Stationary Graph Signals Stationary Graph Signals using an Isometric Graph Translation Benjamin Girault 1 École Normale Supérieure de Lyon, Université de Lyon IXXI – UMR 5668 (CNRS – ENS Lyon – UCB Lyon 1 – Inria) September 02, 2015 1 under the supervision of Éric Fleury and Paulo Gonçalves Benjamin Girault (ENS de Lyon, FRANCE) Eusipco 2015 September 02, 2015 1 / 14
Stationary Graph Signals Section: Stationarity Signal Processing over Graphs G = ( V , E ) a symmetric graph Adjacency matrix: A ij = w ij , ij ∈ E 2 . 5 Degree matrix: D = diag ( d 1 ,..., d N ) , d i = � j w ij Laplacian matrix: L = D − A Normalized Laplacian: L = D − 1 / 2 LD − 1 / 2 0 . 0 X : V → R or C a graph signal − 2 . 5 Fourier: F = [ χ 0 ··· χ N − 1 ] ∗ , � X = FX Example: L = F ∗ Λ F Some successes (non-exhaustive growing list): Wavelets [Hammond et al. 2011] Filter Banks [Narang & Ortega 2012-2013] Graph Shift [Sandryhaila & Moura 2013-2014] Vertex-Frequency [Shuman et al. 2014] Multiscale Community Mining [Tremblay & Borgnat 2015] Benjamin Girault (ENS de Lyon, FRANCE) Eusipco 2015 September 02, 2015 2 / 14
Stationary Graph Signals Section: Stationarity Why (Non-)Stationarity? Stochastic signals Toy Example: Denoising ⇒ Noise statistical invariance to an absolute time Non-stationarity: phase changes ⇒ Ruptures, anomalies, failure diagnosis, etc... Which tools for (non-)stationary graph signals? Benjamin Girault (ENS de Lyon, FRANCE) Eusipco 2015 September 02, 2015 3 / 14
Stationary Graph Signals Section: Stationarity Stationary Temporal Signals Stationary: Statistical invariance to an absolute time P [ x ( t ) = x ] = P [ x ( t − τ ) = x ] and for higher orders: P [ x ( t 1 ) = x 1 and ··· and x ( t k ) = x k ] = P [ x ( t 1 − τ ) = x 1 and ··· and x ( t k − τ ) = x k ] . ⇒ Strict-Sense Stationary (SSS) signal x Time shift T τ { x } ( t ) = x ( t − τ ) invariance: x and T τ { x } are statistically equal Benjamin Girault (ENS de Lyon, FRANCE) Eusipco 2015 September 02, 2015 4 / 14
Stationary Graph Signals Section: Stationarity Stationary Graph Signals? Straightforward definition: x and T τ { x } are statistically equal Question: What is T τ ? T τ i = ∆ i , the Generalized Translation? [Shuman et al. 2013] T τ = A τ , the Graph Shift? [Sandryhaila et al. 2013] ⇒ These operators are not isometric : � AX � 2 �= � X � 2 �= � ∆ i X � 2 Definition (Graph Translation) New isometric operator: � T τ G = e − ı τ L Benjamin Girault (ENS de Lyon, FRANCE) Eusipco 2015 September 02, 2015 5 / 14
Stationary Graph Signals Section: Stationarity Remarks Coherence with the Time Shift: G χ l = e − ı τ � λ l χ l T τ { e ω } = e − ı τω e ω T τ ⇒ G = diag ( e − ı � λ 0 ,..., e − ı � T τ � λ N − 1 ) Supports other graph Fourier transform: � � T τ G = e − ı τ L Complex operator � Not a real problem for stationarity � Analytical signals? � T G is NOT an operator shifting energy from one vertex to another. Benjamin Girault (ENS de Lyon, FRANCE) Eusipco 2015 September 02, 2015 6 / 14
Stationary Graph Signals Section: Stationarity Definition of Stationary Graph Signals Definition (SSS Graph Signal) A stochastic graph signal X verifying ∀ τ , X d = T τ G X . Definition (WSS Graph Signal) A stochastic graph signal X is Wide Sense Stationary iff: � µ X = E [ X ] = E [ T G X ] = µ T G X R X = E [ XX ∗ ] = E [( T G X )( T G X ) ∗ ] = R T G X Benjamin Girault (ENS de Lyon, FRANCE) Eusipco 2015 September 02, 2015 7 / 14
