Graph Signal Processing

Transcription

1 Graph Signal Processing Rahul Singh Data Science Reading Group Iowa State University March, 07

2 Outline Graph Signal Processing Background Graph Signal Processing Frameworks Laplacian Based Discrete Signal Processing on Graphs (DSP G ) Framework Weight Matrix Based Directed Laplacian Based Spectral Graph Wavelets Conclusions Graph Signal Processing Data Science Reading Group, ISU March, 07 /9

3 Background Classical Signal Processing Introduction t t t t N Structure behind time-series Graph Signal Processing Data Science Reading Group, ISU March, 07 /9

4 Background Classical Signal Processing Introduction t t t t N Structure behind time-series Structure behind image Graph Signal Processing Data Science Reading Group, ISU March, 07 /9

5 Background Classical Signal Processing Introduction Translation, filtering, convolution, modulation, Fourier transform, wavelets, sparse representations... t t t t N Structure behind time-series Structure behind image Graph Signal Processing Data Science Reading Group, ISU March, 07 /9

6 Background Graph Signal Processing Introduction Graph Signal Processing Data Science Reading Group, ISU March, 07 /9

7 Background Graph Signal Processing Introduction Graph Signal Processing Data Science Reading Group, ISU March, 07 /9

8 Background Introduction Graph Signal Processing A graph signal 5 - Graph Signal Processing Data Science Reading Group, ISU March, 07 /9

9 Background Introduction Graph Signal Processing Applications Graph Signal Processing Data Science Reading Group, ISU March, 07 5/9

10 Difficulty in GSP Background Introduction Translation is simple in classical signal processing Graph Signal Processing Data Science Reading Group, ISU March, 07 6/9

11 Difficulty in GSP Background Introduction Translation is simple in classical signal processing What does it mean to translate the signal to vertex 50? Challenging in GSP Graph Signal Processing Data Science Reading Group, ISU March, 07 6/9

12 Notation Background Notation 5 A graph signal f Graph G = (V, W) f = Weight matrix 0 W = Degree matrix D = Laplacian matrix 0 0 L = D W = Graph Signal Processing Data Science Reading Group, ISU March, 07 7/9

13 GSP Frameworks Existing Graph Signal Processing (GSP) Frameworks GSP based on Laplacian Discrete Signal Processing on Graphs (DSP G ) Framework Shift Operator Not defined The weight matrix LSI Filters Not applicable Applicable Applicability Only undirected graphs Directed graphs Frequencies Harmonics Frequency Ordering Multiscale Analysis Eigenvalues of the Laplacian matrix Eigenvectors of the Laplacian matrix Laplacian quadratic form Exists (SGWT) Eigenvalues of the weight matrix Eigenvectors of the weight matrix Total variation Does not exist David K Hammond, Pierre Vandergheynst, and Rémi Gribonval. Wavelets on graphs via spectral graph theory. In: Applied and Computational Harmonic Analysis 0. (0), pp A. Sandryhaila and J.M.F. Moura. Discrete Signal Processing on Graphs: Frequency Analysis. In: Signal Processing, IEEE Transactions on 6. (0), pp Graph Signal Processing Data Science Reading Group, ISU March, 07 8/9

14 GSP Frameworks Laplacian Based Classical vs Graph Signal Processing (Laplacian Based) Operator/ Transform Classical Signal Processing Graph Signal Processing Fourier Transform ˆx(ω) = x(t)e jωt dt Frequency: ω can take any value Fourier basis: Complex exponentials e jωt ˆf(λl ) = N n= f(n)u l (n) Frequency: Eigenvalues of the graph Laplacian (λ l ) Fourier basis: Eigenvectors of the graph Laplacian (u l ) Convolution In time domain: x(t) y(t) = x(τ)y(t τ)dτ In frequency domain: x(t) y(t) = ˆx(ω)ŷ(ω) Defined through Graph Fourier Transform ) f g = U (ˆf.ĝ Translation Can be defined using convolution T τ x(t) = x(t τ) = x(t) δ τ (t) Defined through graph convolution T i f(n) = N(f δ i )(n) = N N ˆf (λ l )u l=0 l (i)u l(n) Modulation Multiplication with the complex exponential M ωx(t) = e jωt x(t) Multiplication with the eigenvector of the graph Laplacian M k f(n) = Nu k (n)f(n) Graph Signal Processing Data Science Reading Group, ISU March, 07 9/9

