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Graph Signal Processing
Rahul Singh
Data Science Reading GroupIowa State University
March 24, 2017
Outline
1 Graph Signal Processing Background
2 Graph Signal Processing FrameworksLaplacian BasedDiscrete Signal Processing on Graphs (DSPG) Framework
Weight Matrix Based
Directed Laplacian Based
3 Spectral Graph Wavelets
4 Conclusions
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 2/39
Background Introduction
Classical Signal Processing
Translation, filtering, convolution, modulation, Fouriertransform, wavelets, sparse representations . . .
t1 t2 t3 tN
Structure behind time-series
Structure behind image
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 3/39
Background Introduction
Classical Signal Processing
Translation, filtering, convolution, modulation, Fouriertransform, wavelets, sparse representations . . .
t1 t2 t3 tN
Structure behind time-series Structure behind image
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 3/39
Background Introduction
Classical Signal Processing
Translation, filtering, convolution, modulation, Fouriertransform, wavelets, sparse representations . . .
t1 t2 t3 tN
Structure behind time-series Structure behind image
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 3/39
Background Introduction
Graph Signal Processing
A graph signal
12
3
4
52
3 2
4
-3
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 4/39
Background Introduction
Graph Signal Processing
A graph signal
12
3
4
52
3 2
4
-3
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 4/39
Background Introduction
Graph Signal Processing
A graph signal
12
3
4
52
3 2
4
-3
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 4/39
Background Introduction
Graph Signal Processing Applications
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 5/39
Background Introduction
Difficulty in GSP
Translation is simple in classicalsignal processing
0 2 4 6 8 10 12 14 16 18 200
2
4
6
8
10
12
What does it mean to translate thesignal to ‘vertex 50’?
Challenging in GSP
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 6/39
Background Introduction
Difficulty in GSP
Translation is simple in classicalsignal processing
0 2 4 6 8 10 12 14 16 18 200
2
4
6
8
10
12
What does it mean to translate thesignal to ‘vertex 50’?
Challenging in GSP
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 6/39
Background Notation
Notation
12
3
4
52
3 2
4
-3
A graph signal f
Graph G = (V, W)
f =
234−32
Weight matrix W =
0 1 1 0 01 0 1 1 11 1 0 1 00 1 1 0 10 1 0 1 0
Degree matrix D =
2 0 0 0 00 4 0 0 00 0 3 0 00 0 0 3 00 0 0 0 2
Laplacian matrix
L = D−W =
2 −1 −1 0 0−1 4 −1 −1 −1−1 −1 3 −1 00 −1 −1 3 −10 −1 0 −1 2
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 7/39
GSP Frameworks
Existing Graph Signal Processing (GSP) Frameworks
GSP based on Laplacian1 Discrete Signal Processing onGraphs (DSPG) Framework2
Shift Operator Not defined The weight matrix
LSI Filters Not applicable Applicable
Applicability Only undirected graphs Directed graphs
Frequencies Eigenvalues of the Laplacian ma-trix Eigenvalues of the weight matrix
Harmonics Eigenvectors of the Laplacianmatrix
Eigenvectors of the weight ma-trix
FrequencyOrdering Laplacian quadratic form Total variation
MultiscaleAnalysis Exists (SGWT) Does not exist
1David K Hammond, Pierre Vandergheynst, and Remi Gribonval. “Wavelets on graphs via spectral graphtheory”. In: Applied and Computational Harmonic Analysis 30.2 (2011), pp. 129–150.
2A. Sandryhaila and J.M.F. Moura. “Discrete Signal Processing on Graphs: Frequency Analysis”. In: SignalProcessing, IEEE Transactions on 62.12 (2014), pp. 3042–3054.
