A Digraph Fourier Transform with Spread Frequency Components Gonzalo Mateos Dept. of Electrical and Computer Engineering University of Rochester gmateosb@ece.rochester.edu http://www.ece.rochester.edu/~gmateosb/ GlobalSIP, November 14, 2017 A Digraph Fourier Transform with Spread Frequency Components GlobalSIP 2017 1

Co-authors Rasoul Shafipour Ali Khodabakhsh Evdokia Nikolova University of Rochester University of Texas at Austin University of Texas at Austin A Digraph Fourier Transform with Spread Frequency Components GlobalSIP 2017 2

Network Science analytics Online social media Internet Clean energy and grid analy,cs ◮ Network as graph G = ( V , E ): encode pairwise relationships ◮ Desiderata: Process, analyze and learn from network data [Kolaczyk’09] A Digraph Fourier Transform with Spread Frequency Components GlobalSIP 2017 3

Network Science analytics Online social media Internet Clean energy and grid analy,cs ◮ Network as graph G = ( V , E ): encode pairwise relationships ◮ Desiderata: Process, analyze and learn from network data [Kolaczyk’09] ◮ Interest here not in G itself, but in data associated with nodes in V ⇒ The object of study is a graph signal ◮ Ex: Opinion profile, buffer congestion levels, neural activity, epidemic A Digraph Fourier Transform with Spread Frequency Components GlobalSIP 2017 4

Graph signal processing and Fourier transform ◮ Directed graph (digraph) G with adjacency matrix A 2 ⇒ A ij = Edge weight from node i to node j ◮ Define a signal x ∈ R N on top of the graph 1 4 3 ⇒ x i = Signal value at node i A Digraph Fourier Transform with Spread Frequency Components GlobalSIP 2017 5

Graph signal processing and Fourier transform ◮ Directed graph (digraph) G with adjacency matrix A 2 ⇒ A ij = Edge weight from node i to node j ◮ Define a signal x ∈ R N on top of the graph 1 4 3 ⇒ x i = Signal value at node i ◮ Associated with G is the underlying undirected G u ⇒ Laplacian marix L = D − A u , eigenvectors V = [ v 1 , · · · , v N ] ◮ Graph Signal Processing (GSP): exploit structure in A or L to process x ◮ Graph Fourier Transform (GFT): ˜ x = V T x for undirected graphs ⇒ Decompose x into different modes of variation ⇒ Inverse (i)GFT x = V ˜ x , eigenvectors as frequency atoms A Digraph Fourier Transform with Spread Frequency Components GlobalSIP 2017 6

GFT: Motivation and context ◮ Spectral analysis and filter design [Tremblay et al’17], [Isufi et al’16] ◮ Promising tool in neuroscience [Huang et al’16] ⇒ Graph frequency analyses of fMRI signals A Digraph Fourier Transform with Spread Frequency Components GlobalSIP 2017 7

GFT: Motivation and context ◮ Spectral analysis and filter design [Tremblay et al’17], [Isufi et al’16] ◮ Promising tool in neuroscience [Huang et al’16] ⇒ Graph frequency analyses of fMRI signals ◮ Noteworthy GFT approaches ◮ Eigenvectors of the Laplacian L [Shuman et al’13] ◮ Jordan decomposition of A [Sandryhaila-Moura’14], [Deri-Moura’17] ◮ Lova´ sz extension of the graph cut size [Sardellitti et al’17] A Digraph Fourier Transform with Spread Frequency Components GlobalSIP 2017 8

GFT: Motivation and context ◮ Spectral analysis and filter design [Tremblay et al’17], [Isufi et al’16] ◮ Promising tool in neuroscience [Huang et al’16] ⇒ Graph frequency analyses of fMRI signals ◮ Noteworthy GFT approaches ◮ Eigenvectors of the Laplacian L [Shuman et al’13] ◮ Jordan decomposition of A [Sandryhaila-Moura’14], [Deri-Moura’17] ◮ Lova´ sz extension of the graph cut size [Sardellitti et al’17] ◮ Our contribution: design a novel digraph (D)GFT such that ◮ Bases offer notions of frequency and signal variation ◮ Frequencies are (approximately) equidistributed in [0 , f max ] ◮ Bases are orthonormal, so Parseval’s identity holds A Digraph Fourier Transform with Spread Frequency Components GlobalSIP 2017 9

Signal variation on digraphs ◮ Total variation of signal x with respect to L N � TV( x ) = x T Lx = A u ij ( x i − x j ) 2 i , j =1 , j > i ⇒ Smoothness measure on the graph G u ◮ For Laplacian eigenvectors V = [ v 1 , · · · , v N ] ⇒ TV( v k ) = λ k ⇒ 0 = λ 1 < · · · ≤ λ N can be viewed as frequencies A Digraph Fourier Transform with Spread Frequency Components GlobalSIP 2017 10

Signal variation on digraphs ◮ Total variation of signal x with respect to L N � TV( x ) = x T Lx = A u ij ( x i − x j ) 2 i , j =1 , j > i ⇒ Smoothness measure on the graph G u ◮ For Laplacian eigenvectors V = [ v 1 , · · · , v N ] ⇒ TV( v k ) = λ k ⇒ 0 = λ 1 < · · · ≤ λ N can be viewed as frequencies ◮ Def: Directed variation for signals over digraphs ([ x ] + = max(0 , x )) N � A ij [ x i − x j ] 2 DV( x ) := + i , j =1 ⇒ Captures signal variation (flow) along directed edges ⇒ Consistent, since DV( x ) ≡ TV( x ) for undirected graphs A Digraph Fourier Transform with Spread Frequency Components GlobalSIP 2017 11