15 GSP Frameworks Graph Fourier Transform Laplacian Based Analogy from classical signal processing Classical Fourier basis: Complex exponentials Complex exponentials are Eigenfunctions of -D Laplacian operator, (e jωt ) = t ejωt = (ω) e jωt. Graph Signal Processing Data Science Reading Group, ISU March, 07 0/9

16 GSP Frameworks Graph Fourier Transform Laplacian Based Analogy from classical signal processing Classical Fourier basis: Complex exponentials Complex exponentials are Eigenfunctions of -D Laplacian operator, Graph Fourier Transform (e jωt ) = t ejωt = (ω) e jωt. Graph Fourier basis are Eigenfunctions of the Laplacian matrix (operator) Graph Frequencies: Eigenvalues of the Laplacian matrix L Graph Harmonics: Eigenvectors of the Laplacian matrix L L = UΛU T GFT ˆf = U T f, IGFT f = Uˆf Graph Signal Processing Data Science Reading Group, ISU March, 07 0/9

17 GSP Frameworks Graph Signal in Two Domains Laplacian Based GFT coefficients ˆf(λ l) A graph signal in vertex domain and spectral domain λ l Eigenvalues of graph Laplacian Graph Signal Processing Data Science Reading Group, ISU March, 07 /9

18 GSP Frameworks Laplacian Based Laplacian Eigenvectors as GFT Basis u 0 u u 5 6 u 5 6 u u Graph Signal Processing Data Science Reading Group, ISU March, 07 /9

19 Graph Convolution 5 GSP Frameworks Laplacian Based L = Eigenvalues: 0, 0.899,.6889,, U = Graph Signal Processing Data Science Reading Group, ISU March, 07 /9

20 GSP Frameworks Graph Convolution(cont d... ) Laplacian Based 5 5 g = [,,,, ] T f = [,, 6,, ] T h = f g = IGFT(ˆf.ĝ).9 5 h = [.9,.9,.08,.7, 7.80] T Graph Signal Processing Data Science Reading Group, ISU March, 07 /9

21 Graph Translation GSP Frameworks Laplacian Based 5 f = [,, 6,, ] T Translation to node i : T i (f) = N (f δ i ) = N IGFT(ˆf.U T (:, i)). 5 T f = [., 5.08, 5.08,.7, 0.69] T T f = [5.08,.50,.66,.56,.] T.7 5 T f = [.7,.56,.56,.08, 5.08] T Graph Signal Processing Data Science Reading Group, ISU March, 07 5/9

22 GSP Frameworks DSP G Discrete Signal Processing on Graphs (DSP G ) Framework Shift Operator Graph Fourier Transform LSI Filters Graph harmonics are eigenfunctions of LSI filters Graph Frequencies Graph Harmonics Total Variation Total Variation is used for frequency ordering Frequency Ordering Concepts in the DSP G framework Graph Signal Processing Data Science Reading Group, ISU March, 07 6/9

23 GSP Frameworks DSP G DSP G Framework (Cont d... ) Shift operator Weight matrix W of the graph Shifted graph signal f = Wf Example: shifting discrete-time signal (one unit right) f in H A linear graph filter f out = Hf in 5 x = [9, 7, 5, 0, 6] T x = Wx = = Linear Shift Invariant (LSI) filters H(Wf in) = W(Hf in) Polynomials in W M H = h(w) = h mw m m=0 = h 0 I + h W h M W M Graph Signal Processing Data Science Reading Group, ISU March, 07 7/9

24 GSP Frameworks DSP G DSP G Framework (Cont d... ) Analogy from classical signal processing Classical Fourier basis: Complex exponentials Complex exponentials are Eigenfunctions of Linear Time Invariant (LTI) filters Graph Fourier Transform Graph Fourier basis are Eigenfunctions of Linear Shift Invariant (LSI) graph filters Graph Frequencies: Eigenvalues of the weight matrix W Graph Harmonics: Eigenvectors of the weight matrix W W = VΣV GFT ˆf = V f, IGFT f = V ˆf Graph Signal Processing Data Science Reading Group, ISU March, 07 8/9

25 GSP Frameworks DSP G DSP G Framework (Cont d... ) Total Variation in classical signal processing TV(x) = x[n] x[n ] = x x, where x[n] = x[n ] n Analogy from classical signal processing Total Variation on graphs TV G (f) = f f = f Wf Im Frequency ordering: Based on Total Variation Eigenvalue with largest magnitude: Lowest frquency σ max HF σ LF σ max σ N σ 0 σ Re Graph Signal Processing Data Science Reading Group, ISU March, 07 9/9