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 8/39
GSP Frameworks Laplacian Based
Classical vs Graph Signal Processing (Laplacian Based)Operator/ Transform Classical Signal Processing Graph Signal Processing
Fourier Transformx(ω) =
∫∞−∞
x(t)e−jωt dt
Frequency: ω can take any valueFourier basis: Complex exponentialsejωt
f(λ`) =∑N
n=1f(n)u∗
`(n)
Frequency: Eigenvalues of the graphLaplacian (λ`)Fourier basis: Eigenvectors of thegraph Laplacian (u`)
Convolution
In time domain:x(t)∗y(t) =
∫∞−∞
x(τ)y(t−τ)dτ
In frequency domain:x(t) ∗ y(t) = x(ω)y(ω)
Defined through Graph FourierTransform
f ∗ g = U(
f.g)
Translation Can be defined using convolutionTτ x(t) = x(t − τ) = x(t) ∗ δτ (t)
Defined through graph convolutionTi f(n) =
√N(f ∗ δi )(n)
=√N∑N−1
`=0f (λ`)u∗
`(i)u`(n)
Modulation Multiplication with the complexexponentialMωx(t) = ejωt x(t)
Multiplication with the eigenvectorof the graph LaplacianMk f(n) =
√Nuk (n)f(n)
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 9/39
GSP Frameworks Laplacian Based
Graph Fourier Transform
Analogy from classical signal processingClassical Fourier basis: Complex exponentials
Complex exponentials are Eigenfunctions of 1-D Laplacian operator∆,
−∆(ejωt) = − ∂2
∂t2 ejωt = (ω)2ejωt .
Graph Fourier TransformGraph Fourier basis are Eigenfunctions of the Laplacian matrix(operator)
Graph Frequencies: Eigenvalues of the Laplacian matrix L
Graph Harmonics: Eigenvectors of the Laplacian matrix L
L = UΛUT
GFT f = UTf , IGFT f = Uf
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 10/39
GSP Frameworks Laplacian Based
Graph Fourier Transform
Analogy from classical signal processingClassical Fourier basis: Complex exponentials
Complex exponentials are Eigenfunctions of 1-D Laplacian operator∆,
−∆(ejωt) = − ∂2
∂t2 ejωt = (ω)2ejωt .
Graph Fourier TransformGraph Fourier basis are Eigenfunctions of the Laplacian matrix(operator)
Graph Frequencies: Eigenvalues of the Laplacian matrix L
Graph Harmonics: Eigenvectors of the Laplacian matrix L
L = UΛUT
GFT f = UTf , IGFT f = Uf
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 10/39
GSP Frameworks Laplacian Based
Graph Signal in Two Domains
0 1 2 3 4 5 6 7 8 9
−2
0
2
λℓ
f(λ
ℓ)
GFT coefficients
Eigenvalues of graph Laplacian
A graph signal in vertex domain and spectral domain
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 11/39
GSP Frameworks Laplacian Based
Laplacian Eigenvectors as GFT Basis
1
23
45
6
u0
1
23
45
6
u1
1
23
45
6
u2
1
23
45
6
u3
1
23
45
6
u4
1
23
45
6
u5
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 12/39
GSP Frameworks Laplacian Based
Graph Convolution
1 3
4
2
5
L =
2 −1 −1 0 0−1 3 −1 −1 0−1 −1 3 −1 00 −1 −1 3 −10 0 0 −1 1
Eigenvalues: 0, 0.8299, 2.6889, 4, 4.4812
U =
0.4472 0.4375 0.7031 0 0.33800.4472 0.2560 −0.2422 0.7071 −0.41930.4472 0.2560 −0.2422 −0.7071 −0.41930.4472 −0.1380 −0.5362 0 0.70240.4472 −0.8115 0.3175 0 −0.2018
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 13/39
GSP Frameworks Laplacian Based
Graph Convolution(cont’d. . . )
1
23
4 5
3
f = [3, 4, 6, 3, 1]T
1
23
4 5
4
g = [4, 2, 4, 2, 2]T
h = f ∗ g = IGFT(f.g)
1
23
4 5
21.92
h = [21.92, 23.92, 21.08, 21.72, 17.80]T
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 14/39
GSP Frameworks Laplacian Based
Graph Translation
1
23
4 5
3
f = [3, 4, 6, 3, 1]T
Translation to node i :
Ti (f) =√
N (f ∗ δi ) =√
N IGFT(f.UT (:, i))
1
23
4 5
2.44
T1f = [2.44, 5.08, 5.08, 3.72, 0.69]T
1
23
4 5
5.08
T2f = [5.08, 1.50, 4.66, 3.56, 2.21]T
1
23
4 5
3.72
T4f = [3.72, 3.56, 3.56, 1.08, 5.08]T
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 15/39
GSP Frameworks DSPG
Discrete Signal Processing on Graphs (DSPG) Framework
Shift Operator
LSI Filters
Total Variation
Graph FourierTransform
Graph Frequencies
Graph Harmonics
FrequencyOrdering
Graph harmonicsare eigenfunctionsof LSI filters
Total Variation isused for frequencyordering
Concepts in the DSPG framework
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 16/39
GSP Frameworks DSPG
DSPG Framework (Cont’d. . . )
Shift operatorWeight matrix W of the graph
Shifted graph signal f = Wf
