analysis meets graphs · 2019-02-28 · functional analysis compressive sensing stochastic...
TRANSCRIPT
![Page 1: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/1.jpg)
Analysis meets Graphs
Matthew Begue
Norbert Wiener CenterDepartment of Mathematics
University of Maryland, College Park
![Page 2: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/2.jpg)
Why study graphs?
Graph theory has developed into a useful tool in appliedmathematics.Vertices correspond to different sensors, observations, or datapoints. Edges represent connections, similarities, or correlationsamong those points.
![Page 3: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/3.jpg)
Wikipedia graph
![Page 4: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/4.jpg)
Graphs in pure mathematics too!
![Page 5: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/5.jpg)
Why study Analysis?
![Page 6: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/6.jpg)
Why study Analysis?
Real AnalysisComplex AnalysisPartial Differential EquationsOrdinary Differential EquationsHARMONIC ANALYSISProbabilityDifferential GeometryFunctional AnalysisCompressive SensingStochastic ProcessesMeasure TheoryBrownian MotionOperator TheorySpectral TheoryMathematical ModelingNumerical Analysis
![Page 7: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/7.jpg)
Why study Analysis?
Real AnalysisComplex AnalysisPartial Differential EquationsOrdinary Differential EquationsHARMONIC ANALYSISProbabilityDifferential GeometryFunctional AnalysisCompressive SensingStochastic ProcessesMeasure TheoryBrownian MotionOperator TheorySpectral TheoryMathematical ModelingNumerical Analysis
![Page 8: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/8.jpg)
Why study Analysis?
Real AnalysisComplex AnalysisPartial Differential EquationsOrdinary Differential EquationsHARMONIC ANALYSISProbabilityDifferential GeometryFunctional AnalysisCompressive SensingStochastic ProcessesMeasure TheoryBrownian MotionOperator TheorySpectral TheoryMathematical ModelingNumerical Analysis
![Page 9: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/9.jpg)
Why study Analysis?
Real AnalysisComplex AnalysisPartial Differential EquationsOrdinary Differential EquationsHARMONIC ANALYSISProbabilityDifferential GeometryFunctional AnalysisCompressive SensingStochastic ProcessesMeasure TheoryBrownian MotionOperator TheorySpectral TheoryMathematical ModelingNumerical Analysis
![Page 10: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/10.jpg)
Why study Analysis?
Real AnalysisComplex AnalysisPartial Differential EquationsOrdinary Differential EquationsHARMONIC ANALYSISProbabilityDifferential GeometryFunctional AnalysisCompressive SensingStochastic ProcessesMeasure TheoryBrownian MotionOperator TheorySpectral TheoryMathematical ModelingNumerical Analysis
![Page 11: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/11.jpg)
Why study Analysis?
Real AnalysisComplex AnalysisPartial Differential EquationsOrdinary Differential EquationsHARMONIC ANALYSISProbabilityDifferential GeometryFunctional AnalysisCompressive SensingStochastic ProcessesMeasure TheoryBrownian MotionOperator TheorySpectral TheoryMathematical ModelingNumerical Analysis
![Page 12: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/12.jpg)
Why study Analysis?
Real AnalysisComplex AnalysisPartial Differential EquationsOrdinary Differential EquationsHARMONIC ANALYSISProbabilityDifferential GeometryFunctional AnalysisCompressive SensingStochastic ProcessesMeasure TheoryBrownian MotionOperator TheorySpectral TheoryMathematical ModelingNumerical Analysis
![Page 13: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/13.jpg)
Why study Analysis?
Real AnalysisComplex AnalysisPartial Differential EquationsOrdinary Differential EquationsHARMONIC ANALYSISProbabilityDifferential GeometryFunctional AnalysisCompressive SensingStochastic ProcessesMeasure TheoryBrownian MotionOperator TheorySpectral TheoryMathematical ModelingNumerical Analysis
![Page 14: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/14.jpg)
Why study Analysis?