26 GSP Frameworks DSP G Problems in Weight Matrix based DSP G Im Constant graph signal σ max HF σ N σ LF σ max σ 0 σ Re ˆf = Graph frequencies: -.6, -.7, -0.6, 0.6,.9 Weight matrix based DSP G Does not provide natural frequency ordering Even a constant signal has high frequency components Graph Signal Processing Data Science Reading Group, ISU March, 07 0/9

27 Graph Fourier Transform based on Directed Laplacian

28 Graph Fourier Transform based on Directed Laplacian Graph Fourier Transform based on Directed Laplacian Redefines Graph Fourier Transform under DSP G Shift operator: Derived from directed Laplacian Linear Shift Invariant filters: Polynomials in the directed Laplacian Graph frequencies: Eigenvalues of the directed Laplacian Graph harmonics: Eigenvectors of the directed Laplacian Natural frequency ordering Better intuition of frequency as compared to the weight matrix based approach Coincides with the Laplacian based approach for undirected graphs Rahul Singh, Abhishek Chakraborty, and BS Manoj. Graph Fourier transform based on directed Laplacian. In: Signal Processing and Communications (SPCOM), 06 International Conference on. IEEE. 06, pp. 5. Graph Signal Processing Data Science Reading Group, ISU March, 07 /9

29 Graph Fourier Transform based on Directed Laplacian Directed Laplacian Matrix Directed Laplacian Matrix Basic matrices of a directed graph Weight matrix: W w ij is the weight of the directed edge from node j to node i In-degree matrix: D in = diag({di in } i=,,...,n ), di in = N j= wij Out-degree matrix: D out = diag({di out } i=,,...,n ), di out = N i= wij Directed Laplacian matrix L = D in W Sum of each row is zero λ = 0 is surely an eigenvalue W = [ 0 ] D in = [ ] L = [ ] Weight matrix In-degree matrix Directed Laplacian matrix A directed graph Graph Signal Processing Data Science Reading Group, ISU March, 07 /9

30 Graph Fourier Transform based on Directed Laplacian Shift Operator (Proposed) 5 A directed cyclic (ring) graph Shift Operator L = Laplacian matrix A signal x = [ ] T ; shifted by one unit to the right x = [ ] T x = Sx = (I L)x = = S = (I L) is the shift operator Shifted graph signal: f = Sf = (I L)f Graph Signal Processing Data Science Reading Group, ISU March, 07 /9

31 Graph Fourier Transform based on Directed Laplacian LSI Filters LSI Filters f in H A linear graph filter f out = Hf in Theorem Linear Shift-Invariant (LSI) filter: S(f out ) = H(Sf in ) A graph filter H is LSI if the following conditions are satisfied. Geometric multiplicity of each distinct eigenvalue of the graph Laplacian is one. The graph filter H is a polynomial in L, i.e., if H can be written as H = h(l) = h 0 I + h L h m L m where, h 0, h,..., h m C are called filter taps. Graph Signal Processing Data Science Reading Group, ISU March, 07 /9

32 Graph Fourier Transform based on Directed Laplacian Graph Frequencies and Harmonics Graph Fourier Transform based on Directed Laplacian Jordan decomposition of the directed Laplacian: L = VJV Graph Fourier basis: Columns of V (Jordan Eigenvectors of L) Graph frequencies: Eigenvalues of L (diagonal entries of Jordan blocks in J) GFT ˆf = V f and IGFT: f = Vˆf Frequency Ordering: based on Total Variation Total Variation: TV G (f) = f Sf = f (I L)f TV G (f) = Lf Theorem TV of an eigenvector v r is proportional to the absolute value of the corresponding eigenvalue TV(v r ) λ r Graph Signal Processing Data Science Reading Group, ISU March, 07 5/9

33 Graph Fourier Transform based on Directed Laplacian Frequency Ordering Im Frequency Ordering Im HF λ λ N λ λ N LF λ 0 λ HF Re LF λ 0 λ HF Re λ λ N λ λ N Arbitrary graph Graph with positive edge weights Im Im HF λ N λ LF λ 0 λ N HF Re LF HF λ 0 λ λ N λ N Re Undirected graph with real edge weights. Undirected graph with real and non-negative edge weights. Graph Signal Processing Data Science Reading Group, ISU March, 07 6/9