DGFT with spread frequeny components ◮ Goal: find N orthonormal bases capturing different modes of DV on G ◮ Collect the desired bases in a matrix U = [ u 1 , · · · , u N ] ∈ R N × N ⇒ u k represents the k th frequency component with f k := DV( u k ) A Digraph Fourier Transform with Spread Frequency Components GlobalSIP 2017 12

DGFT with spread frequeny components ◮ Goal: find N orthonormal bases capturing different modes of DV on G ◮ Collect the desired bases in a matrix U = [ u 1 , · · · , u N ] ∈ R N × N ⇒ u k represents the k th frequency component with f k := DV( u k ) ◮ Similar to the DFT, seek N equidistributed graph frequencies f k = DV( u k ) = k − 1 N − 1 f max , k = 1 , . . . , N ⇒ f max is the maximum DV of a unit-norm graph signal on G ◮ Q : Why spread frequencies? ⇒ To better capture low, medium, and high frequencies ⇒ Aid filter design in the graph spectral domain A Digraph Fourier Transform with Spread Frequency Components GlobalSIP 2017 13

Motivation for spread frequencies ◮ Ex: Directed variation minimization [Sardellitti et al’17] 2 � N min i , j =1 A ij [ u i − u j ] + U 1 U T U = I s.t. 4 3 A Digraph Fourier Transform with Spread Frequency Components GlobalSIP 2017 14

Motivation for spread frequencies ◮ Ex: Directed variation minimization [Sardellitti et al’17] 2 � N min i , j =1 A ij [ u i − u j ] + U 1 U T U = I s.t. 4 3 √ √ ◮ U ∗ is the optimum basis where a = 1+ 5 , b = 1 − 5 , and c = − 0 . 5 4 4 ◮ All columns of U ∗ satisfy DV( u ∗ k ) = 0 , k = 1 , . . . , 4 ⇒ Expansion x = U ∗ ˜ x fails to capture different modes of variation ◮ Q: Can we always find equidistributed frequencies? A Digraph Fourier Transform with Spread Frequency Components GlobalSIP 2017 15

Challenges: Maximum directed variation ◮ Finding f max is in general challenging ◮ Solve the (non-convex) spherically-constrained problem u max = argmax DV( u ) and f max := DV( u max ) . � u � =1 u max with approximate ˜ ◮ Q : Can we find a basis ˜ f max ≈ f max ? A Digraph Fourier Transform with Spread Frequency Components GlobalSIP 2017 16

Challenges: Maximum directed variation ◮ Finding f max is in general challenging ◮ Solve the (non-convex) spherically-constrained problem u max = argmax DV( u ) and f max := DV( u max ) . � u � =1 u max with approximate ˜ ◮ Q : Can we find a basis ˜ f max ≈ f max ? Proposition: For a digraph G , recall G u and its Laplacian L . Let v N be the dominant eigenvector of L . Then, f max := max { DV( v N ) , DV( − v N ) } ≥ f max ˜ 2 ◮ We can 1/2-approximate f max with ˜ u max = argmax DV( v ) v ∈{ v N , − v N } A Digraph Fourier Transform with Spread Frequency Components GlobalSIP 2017 17

Challenges: Equidistributed frequencies ◮ Equidistributed f k = k − 1 N − 1 f max may not be feasible. Ex: In undirected G u N N � � f u max = λ max & f k = TV( v k ) = trace( L ) k =1 k =1 A Digraph Fourier Transform with Spread Frequency Components GlobalSIP 2017 18

Challenges: Equidistributed frequencies ◮ Equidistributed f k = k − 1 N − 1 f max may not be feasible. Ex: In undirected G u N N � � f u max = λ max & f k = TV( v k ) = trace( L ) k =1 k =1 ◮ Idea: Set u 1 = u min := 1 N 1 N and u N = ˜ u max and minimize √ N − 1 � [DV( u i +1 ) − DV( u i )] 2 δ ( U ) := i =1 ⇒ δ ( U ) is the spectral dispersion function ⇒ δ ( U ) is minimized if the free DV values form an arithmetic sequence ⇒ Consistent with our design criteria A Digraph Fourier Transform with Spread Frequency Components GlobalSIP 2017 19

Spectral dispersion minimization ◮ We cast the optimization problem of finding spread frequencies as N − 1 � [DV( u i +1 ) − DV( u i )] 2 min U i =1 U T U = I subject to u 1 = u min u N = ˜ u max ⇒ Tackle via feasible optimization method in the Stiefel manifold ◮ Here instead we resort to a simple yet efficient heuristic A Digraph Fourier Transform with Spread Frequency Components GlobalSIP 2017 20

A DGFT construction heuristic ◮ Use eigenvectors of L , the Laplacian of G u , to construct U ◮ Fix f 1 = 0 ( u 1 = u min ) and ˜ f N = ˜ f max ( u N = ˜ u max ) ◮ Let f i := DV( v i ) and f i := DV( − v i ), where v i is the i th eigenvector of L A Digraph Fourier Transform with Spread Frequency Components GlobalSIP 2017 21

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