Real AnalysisComplex AnalysisPartial Differential EquationsOrdinary Differential EquationsHARMONIC ANALYSISProbabilityDifferential GeometryFunctional AnalysisCompressive SensingStochastic ProcessesMeasure TheoryBrownian MotionOperator TheorySpectral TheoryMathematical ModelingNumerical Analysis
![Page 15: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/15.jpg)
Why study Analysis?
Real AnalysisComplex AnalysisPartial Differential EquationsOrdinary Differential EquationsHARMONIC ANALYSISProbabilityDifferential GeometryFunctional AnalysisCompressive SensingStochastic ProcessesMeasure TheoryBrownian MotionOperator TheorySpectral TheoryMathematical ModelingNumerical Analysis
![Page 16: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/16.jpg)
Why study Analysis?
Real AnalysisComplex AnalysisPartial Differential EquationsOrdinary Differential EquationsHARMONIC ANALYSISProbabilityDifferential GeometryFunctional AnalysisCompressive SensingStochastic ProcessesMeasure TheoryBrownian MotionOperator TheorySpectral TheoryMathematical ModelingNumerical Analysis
![Page 17: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/17.jpg)
Why study Analysis?
Real AnalysisComplex AnalysisPartial Differential EquationsOrdinary Differential EquationsHARMONIC ANALYSISProbabilityDifferential GeometryFunctional AnalysisCompressive SensingStochastic ProcessesMeasure TheoryBrownian MotionOperator TheorySpectral TheoryMathematical ModelingNumerical Analysis
![Page 18: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/18.jpg)
Why study Analysis?
Real AnalysisComplex AnalysisPartial Differential EquationsOrdinary Differential EquationsHARMONIC ANALYSISProbabilityDifferential GeometryFunctional AnalysisCompressive SensingStochastic ProcessesMeasure TheoryBrownian MotionOperator TheorySpectral TheoryMathematical ModelingNumerical Analysis
![Page 19: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/19.jpg)
Why study Analysis?
Real AnalysisComplex AnalysisPartial Differential EquationsOrdinary Differential EquationsHARMONIC ANALYSISProbabilityDifferential GeometryFunctional AnalysisCompressive SensingStochastic ProcessesMeasure TheoryBrownian MotionOperator TheorySpectral TheoryMathematical ModelingNumerical Analysis
![Page 20: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/20.jpg)
Why study Analysis?
Real AnalysisComplex AnalysisPartial Differential EquationsOrdinary Differential EquationsHARMONIC ANALYSISProbabilityDifferential GeometryFunctional AnalysisCompressive SensingStochastic ProcessesMeasure TheoryBrownian MotionOperator TheorySpectral TheoryMathematical ModelingNumerical Analysis
![Page 21: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/21.jpg)
Why study Analysis?
Real AnalysisComplex AnalysisPartial Differential EquationsOrdinary Differential EquationsHARMONIC ANALYSISProbabilityDifferential GeometryFunctional AnalysisCompressive SensingStochastic ProcessesMeasure TheoryBrownian MotionOperator TheorySpectral TheoryMathematical ModelingNumerical Analysis
![Page 22: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/22.jpg)
Analysis toolbox
TranslationConvolutionModulationFourier Transform
![Page 23: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/23.jpg)
Analysis toolbox
TranslationConvolutionModulationFourier Transform
![Page 24: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/24.jpg)
Analysis toolbox
TranslationConvolutionModulationFourier Transform
![Page 25: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/25.jpg)
Analysis toolbox
TranslationConvolutionModulationFourier Transform
![Page 26: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/26.jpg)
Outline
1 Fourier Transform
2 Modulation
3 Convolution
4 Translation
![Page 27: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/27.jpg)
Motivation
In the classical setting, the Fourier transform on R is given by
f (ξ) =
∫R
f (t)e−2πiξt dt = 〈f ,e2πiξt〉.
This is precisely the inner product of f with the eigenfunctions ofthe Laplace operator.
