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Localization Bounds for the Graph Translation Benjamin Girault , - - PowerPoint PPT Presentation

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Localization Bounds Motivation Case Study #1 Case Study #2 Conclusion Localization Bounds for the Graph Translation Benjamin Girault , Paulo Gonalves , Shrikanth S. Narayanan , Antonio Ortega University of Southern

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Localization Bounds Motivation Case Study #1 Case Study #2 Conclusion

Localization Bounds for the Graph Translation

Benjamin Girault⋆, Paulo Gonçalves⋄, Shrikanth S. Narayanan⋆, Antonio Ortega⋆

⋆University of Southern California, USA ⋄Université de Lyon, Inria, ENS de Lyon, CNRS, UCB Lyon 1, FRANCE

December 8, 2016

Benjamin Girault (USC) IEEE GlobalSIP 2016 December 8, 2016 1 / 15

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Localization Bounds Motivation Case Study #1 Case Study #2 Conclusion

Context: Signal Processing over Graphs

Grand Goal: Interpretation of variations over a discrete structure. Structure: Vertices linked by edges. Signal: Values carried by vertices. Assumption: The structure explains variations. Tools: Fourier transform, wavelet transform, filtering, sampling...

Benjamin Girault (USC) IEEE GlobalSIP 2016 December 8, 2016 2 / 15

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Localization Bounds Motivation Case Study #1 Case Study #2 Conclusion

Fundamental Question: Time Shift Equivalent?

Observation: Time shift is at the core of Temporal Signal Processing. Examples: Fourier, Wavelets, Time-Frequency, Stationarity... Time Shift properties: Linear Operator Delta signal mapped to a delta signal

δ1

Tδ1 = δ2 T 5δ1 = δ6

Benjamin Girault (USC) IEEE GlobalSIP 2016 December 8, 2016 3 / 15

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Localization Bounds Motivation Case Study #1 Case Study #2 Conclusion

Fundamental Question: Time Shift Equivalent?

Observation: Time shift is at the core of Temporal Signal Processing. Examples: Fourier, Wavelets, Time-Frequency, Stationarity... Time Shift properties: Linear Operator Delta signal mapped to a delta signal Fourier mode: phase shifted

⇒ Convolutive operator

Energy invariance e1 Te1 T 5e1

Benjamin Girault (USC) IEEE GlobalSIP 2016 December 8, 2016 3 / 15

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Localization Bounds Motivation Case Study #1 Case Study #2 Conclusion

Fundamental Question: Time Shift Equivalent? (cont’d)

State of the Art of its equivalent Graph operator: Graph Shift

[Sandryhaila & Moura TSP‘13]

0.000 0.500 1.000

|Aδ1|

Generalized Translation

[Shuman et al. SPM‘13]

0.0 0.1 0.2

|T2h|

Graph Translation

[Girault et al. SPL‘15]

0.0 0.2 0.4

|TG δ1|

Benjamin Girault (USC) IEEE GlobalSIP 2016 December 8, 2016 4 / 15

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Localization Bounds Motivation Case Study #1 Case Study #2 Conclusion

Fundamental Question: Time Shift Equivalent? (cont’d)

State of the Art of its equivalent Graph operator: Graph Shift

[Sandryhaila & Moura TSP‘13]

700 1,400

|A5δ1|

Generalized Translation

[Shuman et al. SPM‘13]

0.0 0.2 0.3

|T2δ1|

Graph Translation

[Girault et al. SPL‘15]

0.0 0.2 0.4

|T 5

G δ1|

⇒ Time Shift transposes to Diffusion.

Is the Graph Translation formally a diffusion operator?

Benjamin Girault (USC) IEEE GlobalSIP 2016 December 8, 2016 4 / 15

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Localization Bounds Motivation Case Study #1 Case Study #2 Conclusion

Graph Translation: well defined isometric operator in the Fourier domain. TG = exp

  • −ıπ
  • L

ρG

  • ⇒

TG χl = e−ıπ

λl/ρG χl.

Question: vertex domain behavior? Some evidence of diffusive behavior (|TG δ1|):

0.0 0.2 0.4

2 4 6 8 70 80 90 100 Hops Energy (%) TG δ1

Benjamin Girault (USC) IEEE GlobalSIP 2016 December 8, 2016 5 / 15

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Localization Bounds Motivation Case Study #1 Case Study #2 Conclusion

Graph Translation: well defined isometric operator in the Fourier domain. TG = exp

  • −ıπ
  • L

ρG

  • ⇒

TG χl = e−ıπ

λl/ρG χl.

Question: vertex domain behavior? Some evidence of diffusive behavior (|T 10

G δ1|):

0.0 0.2 0.4

2 4 6 8 70 80 90 100 Hops Energy (%) TG δ1 T 10

G δ1

Benjamin Girault (USC) IEEE GlobalSIP 2016 December 8, 2016 5 / 15

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Localization Bounds Motivation Case Study #1 Case Study #2 Conclusion

Aims and Outline

Aim: Study the diffusion properties through the impulse response Context: Operators verifying H = f (M) (M a local operator) Premise: If f is a polynomial of degree K, the energy of the impulse response is located within K-hops of the impulse Method: Approximation using a truncation of the analytical form of f Fundamental result (Theorem 1): If pK is a polynomial of degree K and

|f −pK| ≤ κ(K) , then the energy of f (M)δi outside the K-hops

neighborhood of i is at most κ(K). Outline

1 Case Study #1: Simple analytical form of f 2 Case Study #2: More complex analytical form involving composition

Benjamin Girault (USC) IEEE GlobalSIP 2016 December 8, 2016 6 / 15

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Localization Bounds Motivation Case Study #1 Case Study #2 Conclusion

Adjacency-Based Translation Operator

(GFT based on [Sandryhaila & Moura 2013].) Adjacency matrix A = UΓU∗, with Γ = diag(γ0,...,γN−1) Graph frequencies π(1−γl/γmax) ∈ [0,2π], Fourier modes Ul.