Example: shifting discrete-timesignal (one unit right)
1 2 3 4 5
x = [9, 7, 5, 0, 6]T
x = Wx =
0 0 0 0 11 0 0 0 00 1 0 0 00 0 1 0 00 0 0 1 0
97506
=
69750
Hfin fout = Hf in
A linear graph filter
Linear Shift Invariant (LSI) filtersH(Wf in) = W(Hf in)
Polynomials in W
H = h(W) =M−1∑m=0
hmWm
= h0I + h1W + . . .+ hM−1WM−1
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 17/39
GSP Frameworks DSPG
DSPG Framework (Cont’d. . . )
Analogy from classical signal processingClassical Fourier basis: Complex exponentials
Complex exponentials are Eigenfunctions of Linear Time Invariant(LTI) filters
Graph Fourier TransformGraph Fourier basis are Eigenfunctions of Linear Shift Invariant(LSI) graph filters
Graph Frequencies: Eigenvalues of the weight matrix W
Graph Harmonics: Eigenvectors of the weight matrix W
W = VΣV−1
GFT f = V−1f , IGFT f = V−1 f
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 18/39
GSP Frameworks DSPG
DSPG Framework (Cont’d. . . )Total Variation in classical signal processing
TV(x) =∑
n x [n]− x [n − 1] = ||x− x||1 , where x [n] = x [n − 1]
Analogy from classical signal processing
Total Variation on graphs TVG(f) = ||f − f||1 = ||f −Wf||1
Frequency ordering: Based on TotalVariationEigenvalue with largest magnitude:Lowest frquency
Re
Im
σN−1|σmax|−|σmax|
σ1
σ2σ0
LFHF
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 19/39
GSP Frameworks DSPG
Problems in Weight Matrix based DSPG
Re
Im
σN−1|σmax|−|σmax|
σ1
σ2σ0
LFHF
Constant graph signal
12
3
4
5
11
1
1
1
Graph frequencies:-1.62, -1.47, -0.46, 0.62, 2.94
f =
0
0.360.16
02.20
Weight matrix based DSPG
Does not provide “natural” frequency ordering
Even a constant signal has high frequency components
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 20/39
Graph Fourier Transform based on Directed Laplacian
Graph Fourier Transform based on Directed Laplacian
Graph Fourier Transform based on Directed Laplacian3
Redefines Graph Fourier Transform under DSPG
Shift operator: Derived from directed LaplacianLinear Shift Invariant filters: Polynomials in the directed LaplacianGraph frequencies: Eigenvalues of the directed LaplacianGraph harmonics: Eigenvectors of the directed Laplacian
“Natural” frequency ordering
Better intuition of frequency as compared to the weightmatrix based approach
Coincides with the Laplacian based approach for undirectedgraphs
3Rahul Singh, Abhishek Chakraborty, and BS Manoj. “Graph Fourier transform based on directed Laplacian”.In: Signal Processing and Communications (SPCOM), 2016 International Conference on. IEEE. 2016, pp. 1–5.
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 21/39
Graph Fourier Transform based on Directed Laplacian Directed Laplacian Matrix
Directed Laplacian Matrix
Basic matrices of a directed graphWeight matrix: W
wij is the weight of the directed edge from node j to node i
In-degree matrix: Din = diag({d ini }i=1,2,...,N), d in
i =∑N
j=1 wij
Out-degree matrix: Dout = diag({douti }i=1,2,...,N), dout
i =∑N
i=1 wij
Directed Laplacian matrix L = Din −WSum of each row is zeroλ = 0 is surely an eigenvalue
1
2
3
4
1
1
11
1
1
1
A directed graph
W =
[0 0 0 11 0 1 11 1 0 01 0 0 0
]Weight matrix
Din =
[1 0 0 00 3 0 00 0 2 00 0 0 1
]In-degree matrix
L =
[1 0 0 −1−1 3 −1 −1−1 −1 2 0−1 0 0 1
]Directed Laplacian matrix
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 22/39
Graph Fourier Transform based on Directed Laplacian Shift Operator
Shift Operator (Proposed)
1 2 3 4 5
A directed cyclic (ring) graph
L =
1 0 0 0 −1−1 1 0 0 00 −1 1 0 00 0 −1 1 00 0 0 −1 1
Laplacian matrix
A signal x = [9 7 1 0 6]T ; shifted by one unit to the right x = [6 9 7 1 0]T
x = Sx = (I− L)x =
0 0 0 0 11 0 0 0 00 1 0 0 00 0 1 0 00 0 0 1 0
97106
=
69710
S = (I− L) is the shift operator
Shifted graph signal: f = Sf = (I− L)f
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 23/39
Graph Fourier Transform based on Directed Laplacian LSI Filters
LSI Filters
Hfin fout = Hf in
A linear graph filter
Linear Shift-Invariant (LSI) filter: S(fout) = H(Sfin)
TheoremA graph filter H is LSI if the following conditions are satisfied.