34 Graph Fourier Transform based on Directed Laplacian Example Frequency Ordering Graph signal f = [ ] T defined on the directed graph ˆf(λl) Im(λ l ) A weighted directed graph 0 0 Re(λ l ) 6 8 Spectrum of the signal f = [ ] T Graph Signal Processing Data Science Reading Group, ISU March, 07 7/9

35 Graph Fourier Transform based on Directed Laplacian Example Frequency Ordering Graph signal f = [ ] T defined on the directed graph.5 High frequency component Low frequency component Equal TV Im(λ l ) A weighted directed graph 0 0 Re(λ l ) ˆf(λl) Spectrum of the signal f = [ ] T Graph Signal Processing Data Science Reading Group, ISU March, 07 7/9

36 Graph Fourier Transform based on Directed Laplacian Example: Zero Frequency Frequency Ordering Eigenvector corresponding to λ 0 is v 0 = N [,,... ] T TV of v 0 is zero For a constant graph signal f = [k, k,...] T, GFT is ˆf = [(k N), 0,...] T Only zero frequency component 5 A weighted directed graph ˆf(λl) Im(λ l) 0 0 Re(λ l) Spectrum of the constant signal f = [ ] T Graph Signal Processing Data Science Reading Group, ISU March, 07 8/9

37 Graph Fourier Transform based on Directed Laplacian Example: Zero Frequency Frequency Ordering Eigenvector corresponding to λ 0 is v 0 = N [,,... ] T TV of v 0 is zero For a constant graph signal f = [k, k,...] T, GFT is ˆf = [(k N), 0,...] T Only zero frequency component The weight matrix based approach of GFT fails to give this basic intuition 5 A weighted directed graph ˆf(λl) Only zero frequency component Im(λ l) 0 0 Re(λ l) Spectrum of the constant signal f = [ ] T Graph Signal Processing Data Science Reading Group, ISU March, 07 8/9

38 Graph Fourier Transform based on Directed Laplacian Comparison Comparison of the GSP Frameworks GSP based on Laplacian Based on Weight Matrix DSP G Framework Based on Directed Laplacian Shift Operator Not defined The weight matrix W Derived from directed Laplacian (I L) LSI Filters Not applicable Applicable Applicable Applicability Only undirected graphs Directed graphs Directed graphs Frequencies Eigenvalues of the Laplacian (real) Harmonics Eigenvectors of the Laplacian matrix (real) Eigenvalues of the weight matrix Eigenvectors of the weight matrix Eigenvalues of the directed Laplacian Eigenvectors of the directed Laplacian Frequency Ordering Laplacian quadratic form (natural) Total variation (not natural) Total variation (natural) Multiscale Analysis Exists (Spectral Graph Wavelet Transform) Does not exist Possible Graph Signal Processing Data Science Reading Group, ISU March, 07 9/9

39 SGWT Spectral Graph Wavelet Transform Classical wavelets Wavelets at different scales and locations are constructed by scaling and translating a single mother wavelet ψ ψ s,a(x) = s ψ ( x a s ) Scaling in Fourier domain ψ s,a(x) = π ejωx ˆψ(sω)e jωa dω Scaling ψ by /s corresponds to scaling ˆψ with s Term e jωa comes from localization of the wavelet at location a Spectral Graph Wavelets Graph wavelet at scale t and centered at node n N ψ t,n (m) = u l (m)g(tλ l )ul (n) l=0 Frequency ω is replaced with eigenvalues of graph Laplacian λ l Translating to node n corresponds to multiplication by u l (n) g acts as a scaled bandpass filter, replacing ˆψ Graph Signal Processing Data Science Reading Group, ISU March, 07 0/9

40 SGWT Wavelet Generating Kernel Generating Kernel ψ t,n (m) = N l=0 u l (m)g(tλ l )u l (n) g acts as a scaled bandpass filter: wavelet generating kernel x α x α for x < x g(x; α, β, x, x ) = p(x) for x x x x β x β for x > x, x is the distance from origin (zero frequency) α and β are integer parameters of g x and x determine transition regions p(x) is a cubic spline that ensures continuity in g A possible choice of these parameters: α = β =, x =, x =, and p(x) = 5 + x 6x + x Graph Signal Processing Data Science Reading Group, ISU March, 07 /9

