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
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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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
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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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]:
maximize�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
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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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
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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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
enva
lues
of L
apla
cian
Mat
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
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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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
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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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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(xii) Learned with k = 2
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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:
Thanks
For more information visit:
https://www.danielppalomar.com
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