1 Geometric multiplicity of each distinct eigenvalue of the graphLaplacian is one.
2 The graph filter H is a polynomial in L, i.e., if H can be written as
H = h(L) = h0I + h1L + . . .+ hmLm
where, h0, h1, . . . , hm ∈ C are called filter taps.Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 24/39
Graph Fourier Transform based on Directed Laplacian Graph Frequencies and Harmonics
Graph Fourier Transform based on Directed Laplacian
Jordan decomposition of the directed Laplacian: L = VJV−1
Graph Fourier basis: Columns of V (Jordan Eigenvectors of L)
Graph frequencies: Eigenvalues of L (diagonal entries of Jordanblocks in J)
GFT f = V−1f and IGFT: f = Vf
Frequency Ordering: based on Total Variation
Total Variation: TVG(f) = ||f − Sf||1 = ||f − (I− L)f||1TVG(f) = ||Lf||1
TheoremTV of an eigenvector vr is proportional to the absolute value of thecorresponding eigenvalue
TV(vr ) ∝ |λr |
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 25/39
Graph Fourier Transform based on Directed Laplacian Frequency Ordering
Frequency Ordering
Re
Im
λ0 λ1
λ2
λ3
λN−1
λN−2
LFHF
HF
Arbitrary graph
Re
Im
λ0 λ1
λ2
λ3
λN−1
λN−2
LF HF
Graph with positive edge weights
Re
Im
λ0λ1λN−1 λN−2
LFHFHF
Undirected graph with real edge weights.
Re
Im
λ0 λ1 λN−1λN−2
LFHF
Undirected graph with real andnon-negative edge weights.
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 26/39
Graph Fourier Transform based on Directed Laplacian Frequency Ordering
Example
Graph signal f = [0.1189 0.3801 0.8128 0.2441 0.8844]T defined onthe directed graph
1
2
3
4
5
1
2
3
1
2
33
4
3
A weighted directed graph 02
46
8
−10
−5
0
5
100
0.5
1
1.5
Re(λℓ)Im(λℓ)
|f(λ
ℓ)|
High frequency component
Equal TV
Low frequency component
Spectrum of the signal f = [0.1189 0.3801 0.8128 0.2441 0.8844]T
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 27/39
Graph Fourier Transform based on Directed Laplacian Frequency Ordering
Example
Graph signal f = [0.1189 0.3801 0.8128 0.2441 0.8844]T defined onthe directed graph
1
2
3
4
5
1
2
3
1
2
33
4
3
A weighted directed graph 02
46
8
−10
−5
0
5
100
0.5
1
1.5
Re(λℓ)Im(λℓ)
|f(λ
ℓ)|
High frequency component
Equal TV
Low frequency component
Spectrum of the signal f = [0.1189 0.3801 0.8128 0.2441 0.8844]T
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 27/39
Graph Fourier Transform based on Directed Laplacian Frequency Ordering
Example: Zero Frequency
Eigenvector corresponding to λ0 is v0 = 1√N [1, 1, . . . 1]T
TV of v0 is zeroFor a constant graph signal f = [k, k, . . .]T , GFT is f = [(k
√N), 0, . . .]T
Only zero frequency component
The weight matrix based approach of GFT fails to give this basicintuition
1
2
3
4
5
1
2
3
1
2
33
4
3
A weighted directed graph0
24
68
−10
−5
0
5
100
1
2
3
Re(λℓ)Im(λℓ)
|f(λ
ℓ)|
Only zero frequency component
Spectrum of the constant signal f = [1 1 1 1 1]T
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 28/39
Graph Fourier Transform based on Directed Laplacian Frequency Ordering
Example: Zero Frequency
Eigenvector corresponding to λ0 is v0 = 1√N [1, 1, . . . 1]T
TV of v0 is zeroFor a constant graph signal f = [k, k, . . .]T , GFT is f = [(k
√N), 0, . . .]T