41 SGWT Generating Kernel Wavelet Generating Kernel cont d... ψ t,n (m) = N l=0 Scale t is a continuous variable u l (m)g(tλ l )u l (n) For practical purposes, choose J number of logarithmically equally spaced scales t,..., t J t min = λ max /K, λ max is the eigenvalue of L with largest magnitude and K is a design parameter t J = x / λ max and t = x /t min Graph Signal Processing Data Science Reading Group, ISU March, 07 /9

42 SGWT Generating Kernel Wavelet Generating Kernel cont d... λ max = 0, α = β =, x =, x =, K = 0, and J = g(tλ) 0.5 g(tλ) (λ) Kernel at t = (λ) Kernel at t = g(tλ) 0.5 g(tλ) (λ) Kernel at t = (λ) Kernel at t = 0.. As t increases, the kernel becomes increasingly confined to low frequencies Graph Signal Processing Data Science Reading Group, ISU March, 07 /9

43 SGWT Examples SGWT Examples Wavelets at different scales t = t = t =.869 t = 0.57 Large scale Low frequency Spread of the wavelet over the graph is high Graph Signal Processing Data Science Reading Group, ISU March, 07 /9

44 SGWT Examples SGWT Examples Wavelets at different scales t = 5.07 t = t =.869 t = 0.57 Large scale Low frequency Spread of the wavelet over the graph is high Graph Signal Processing Data Science Reading Group, ISU March, 07 5/9

45 SGWT Examples SGWT Examples wavelet ψ(t) wavelet ψ(t) index t = index t =.869 wavelet ψ(t) wavelet ψ(t) index t = index t = 0.57 Graph Signal Processing Data Science Reading Group, ISU March, 07 6/9

46 Matrix Form of SGWT SGWT Matrix Form Wavelet basis at scale t = collection of N number of wavelets (each wavelet centered at a particular node of the graph) Ψ t = [ψ t, ψ t,... ψ t,n ] g(tλ 0 ) g(tλ ) = U... g(tλn ) UT = UG t U T Column of U are eigenvectors of L g(tλ l ) is the sampled value of g(tλ) at frequency λ l Wavelet coefficient at scale t and centered at node n of a graph signal f W f (t, n) = ψ t,n, f = ψ T t,nf Graph Signal Processing Data Science Reading Group, ISU March, 07 7/9

47 Conclusions Conclusions Introduction to Graph Signal Processing (GSP) Graph Signal Processing Frameworks Laplacian Based Discrete Signal Processing on Graphs (DSP G ) Framework Weight Matrix Based Directed Laplacian Based Spectral Graph Wavelet Transform (SGWT) Research oppurtunities are plenty Graph Signal Processing Data Science Reading Group, ISU March, 07 8/9

48 References Conclusions [] Fan RK Chung. Spectral graph theory. Vol. 9. American Mathematical Soc., 997. [] David K Hammond, Pierre Vandergheynst, and Rémi Gribonval. Wavelets on graphs via spectral graph theory. In: Applied and Computational Harmonic Analysis 0. (0), pp [] Sunil K Narang and Antonio Ortega. Perfect reconstruction two-channel wavelet filter banks for graph structured data. In: Signal Processing, IEEE Transactions on 60.6 (0), pp [] A. Sandryhaila and J.M.F. Moura. Big Data Analysis with Signal Processing on Graphs: Representation and processing of massive data sets with irregular structure. In: Signal Processing Magazine, IEEE.5 (0), pp [5] A. Sandryhaila and J.M.F. Moura. Discrete Signal Processing on Graphs. In: Signal Processing, IEEE Transactions on 6.7 (0), pp [6] A. Sandryhaila and J.M.F. Moura. Discrete Signal Processing on Graphs: Frequency Analysis. In: Signal Processing, IEEE Transactions on 6. (0), pp [7] Aliaksei Sandryhaila and José MF Moura. Discrete signal processing on graphs: Graph fourier transform. In: ICASSP. 0, pp [8] David I Shuman, Benjamin Ricaud, and Pierre Vandergheynst. Vertex-frequency analysis on graphs. In: Applied and Computational Harmonic Analysis 0. (06), pp [9] David I Shuman et al. The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains. In: Signal Processing Magazine, IEEE 0. (0), pp [0] Rahul Singh, Abhishek Chakraborty, and BS Manoj. Graph Fourier transform based on directed Laplacian. In: Signal Processing and Communications (SPCOM), 06 International Conference on. IEEE. 06, pp. 5. Graph Signal Processing Data Science Reading Group, ISU March, 07 9/9

49 Thank You.