Definition (Adjacency-Based Isometric Translation Operator)

A = exp

  • −ıπ(I −A/γmax)
  • With M = I −A/γmax, we obtain A = f (M) and:

f (x) = exp(−ıπx) = cos(πx)− ı sin(πx). Question: Polynomial approximation of f ?

Benjamin Girault (USC) IEEE GlobalSIP 2016 December 8, 2016 7 / 15

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Localization Bounds Motivation Case Study #1 Case Study #2 Conclusion

Polynomial Approximation

Analytical form of f : f (x) =

∞

  • k=0

(−1)k

  • π2k

(2k)!x2k − ı

π2k+1

(2k +1)!x2k+1

  • Lemma (Alternating Series Approximation (Lemma 2))

f (x) =

  • k

(−1)kfkxk, fk ≥ 0 truncated at K leads to the polynomial pK verifying

|f (x)−pK(x)| ≤ κ(K) = |fK+1xK+1|.

Benjamin Girault (USC) IEEE GlobalSIP 2016 December 8, 2016 8 / 15

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Localization Bounds Motivation Case Study #1 Case Study #2 Conclusion

Approximation Curve

κ(K) = (2π)2K+2

(2K +2)!

  • 1+

2π 2K +3

  • .

5 10 15 10−12 10−8 10−4 100 K Error upper bound

Important Remark: κ(K) does not depend on the graph.

Benjamin Girault (USC) IEEE GlobalSIP 2016 December 8, 2016 9 / 15

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Localization Bounds Motivation Case Study #1 Case Study #2 Conclusion

Laplacian-Based Translation Operator

Laplacian matrix L = D −A = χΛχ∗, with Λ = diag(λ0,...,λN−1) Graph frequencies π

  • λl/ρG ∈ [0,π], Fourier modes χl.

Definition (Laplacian-Based Isometric Translation Operator)

TG = exp

  • −ıπ
  • L

ρG

  • With M = L/ρG, we obtain TG = f (M) and:

f (x) = exp

  • −ıπ

x

  • = cos
  • π

x

  • − ı sin
  • π

x

  • ,

x ∈ [0,1]. Remark: Complexity due to the square root.

Benjamin Girault (USC) IEEE GlobalSIP 2016 December 8, 2016 10 / 15

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Localization Bounds Motivation Case Study #1 Case Study #2 Conclusion

Dealing with the Square Root

Preliminaries: Rewrite f to obtain polynomial expansions: f (x) = cos

  • π

x

  • − ı

x sin(πx) x

(cos(x): only even degree coeff. / sin(x): only odd degree)

⇒ We are left with approximating x.

Idea: Use the Taylor expansion of

1+y about 0 on [−ǫ,ǫ] and rescale x to approximate it on [̺,1] (spectral gap of M)with ǫ ≤ 1 depending on

the spectral gap.

Benjamin Girault (USC) IEEE GlobalSIP 2016 December 8, 2016 11 / 15

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Localization Bounds Motivation Case Study #1 Case Study #2 Conclusion

Dealing with the Square Root (cont’d)

Approximation bound for x:

Lemma (Non-Alternating Series Approximation (Lemma 1))

Taylor’s expansion f (x) =

k fkxk leads to

κ(K) ≤

1 (K +1)! max

  • |f (K+1)|
  • max
  • |x|K+1

. Approximation bound for the product x sin(πx)

x

:

Lemma (Functional Composition Approximation (Lemma 3))

If f (x) = g(x)h(x) then:

κf (P,Q) ≤ κg(Q)max|h|+κh(P)(max|g|+κg(Q)).

Note: The resulting polynomial in Lemma 3 is of degree P +Q.

Benjamin Girault (USC) IEEE GlobalSIP 2016 December 8, 2016 12 / 15

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Localization Bounds Motivation Case Study #1 Case Study #2 Conclusion

Approximation curve of the Graph Translation

Application to TG δi = cos(π

  • M)δi + ı
  • M sin(π
  • M)
  • M
  • δi −

1

  • N 1
  • κTG (P,Q) = κC(P)+κS(P)+κR(Q)
  • 1+κS(P)
  • 5

10 15 10−12 10−8 10−4 100 P Error upper bound

Q = 1 Q = 5 Q = 2 Q = 10 Q = 3 Q = 15 Q = 4 Q = ∞

Important Remark: κTG (P,Q) depends on the spectral gap of the graph.

Benjamin Girault (USC) IEEE GlobalSIP 2016 December 8, 2016 13 / 15

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Localization Bounds Motivation Case Study #1 Case Study #2 Conclusion

Optimal P +Q for a given error ξ to T α

G

1 2 3 4 5 5 10 15 20 25 30 α P + Q

ξ = 0.5 ξ = 10−1 ξ = 10−2 ξ = 10−3 ξ = 10−4

Benjamin Girault (USC) IEEE GlobalSIP 2016 December 8, 2016 14 / 15

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Localization Bounds Motivation Case Study #1 Case Study #2 Conclusion

Summary and Perspectives

Summary Graph Translation: Approximately a diffusion operator Tools developed: Simple Lemmas to get polynomial approximations bounds on operators and composed operators Perspectives Loose bound: use the weights to better characterize the bound Link with the generalized translation: TG δi = TitG Use this diffusion characterization to interpret stationary graph signals

Benjamin Girault (USC) IEEE GlobalSIP 2016 December 8, 2016 15 / 15