Only zero frequency componentThe weight matrix based approach of GFT fails to give this basicintuition
1
2
3
4
5
1
2
3
1
2
33
4
3
A weighted directed graph0
24
68
−10
−5
0
5
100
1
2
3
Re(λℓ)Im(λℓ)
|f(λ
ℓ)|
Only zero frequency component
Spectrum of the constant signal f = [1 1 1 1 1]T
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 28/39
Graph Fourier Transform based on Directed Laplacian Comparison
Comparison of the GSP Frameworks
GSP based on LaplacianDSPG Framework
Based on WeightMatrix
Based on DirectedLaplacian
ShiftOperator
Not defined The weight matrix W Derived from directedLaplacian (I− L)
LSI Filters Not applicable Applicable Applicable
Applicability Only undirected graphs Directed graphs Directed graphs
Frequencies Eigenvalues of theLaplacian (real)
Eigenvalues of theweight matrix
Eigenvalues of the di-rected Laplacian
Harmonics Eigenvectors of theLaplacian matrix (real)
Eigenvectors of theweight matrix
Eigenvectors of thedirected Laplacian
FrequencyOrdering
Laplacian quadraticform (natural)
Total variation(not natural)
Total variation(natural)
MultiscaleAnalysis
Exists (Spectral GraphWavelet Transform)
Does not exist Possible
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 29/39
SGWT
Spectral Graph Wavelet Transform
Classical waveletsWavelets at different scales and locations are constructed by scalingand translating a single “mother” wavelet ψψs,a(x) = 1
sψ( x−a
s
)Scaling in Fourier domain ψs,a(x) = 1
2π
∫∞−∞ ejωx ψ(sω)e−jωadω
Scaling ψ by 1/s corresponds to scaling ψ with sTerm e−jωa comes from localization of the wavelet at location a
Spectral Graph WaveletsGraph wavelet at scale t and centered at node n
ψt,n(m) =N−1∑`=0
u`(m)g(tλ`)u∗` (n)
Frequency ω is replaced with eigenvalues of graph Laplacian λ`Translating to node n corresponds to multiplication by u∗` (n)g acts as a scaled bandpass filter, replacing ψ
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 30/39
SGWT Generating Kernel
Wavelet Generating Kernel
ψt,n(m) =N−1∑`=0
u`(m)g(tλ`)u∗` (n)
g acts as a scaled bandpass filter: wavelet generating kernel
g(x ;α, β, x1, x2) =
x−α1 xα for x < x1
p(x) for x1 ≤ x ≤ x1
xβ2 x−β for x > x2,
x is the distance from origin (zero frequency)α and β are integer parameters of gx1 and x2 determine transition regionsp(x) is a cubic spline that ensures continuity in g
A possible choice of these parameters: α = β = 2, x1 = 1,x2 = 2, and p(x) = −5 + 11x − 6x2 + x3
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 31/39
SGWT Generating Kernel
Wavelet Generating Kernel cont’d. . .
ψt,n(m) =N−1∑`=0
u`(m)g(tλ`)u∗` (n)
Scale t is a continuous variable
For practical purposes, choose J number of logarithmicallyequally spaced scales t1, . . . , tJ
tmin = |λmax |/K , λmax is the eigenvalue of L with largest magnitudeand K is a design parameter
tJ = x2/|λmax | and t1 = x2/tmin
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 32/39
SGWT Generating Kernel
Wavelet Generating Kernel cont’d. . .|λmax | = 10, α = β = 2, x1 = 1, x2 = 2, K = 20, and J = 4
0 1 2 3 4 5 6 70
0.5
1
(λ)
g(tλ)
Kernel at t1 = 4.0000.
0 1 2 3 4 5 6 70
0.5
1
(λ)
g(tλ)
Kernel at t2 = 1.4736.
0 1 2 3 4 5 6 70
0.5
1
(λ)
g(tλ)
Kernel at t3 = 0.5429.
0 1 2 3 4 5 6 70
0.5
1
(λ)
g(tλ)
Kernel at t4 = 0.2.
