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
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
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 δ 1 Delta signal mapped to a delta signal T δ 1 = δ 2 T 5 δ 1 = δ 6 Benjamin Girault (USC) IEEE GlobalSIP 2016 December 8, 2016 3 / 15
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 e 1 Delta signal mapped to a delta signal Fourier mode: phase shifted Te 1 ⇒ Convolutive operator T 5 e 1 Energy invariance Benjamin Girault (USC) IEEE GlobalSIP 2016 December 8, 2016 3 / 15
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 Translation Graph Shift Generalized Translation [Girault et al. SPL‘15] [Sandryhaila & Moura TSP‘13] [Shuman et al. SPM‘13] 0 . 4 1 . 000 0 . 2 0 . 2 0 . 500 0 . 1 0 . 0 0 . 000 0 . 0 | T G δ 1 | | A δ 1 | | T 2 h | Benjamin Girault (USC) IEEE GlobalSIP 2016 December 8, 2016 4 / 15
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 Translation Graph Shift Generalized Translation [Girault et al. SPL‘15] [Sandryhaila & Moura TSP‘13] [Shuman et al. SPM‘13] 0 . 4 1 , 400 0 . 3 0 . 2 700 0 . 2 0 . 0 0 0 . 0 | T 5 | A 5 δ 1 | G δ 1 | | T 2 δ 1 | ⇒ Time Shift transposes to Diffusion. Is the Graph Translation formally a diffusion operator? Benjamin Girault (USC) IEEE GlobalSIP 2016 December 8, 2016 4 / 15
Localization Bounds Motivation Case Study #1 Case Study #2 Conclusion Graph Translation: well defined isometric operator in the Fourier domain. � T G χ l = e − ı π � � � L λ l / ρ G χ l . T G = exp − ı π ⇒ ρ G Question: vertex domain behavior? Some evidence of diffusive behavior ( | T G δ 1 | ): 100 0 . 4 Energy (%) 90 0 . 2 80 T G δ 1 70 0 2 4 6 8 0 . 0 Hops Benjamin Girault (USC) IEEE GlobalSIP 2016 December 8, 2016 5 / 15
Localization Bounds Motivation Case Study #1 Case Study #2 Conclusion Graph Translation: well defined isometric operator in the Fourier domain. � T G χ l = e − ı π � � � L λ l / ρ G χ l . T G = exp − ı π ⇒ ρ G Question: vertex domain behavior? Some evidence of diffusive behavior ( | T 10 G δ 1 | ): 100 0 . 4 Energy (%) 90 0 . 2 80 T G δ 1 T 10 G δ 1 70 0 2 4 6 8 0 . 0 Hops Benjamin Girault (USC) IEEE GlobalSIP 2016 December 8, 2016 5 / 15
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 p K is a polynomial of degree K and | f − p K | ≤ κ ( K ) , then the energy of f ( M ) δ i 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
Localization Bounds Motivation Case Study #1 Case Study #2 Conclusion Adjacency-Based Translation Operator (GFT based on [Sandryhaila & Moura 2013].) Adjacency matrix A = U Γ U ∗ , with Γ = diag ( γ 0 ,..., γ N − 1 ) Graph frequencies π ( 1 − γ l / γ max ) ∈ [ 0 , 2 π ] , Fourier modes U l . Definition (Adjacency-Based Isometric Translation Operator) � � A = exp − ı π ( I − A / γ max ) With M = I − A / γ max , we obtain A = f ( M ) and: f ( x ) = exp ( − ı π x ) = cos ( π x ) − ı sin ( π x ) . Question: Polynomial approximation of f ? Benjamin Girault (USC) IEEE GlobalSIP 2016 December 8, 2016 7 / 15
