterfly

Ant

Finch

Penguin

Cockroach

Whale

Ostrich

Deer

(xxvi) SGL: proposed (k = 1)

Elephant

Rhino

Horse

Cow

Camel

Giraffe

Chimp

Gorilla

Mouse

Squirrel

Tiger Lion

Cat

Dog

Wolf

Seal

Dolphin

Robin

Eagle

Chicken

Salmon

Trout

Bee

Iguana

Alligator

ButterflyAnt

Finch

PenguinCockroach

Whale

Ostrich

Deer

(xxvii) SGL: proposed (k = 4)

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Resources

An R package “spectralGraphTopology” containing code for all the experimental resultsis available athttps://cran.r-project.org/package=spectralGraphTopology

NeurIPS paper: Sandeep Kumar, Jiaxi Ying, José Vinícius de M. Cardoso, and Daniel P.Palomar, “Structured graph learning via Laplacian spectral constraints,” in Advances inNeural Information Processing Systems (NeurIPS), 2019.https://arxiv.org/pdf/1909.11594.pdf

Extended version paper: Sandeep Kumar, Jiaxi Ying, José Vinícius de M. Cardoso, andDaniel P. Palomar, “A Unified Framework for Structured Graph Learning via SpectralConstraints, (2019).” https://arxiv.org/pdf/1904.09792.pdf

Authors:

https://cran.r-project.org/package=spectralGraphTopology

https://arxiv.org/pdf/1909.11594.pdf

Thanks

For more information visit:

https://www.danielppalomar.com

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References

Absil, P.-A., Mahony, R., and Sepulchre, R. (2009).Optimization algorithms on matrix manifolds.Princeton University Press.

Anandkumar, A., Tan, V. Y., Huang, F., and Willsky, A. S. (2012).High-dimensional gaussian graphical model selection: Walk summability and localseparation criterion.Journal of Machine Learning Research, 13(Aug):2293–2337.

Banerjee, O., Ghaoui, L. E., and d’Aspremont, A. (2008).Model selection through sparse maximum likelihood estimation for multivariateGaussian or binary data.Journal of Machine Learning Research, 9(Mar):485–516.

Benidis, K., Sun, Y., Babu, P., and Palomar, D. P. (2016).Orthogonal sparse pca and covariance estimation via procrustes reformulation.IEEE Transactions on Signal Processing, 64(23):6211–6226.

Bogdanov, A., Mossel, E., and Vadhan, S. (2008).The complexity of distinguishing markov random fields.In Approximation, Randomization and Combinatorial Optimization. Algorithms andTechniques, pages 331–342. Springer.

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References

Chow, C. and Liu, C. (1968).Approximating discrete probability distributions with dependence trees.IEEE transactions on Information Theory, 14(3):462–467.

Chung, F. R. (1997).Spectral graph theory.Number 92. American Mathematical Soc.

Dempster, A. P. (1972).Covariance selection.Biometrics, pages 157–175.

Egilmez, H. E., Pavez, E., and Ortega, A. (2017).Graph learning from data under laplacian and structural constraints.IEEE Journal of Selected Topics in Signal Processing, 11(6):825–841.

Friedman, J., Hastie, T., and Tibshirani, R. (2008).Sparse inverse covariance estimation with the graphical lasso.Biostatistics, 9(3):432–441.

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References

Hao, B., Sun, W. W., Liu, Y., and Cheng, G. (2018).Simultaneous clustering and estimation of heterogeneous graphical models.Journal of Machine Learning Research, 18(217):1–58.

Kumar, S., Ying, J., Cardoso, J. V. d. M., and Palomar, D. (2019).A unified framework for structured graph learning via spectral constraints.arXiv preprint arXiv:1904.09792.

Lake, B. and Tenenbaum, J. (2010).Discovering structure by learning sparse graphs.In Proceedings of the 33rd Annual Cognitive Science Conference.

Lauritzen, S. L. (1996).Graphical models, volume 17.Clarendon Press.

Meinshausen, N. and Bühlmann, P. (2006).High-dimensional graphs and variable selection with the lasso.The annals of statistics, 34(3):1436–1462.

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References

Osherson, D. N., Stern, J., Wilkie, O., Stob, M., and Smith, E. E. (1991).Default probability.Cognitive Science, 15(2):251–269.

Ravikumar, P., Wainwright, M. J., Lafferty, J. D., et al. (2010).High-dimensional ising model selection using `1-regularized logistic regression.The Annals of Statistics, 38(3):1287–1319.

Razaviyayn, M., Hong, M., and Luo, Z.-Q. (2013).A unified convergence analysis of block successive minimization methods fornonsmooth optimization.SIAM Journal on Optimization, 23(2):1126–1153.

Rue, H. and Held, L. (2005).Gaussian Markov random fields: theory and applications.CRC press.

Slawski, M. and Hein, M. (2015).Estimation of positive definite m-matrices and structure learning for attractivegaussian markov random fields.Linear Algebra and its Applications, 473:145–179.

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References

Sun, Y., Babu, P., and Palomar, D. P. (2016).Majorization-minimization algorithms in signal processing, communications, andmachine learning.IEEE Transactions on Signal Processing, 65(3):794–816.

Van Mieghem, P. (2010).Graph spectra for complex networks.Cambridge University Press.

Weinstein, J. N., Collisson, E. A., Mills, G. B., Shaw, K. R. M., Ozenberger, B. A.,Ellrott, K., Shmulevich, I., Sander, C., Stuart, J. M., Network, C. G. A. R., et al.(2013).The cancer genome atlas pan-cancer analysis project.Nature Genetics, 45(10):1113.

Zhao, L., Wang, Y., Kumar, S., and Palomar, D. P. (2019).Optimization algorithms for graph laplacian estimation via admm and mm.IEEE Transactions on Signal Processing, 67(16):4231–4244.

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