As t increases, the kernel becomes increasingly confined to low frequenciesGraph Signal Processing Data Science Reading Group, ISU March 24, 2017 33/39
SGWT Examples
SGWT ExamplesWavelets at different scales
−0.06 0 0.06
t = 5.0742
−0.2 0 0.2
t = 1.8693
−0.5 0 0.5
t = 0.6887
−0.8 0 0.8
t = 0.2537
Large scale ≡ Low frequency ≡ Spread of the wavelet overthe graph is high
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 34/39
SGWT Examples
SGWT Examples
Wavelets at different scales
0.0656
t = 5.0742
0.2173
t = 1.8693
0.5050
t = 0.6887
0.8993
t = 0.2537
Large scale ≡ Low frequency ≡ Spread of the wavelet overthe graph is high
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 35/39
SGWT Examples
SGWT Examples
0 10 20 30 40 50 60 70 80−0.04
−0.02
0
0.02
0.04
0.06
0.08
index
wavelet
ψ(t)
t = 5.0742
0 10 20 30 40 50 60 70 80−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
index
wavelet
ψ(t)
t = 1.8693
0 10 20 30 40 50 60 70 80−0.2
0
0.2
0.4
0.6
index
wavelet
ψ(t)
t = 0.6887
0 10 20 30 40 50 60 70 80−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
index
wavelet
ψ(t)
t = 0.2537
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 36/39
SGWT Matrix Form
Matrix Form of SGWTWavelet basis at scale t = collection of N number of wavelets(each wavelet centered at a particular node of the graph)
Ψt = [ψt,1|ψt,2| . . . |ψt,N ]
= U
g(tλ0)
g(tλ1). . .
g(tλN−1)
UT
= UGtUT
Column of U are eigenvectors of Lg(tλ`) is the sampled value of g(tλ) at frequency λ`Wavelet coefficient at scale t and centered at node n of agraph signal f
Wf (t, n) = 〈ψt,n, f〉 = ψTt,nf
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 37/39
Conclusions
Conclusions
Introduction to Graph Signal Processing (GSP)
Graph Signal Processing FrameworksLaplacian BasedDiscrete Signal Processing on Graphs (DSPG) Framework
Weight Matrix Based
Directed Laplacian Based
Spectral Graph Wavelet Transform (SGWT)Research oppurtunities are plenty
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 38/39
Conclusions
References
[1] Fan RK Chung. Spectral graph theory. Vol. 92. American Mathematical Soc., 1997.
[2] David K Hammond, Pierre Vandergheynst, and Remi Gribonval. “Wavelets on graphs via spectral graphtheory”. In: Applied and Computational Harmonic Analysis 30.2 (2011), pp. 129–150.
[3] Sunil K Narang and Antonio Ortega. “Perfect reconstruction two-channel wavelet filter banks for graphstructured data”. In: Signal Processing, IEEE Transactions on 60.6 (2012), pp. 2786–2799.
[4] A. Sandryhaila and J.M.F. Moura. “Big Data Analysis with Signal Processing on Graphs: Representationand processing of massive data sets with irregular structure”. In: Signal Processing Magazine, IEEE 31.5(2014), pp. 80–90.
[5] A. Sandryhaila and J.M.F. Moura. “Discrete Signal Processing on Graphs”. In: Signal Processing, IEEETransactions on 61.7 (2013), pp. 1644–1656.
[6] A. Sandryhaila and J.M.F. Moura. “Discrete Signal Processing on Graphs: Frequency Analysis”. In: SignalProcessing, IEEE Transactions on 62.12 (2014), pp. 3042–3054.
[7] Aliaksei Sandryhaila and Jose MF Moura. “Discrete signal processing on graphs: Graph fourier transform.”In: ICASSP. 2013, pp. 6167–6170.
[8] David I Shuman, Benjamin Ricaud, and Pierre Vandergheynst. “Vertex-frequency analysis on graphs”. In:Applied and Computational Harmonic Analysis 40.2 (2016), pp. 260–291.
[9] David I Shuman et al. “The emerging field of signal processing on graphs: Extending high-dimensional dataanalysis to networks and other irregular domains”. In: Signal Processing Magazine, IEEE 30.3 (2013),pp. 83–98.
[10] Rahul Singh, Abhishek Chakraborty, and BS Manoj. “Graph Fourier transform based on directedLaplacian”. In: Signal Processing and Communications (SPCOM), 2016 International Conference on.IEEE. 2016, pp. 1–5.
Graph Signal Processing Data Science Reading Group, ISU March 24, 2017 39/39
Thank You.rasingh@iastate.edu
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