![Page 28: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/28.jpg)
Motivation
In the classical setting, the Fourier transform on R is given by
f (ξ) =
∫R
f (t)e−2πiξt dt = 〈f ,e2πiξt〉.
This is precisely the inner product of f with the eigenfunctions ofthe Laplace operator.
![Page 29: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/29.jpg)
Graph Laplacian
DefinitionThe pointwise formulation for the Laplacian acting on a functionf : V → R is
∆f (x) =∑y∼x
f (x)− f (y).
For a finite graph, the Laplacian can be represented as a matrix.Let D denote the N × N degree matrix, D = diag(dx ).Let A denote the N × N adjacency matrix,
A(i , j) =
{1, if xi ∼ xj0, otherwise.
Then the unweighted graph Laplacian can be written as
L = D − A.
![Page 30: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/30.jpg)
Spectrum of the Laplacian
L is a real symmetric matrix and therefore has nonnegativeeigenvalues {λk}N−1
k=0 with associated orthonormal eigenvectors{ϕk}N−1
k=0 .If G is finite and connected, then we have
0 = λ0 < λ1 ≤ λ2 ≤ · · · ≤ λN−1.
Easy to show that ϕ0 ≡ 1/√
N.
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Data Sets - Minnesota Road Network
−98 −97 −96 −95 −94 −93 −92 −91 −90 −8943
44
45
46
47
48
49
50
(a) λ3
−98 −97 −96 −95 −94 −93 −92 −91 −90 −8943
44
45
46
47
48
49
50
(b) λ4
−98 −97 −96 −95 −94 −93 −92 −91 −90 −8943
44
45
46
47
48
49
50
(c) λ5
−98 −97 −96 −95 −94 −93 −92 −91 −90 −8943
44
45
46
47
48
49
50
(d) λ6
−98 −97 −96 −95 −94 −93 −92 −91 −90 −8943
44
45
46
47
48
49
50
(e) λ7
−98 −97 −96 −95 −94 −93 −92 −91 −90 −8943
44
45
46
47
48
49
50
(f) λ8
Figure : Eigenfunctions corresponding to the first six nonzero eigenvalues.Minnesota road graph (2642 vertices)
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Data Sets - Sierpinski gasket graph approximation
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
(a) λ3
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
(b) λ4
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
(c) λ5
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
(d) λ6
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
(e) λ7
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
(f) λ8
Figure : Eigenfunctions corresponding to the first six nonzero eigenvalues.Level-8 graph approximation to Sierpinski gasket (9843 vertices)
![Page 33: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/33.jpg)
Data Sets - Sierpinski gasket graph approximation
−1
−0.5
0
0.5
1
0
0.5
1
1.5
2
−0.02
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
0.02
(a) λ3
−1
−0.5
0
0.5
1
0
0.5
1
1.5
2
−0.02
−0.015
−0.01
−0.005
0
0.005
0.01
(b) λ4
−1
−0.5
0
0.5
1
0
0.5
1
1.5
2
−0.02
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
(c) λ5
−1
−0.5
0
0.5
1
0
0.5
1
1.5
2
−0.02
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
0.02
(d) λ6
−1
−0.5
0
0.5
1
0
0.5
1
1.5
2
−0.02
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
0.02
(e) λ7
−1
−0.5
0
0.5
1
0
0.5
1
1.5
2
−0.02
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
0.02
(f) λ8
Figure : Eigenfunctions corresponding to the first six nonzero eigenvalues.Level-8 graph approximation to Sierpinski gasket (9843 vertices)
![Page 34: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/34.jpg)
Graph Fourier Transform
DefinitionThe graph Fourier transform is defined as
f (λl ) = 〈f , ϕl〉 =N∑
n=1
f (n)ϕ∗l (n).
Notice that the graph Fourier transform is only defined on values ofσ(L).The inverse Fourier transform is then given by
f (n) =N−1∑l=0
f (λl )ϕl (n).