Stationary Graph Signals Section: Stationarity Spectral Characterisation Theorem A stochastic graph signal X is WSS iff ∀ l , λ l �= 0 ⇒ E [ � X ( l )] = 0 (i) X ∗ ( k )] = 0 ∀ l , λ l �= λ k ⇒ E [ � X ( l ) � (ii) Remark on (i) (First Moment) First moment condition: E [ X ] ∝ χ 0 � 1 ,..., 1 � T L = F ∗ Λ F ⇒ χ 0 = � � � 1 / � T L = F ∗ Λ F ⇒ χ 0 = d 1 ,..., 1 / d N Remark on (ii) (Second Moment) if ∀ l �= k , λ l �= λ k , then (ii) ⇔ S = E [ � X � X ∗ ] diagonal Benjamin Girault (ENS de Lyon, FRANCE) Eusipco 2015 September 02, 2015 8 / 14
Stationary Graph Signals Section: Stationarity Discussion Complex Operator: But real WSS signals exists R Non Toeplitz: R i , j = E [ X i X ∗ j ] �= E [ X i + k X ∗ j + k ] = R i + k , j + k Alternative Definition of Translation? ⇒ Yes, as long as T = e − ı Ω , and T χ 0 = χ 0 Benjamin Girault (ENS de Lyon, FRANCE) Eusipco 2015 September 02, 2015 9 / 14
Stationary Graph Signals Section: Illustrations on Synthetic Graph Signal Summary before Illustrations Graph Translation � T τ G = e − ı τ L Wide-Sense Stationary Graph Signal X and T G have equal first and second moments: E [ X ] = µ X = µ T G X E [ XX ∗ ] = R X = R T G X Spectral Characterisation � E [ � X � ] if l = 0 E [ � X ( l )] = 0 otherwise E [ � X � X ∗ ] = S X diagonal (if ∀ l �= k , λ l �= λ k ) Benjamin Girault (ENS de Lyon, FRANCE) Eusipco 2015 September 02, 2015 10 / 14
Stationary Graph Signals Section: Illustrations on Synthetic Graph Signal Stationary White Noise on Graph 2 . 5 Signal with flat power spectrum: 0 . 0 X ∗ ] = σ 2 I S = E [ � X � − 2 . 5 A realisation 1 . 0 1 . 0 0 . 0 0 . 0 − 1 . 0 − 1 . 0 Empirical R (50k realisations) Empirical S (50k realisations) Benjamin Girault (ENS de Lyon, FRANCE) Eusipco 2015 September 02, 2015 11 / 14
Stationary Graph Signals Section: Illustrations on Synthetic Graph Signal WSS Graph Signal 0 . 5 Low pass WSS graph signal: � 0 . 0 1 if l ∈ { 1 , 2 , 3 } X l | 2 ] = S ll = E [ | � 0 otherwise − 0 . 5 A realisation 0 . 5 1 . 0 0 . 0 0 . 0 − 0 . 5 − 1 . 0 Empirical R (50k realisations) Empirical S (50k realisations) Benjamin Girault (ENS de Lyon, FRANCE) Eusipco 2015 September 02, 2015 12 / 14
Stationary Graph Signals Section: Illustrations on Synthetic Graph Signal Non Stationary Graph Signal 0 . 5 Same as previously, with additive 0 . 0 noise on X ( 1 ) with distribution N ( 0 , 1 ) . − 0 . 5 A realisation 0 . 5 1 . 0 0 . 0 0 . 0 − 0 . 5 − 1 . 0 Empirical R (50k realisations) Empirical S (50k realisations) Benjamin Girault (ENS de Lyon, FRANCE) Eusipco 2015 September 02, 2015 13 / 14
Stationary Graph Signals Section: Perspectives and conclusion Perspectives Ongoing work Application to weather reports ⇒ Next week at Gretsi 2015, Lyon, FRANCE Statistical test for stationarity Classes of interesting non-stationary signals Open Questions Meaning of a translated signal? Analytic signal? Alternative invariance operator? Benjamin Girault (ENS de Lyon, FRANCE) Eusipco 2015 September 02, 2015 14 / 14
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