Localization Bounds Motivation Case Study #1 Case Study #2 Conclusion Polynomial Approximation Analytical form of f : � � π 2 k π 2 k + 1 ∞ ( 2 k )! x 2 k − ı ( − 1 ) k ( 2 k + 1 )! x 2 k + 1 � f ( x ) = k = 0 Lemma (Alternating Series Approximation (Lemma 2)) ( − 1 ) k f k x k , � f ( x ) = f k ≥ 0 k truncated at K leads to the polynomial p K verifying | f ( x ) − p K ( x ) | ≤ κ ( K ) = | f K + 1 x K + 1 | . Benjamin Girault (USC) IEEE GlobalSIP 2016 December 8, 2016 8 / 15
Localization Bounds Motivation Case Study #1 Case Study #2 Conclusion Approximation Curve κ ( K ) = ( 2 π ) 2 K + 2 2 π � � 1 + . ( 2 K + 2 )! 2 K + 3 10 0 Error upper bound 10 − 4 10 − 8 10 − 12 0 5 10 15 K Important Remark: κ ( K ) does not depend on the graph. Benjamin Girault (USC) IEEE GlobalSIP 2016 December 8, 2016 9 / 15
Localization Bounds Motivation Case Study #1 Case Study #2 Conclusion Laplacian-Based Translation Operator Laplacian matrix L = D − A = χ Λ χ ∗ , with Λ = diag ( λ 0 ,..., λ N − 1 ) � Graph frequencies π λ l / ρ G ∈ [ 0 , π ] , Fourier modes χ l . Definition (Laplacian-Based Isometric Translation Operator) � � � L T G = exp − ı π ρ G With M = L / ρ G , we obtain T G = f ( M ) and: � x � x � x � � � � � � f ( x ) = exp = cos − ı sin x ∈ [ 0 , 1 ] . − ı π π π , Remark: Complexity due to the square root. Benjamin Girault (USC) IEEE GlobalSIP 2016 December 8, 2016 10 / 15
Localization Bounds Motivation Case Study #1 Case Study #2 Conclusion Dealing with the Square Root Preliminaries: Rewrite f to obtain polynomial expansions: � x sin ( π � x ) � x � � f ( x ) = cos π − ı � x (cos ( x ) : only even degree coeff. / sin ( x ) : only odd degree) ⇒ We are left with approximating � x . � 1 + y about 0 on [ − ǫ , ǫ ] and rescale Idea: Use the Taylor expansion of � 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
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)) k f k x k leads to Taylor’s expansion f ( x ) = � 1 � � � | x | K + 1 � | f ( K + 1 ) | κ ( K ) ≤ ( K + 1 )! max max . Approximation bound for the product � x sin ( π � x ) : � x Lemma (Functional Composition Approximation (Lemma 3)) If f ( x ) = g ( x ) h ( x ) then: κ f ( P , Q ) ≤ κ g ( Q ) max | h |+ κ h ( P )( max | g |+ κ g ( Q )) . Note: The resulting polynomial in Lemma 3 is of degree P + Q . Benjamin Girault (USC) IEEE GlobalSIP 2016 December 8, 2016 12 / 15
Localization Bounds Motivation Case Study #1 Case Study #2 Conclusion Approximation curve of the Graph Translation � � � � � M sin ( π M ) 1 Application to T G δ i = cos ( π M ) δ i + ı N 1 δ i − � � M � � κ T G ( P , Q ) = κ C ( P ) + κ S ( P ) + κ R ( Q ) 1 + κ S ( P ) 10 0 Error upper bound 10 − 4 10 − 8 Q = 1 Q = 5 Q = 2 Q = 10 Q = 3 Q = 15 10 − 12 Q = 4 Q = ∞ 0 5 10 15 P Important Remark: κ T G ( P , Q ) depends on the spectral gap of the graph. Benjamin Girault (USC) IEEE GlobalSIP 2016 December 8, 2016 13 / 15
Localization Bounds Motivation Case Study #1 Case Study #2 Conclusion Optimal P + Q for a given error ξ to T α G 30 ξ = 0 . 5 25 ξ = 10 − 1 ξ = 10 − 2 20 ξ = 10 − 3 P + Q ξ = 10 − 4 15 10 5 0 0 1 2 3 4 5 α Benjamin Girault (USC) IEEE GlobalSIP 2016 December 8, 2016 14 / 15
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: T G δ i = T i t G Use this diffusion characterization to interpret stationary graph signals Benjamin Girault (USC) IEEE GlobalSIP 2016 December 8, 2016 15 / 15
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