![Page 35: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/35.jpg)
Outline
1 Fourier Transform
2 Modulation
3 Convolution
4 Translation
![Page 36: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/36.jpg)
Graph Modulation
In Euclidean setting, modulation is multiplication of a Laplacianeigenfunction.
DefinitionFor any k = 0,1, ...,N − 1 the graph modulation operator Mk , isdefined as
(Mk f )(n) =√
Nf (n)ϕk (n).
On R,
modulation in the time domain = translation in the frequencydomain
Mξf (ω) = f (ω − ξ).
![Page 37: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/37.jpg)
Example - Classical Setting
![Page 38: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/38.jpg)
Example - Movie
G =SG6f (λl ) = δ2(l) =⇒ f = ϕ2.
−1
−0.5
0
0.5
1
0
0.5
1
1.5
2
−0.01
−0.005
0
0.005
0.01
0.015
0.02
![Page 39: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/39.jpg)
Example - Movie
G =Minnesotaf (λl ) = δ2(l) =⇒ f = ϕ2.
−98 −97 −96 −95 −94 −93 −92 −91 −90 −8943
44
45
46
47
48
49
50
![Page 40: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/40.jpg)
Outline
1 Fourier Transform
2 Modulation
3 Convolution
4 Translation
![Page 41: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/41.jpg)
Graph Convolution - Motivation and Definition
Classically, for signals f ,g ∈ L2(R) we define the convolution as
f ∗ g(t) =
∫R
f (u)g(t − u) du.
However, there is no clear analogue of translation in the graphsetting. So we exploit the property
( f ∗ g)(ξ) = f (ξ)g(ξ),
and then take inverse Fourier transform.
DefinitionFor f ,g : V → R, we define the graph convolution of f and g as
f ∗ g(n) =N−1∑l=0
f (λl )g(λl )ϕl (n).
![Page 42: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/42.jpg)
Outline
1 Fourier Transform
2 Modulation
3 Convolution
4 Translation
![Page 43: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/43.jpg)
Graph Translation
For signal f ∈ L2(R), the translation operator, Tu, can be thoughtof as a convolution with δu.On R we can calculateδu(k) =
∫R δu(x)e−2πikx dx = e−2πiku(= ϕ∗
k (u)).Then by taking the convolution on R we have
(Tuf )(t) = (f∗δu)(t) =
∫R
f (k)δu(k)ϕk (t) dk =
∫R
f (k)ϕ∗k (u)ϕk (t) dk
DefinitionFor any f : V → R the graph translation operator, Ti , is defined to be
(Ti f )(n) =√
N(f ∗ δi )(n) =√
NN−1∑l=0
f (λl )ϕ∗l (i)ϕl (n).
![Page 44: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/44.jpg)
Example - Classical Setting
Translation in the time domain = Modulation in the frequency domain
![Page 45: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/45.jpg)
Example - Movie
G =Minnesotaf = 11
−98 −97 −96 −95 −94 −93 −92 −91 −90 −8943
44
45
46
47
48
49
50
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
![Page 46: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/46.jpg)
Example - Movie
G =Minnesotaf (λl ) = e−5λl
−98 −97 −96 −95 −94 −93 −92 −91 −90 −89
43
44
45
46
47
48
49
50
−0.1
−0.05
0
0.05
0 1 2 3 4 5 6 7−0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
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Example - Movie
G =SG6f (λl ) = e−5λl
−1
−0.5
0
0.5
1
0
0.5
1
1.5
2
−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1
0 1 2 3 4 5 6−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
![Page 48: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/48.jpg)
Example - Movie
G =Minnesotaf ≡ 1
−98 −97 −96 −95 −94 −93 −92 −91 −90 −8943
44
45
46
47
48
49
50
−5
0
5
10
0 1 2 3 4 5 6−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
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Nice Properties of Graph Translation
Theorem
For any f ,g : V → R and i , j ∈ {1,2, ...,N} then1 Ti (f ∗ g) = (Ti f ) ∗ g = f ∗ (Tig).2 TiTj f = TjTi f .
![Page 50: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/50.jpg)
Not-so-nice Properties of Translation operator
The translation operator is not isometric
‖Ti f‖`2 6= ‖f‖`2
In general, the set of translation operators {Ti}Ni=1 do not form a
group like in the classical Euclidean setting.
TiTj 6= Ti+j
In general, Can we even hope for TiTj = Ti•j for some semigroupoperation, • : {1, ...,N} × {1, ...,N} → {1, ...,N}?
Theorem (B. & O.)
Given a graph, G, with eigenvector matrix Φ = [ϕ0| · · · |ϕN−1]. Graphtranslation on G is a semigroup, i.e. TiTj = Ti•j for some semigroupoperator • : {1, ...,N} × {1, ...,N} → {1, ...,N}, only if Φ = (1/
√N)H,
where H is a Hadamard matrix.
![Page 51: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/51.jpg)
Not-so-nice Properties of Translation operator
The translation operator is not isometric
‖Ti f‖`2 6= ‖f‖`2
In general, the set of translation operators {Ti}Ni=1 do not form a
group like in the classical Euclidean setting.
TiTj 6= Ti+j
In general, Can we even hope for TiTj = Ti•j for some semigroupoperation, • : {1, ...,N} × {1, ...,N} → {1, ...,N}?
Theorem (B. & O.)
Given a graph, G, with eigenvector matrix Φ = [ϕ0| · · · |ϕN−1]. Graphtranslation on G is a semigroup, i.e. TiTj = Ti•j for some semigroupoperator • : {1, ...,N} × {1, ...,N} → {1, ...,N}, only if Φ = (1/
√N)H,
where H is a Hadamard matrix.
![Page 52: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/52.jpg)
Not-so-nice Properties of Translation operator
The translation operator is not isometric
‖Ti f‖`2 6= ‖f‖`2
In general, the set of translation operators {Ti}Ni=1 do not form a
group like in the classical Euclidean setting.
TiTj 6= Ti+j
In general, Can we even hope for TiTj = Ti•j for some semigroupoperation, • : {1, ...,N} × {1, ...,N} → {1, ...,N}?
Theorem (B. & O.)
Given a graph, G, with eigenvector matrix Φ = [ϕ0| · · · |ϕN−1]. Graphtranslation on G is a semigroup, i.e. TiTj = Ti•j for some semigroupoperator • : {1, ...,N} × {1, ...,N} → {1, ...,N}, only if Φ = (1/
√N)H,
where H is a Hadamard matrix.
![Page 53: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/53.jpg)
Not-so-nice Properties of Translation operator
The translation operator is not isometric
‖Ti f‖`2 6= ‖f‖`2
In general, the set of translation operators {Ti}Ni=1 do not form a
group like in the classical Euclidean setting.
TiTj 6= Ti+j
In general, Can we even hope for TiTj = Ti•j for some semigroupoperation, • : {1, ...,N} × {1, ...,N} → {1, ...,N}?
Theorem (B. & O.)
Given a graph, G, with eigenvector matrix Φ = [ϕ0| · · · |ϕN−1]. Graphtranslation on G is a semigroup, i.e. TiTj = Ti•j for some semigroupoperator • : {1, ...,N} × {1, ...,N} → {1, ...,N}, only if Φ = (1/
√N)H,
where H is a Hadamard matrix.
![Page 54: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/54.jpg)
Why should we care?
With these time-frequency operators generalized tovertex-frequency operators, we are able to convert many niceresults and theories from harmonic analysis to the graph setting.
Wavelets and Wavelet TransformWindowed/Short-Time Fourier Transform (STFT)Windowed Fourier Frames
![Page 55: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/55.jpg)
Why should we care?
![Page 56: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/56.jpg)
Why should we care?
![Page 57: Analysis meets Graphs · 2019-02-28 · Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling](https://reader035.vdocument.in/reader035/viewer/2022070901/5f4a118891bb81620f672974/html5/thumbnails/57.jpg)
Thank you!