On Wavelet and Wavelet Packet Transforms on Graphs and … · 2019-07-23 · On Wavelet and Wavelet...

On Wavelet and Wavelet Packet Transforms

on Graphs and Networks

Naoki Saito & Je� Irion

Department of MathematicsUniversity of California, Davis

Organized Session by Activity Group on Wavelet AnalysisJSIAM 2013 Annual Meeting

Fukuoka, JapanSeptember 11, 2013

saito@math.ucdavis.edu (UC Davis) Wavelet Packets on Graphs Sep. 11, 2013 1 / 56

1 Aims & Objectives

2 Basics of Graph Laplacians

3 Hierarchical Graph Laplacian Eigen Transform (HGLET)HGLET Variation 1: Haar-like BasisHGLET Variation 2: Orthonormalized Hierarchical Fiedler Transform(OHFT)

4 Approximation ExperimentsDiscussions

5 Bonus: Simultaneous Signal Segmentation & Compression

6 Summary and Future Work

7 References

Aims & Objectives

1 Aims & Objectives

7 References

Aims & Objectives

Wavelets

Successful on regular domains

Extend to irregular domains ⇒ �2nd Generation Wavelets�

For example,

Hammond, Vandergheynst, and Gribonval (2011): wavelets viaspectral graph theory

Coifman and Maggioni (2006): di�usion wavelets

Bremer et al. (2006): di�usion wavelet packets

Aims & Objectives

Wavelets

Successful on regular domains

Extend to irregular domains ⇒ �2nd Generation Wavelets�

For example,

Hammond, Vandergheynst, and Gribonval (2011): wavelets viaspectral graph theory

Coifman and Maggioni (2006): di�usion wavelets

Bremer et al. (2006): di�usion wavelet packets

Aims & Objectives

Step 1. Develop and implement multiscale transforms for data on graphsand point clouds.

Step 2. Investigate usefulness for:

1 Approximation/Denoising.

Smoothing crime rate data

2 Classi�cation.

Twitter spam accountclassi�cation/detection

Aims & Objectives

2 Classi�cation.

Aims & Objectives

https://www.ncjrs.gov

2 Classi�cation.

Aims & Objectives

https://www.ncjrs.gov

2 Classi�cation.

http://beautifuldata.net

Basics of Graph Laplacians

1 Aims & Objectives

7 References

Basic De�nitions and Notation

Let G be a graph.

If G is a connected graph without cycles/loops, then it is called a tree.

Let V =V (G) = {v1, . . . , vN } be a set of vertices representing some data.

Let |V (G)| = N , and let 0 =λ0(G) ≤λ1(G) ≤ ·· · ≤λN−1(G) be the sortedeigenvalues of L(G).

Let E = E(G) = {e1, . . . ,eN ′} be a set of edges where ek = (vi , v j )represents an edge (or line segment) connecting between adjacentvertices vi , v j for some 1 ≤ i , j ≤ N . Note that if G is a tree, then|E(G)| = |V (G)|−1.

Let d(vk ) = dvk be the degree of the vertex vk .

Graph LaplaciansL(G) :=D(G)−W (G) the Laplacian matrix

W (G) = (wi j ) the weight matrix

D(G) :=diag(dv1 , . . . ,dvn ) the degree matrix, where dvi :=∑Nj=1 wi j .

Note that there are many ways to de�ne wi j .For example, for unweighted graphs, we typically use

wi j :={

1 if vi ∼ v j (i.e., vi and v j are adjacent);

0 otherwise.

This is often referred to as the adjacency matrix and denoted by A(G).

For weighted graphs, wi j should re�ect the similarity (or a�nity) of informationat vi and v j , e.g., if vi ∼ v j , then

wi j := 1/dist(vi , v j ) or exp(−dist(vi , v j )2/ε2),

where dist(·, ·) is a certain measure of dissimilarity and ε> 0 is an appropriate

scale parameter.saito@math.ucdavis.edu (UC Davis) Wavelet Packets on Graphs Sep. 11, 2013 8 / 56

Why Graph Laplacians?

Let f ∈ L2(V ). Then

L(G) f (vi ) = dvi f (vi )− ∑j 6=i

wi j f (v j ),

i.e., this is a generalization of the �nite di�erence approximation to theLaplace operator.

After all, sines (cosines) are the eigenfunctions of the Laplacian on therectangular domain with Dirichlet (Neumann) boundary conditions.

Spherical harmonics, Bessel functions, and Prolate Spheroidal WaveFunctions are part of the eigenfunctions of the Laplacian for the spherical,cylindrical, and spheroidal domains, respectively.

Hence, the eigenfunction expansion of data measured at the vertices usingthe eigenfunctions (in fact, eigenvectors) of a graph Laplacian correspondsto Fourier (or spectral) analysis of the data on that graph.

They also play a useful role in understanding a graph (e.g., the discretenodal domain theorem useful for grouping vertices; see B�y�ko§lu, Leydold, &Stadler, LNM, Springer, 2007)

Why Graph Laplacians? . . .

Furthermore, the eigenvalues of L(G) re�ect various intrinsic geometric andtopological information about the graph including:

connectivity or the number of separated componentsdiameter (the maximum distance over all pairs of vertices)mean distance, . . .Fan Chung: Spectral Graph Theory, AMS, 1997, says: �Thismonograph is an intertwined tale of eigenvalues and their use inunlocking a thousand secrets about graphs.

However, eigenvalues of L(G) cannot uniquely determine the graph G.∼ Kac (1966): �Can one hear the shape of a drum?�⇒ Gordon, Webb, & Wolpert (1992): �One cannot hear the shape of adrum.�

An example of �isospectral� graphs (Tan, 1998; Fujii & Katsuda, 1999):

A Simple Yet Important Example: A Path Graph

1 −1−1 2 −1

−1 2 −1

. . .. . .

−1 2 −1−1 1

︸︷︷︸

0 11 0 1

. . .. . .

1 0 11 0

︸︷︷︸

The eigenvectors of this matrix are exactly the DCT Type II basis vectorsused for the JPEG image compression standard! (See e.g., Strang, SIAMReview, 1999).

λk = 2−2cos(πk/N ) = 4sin2(πk/2N ), k = 0,1, . . . , N −1.

φk (`) =p2/N cos

(πk(`+ 1

2 )/N), k,`= 0,1, . . . , N −1.

In this simple case, λ (eigenvalue) is a monotonic function w.r.t. k(frequency). However, for general graphs, λ does not have a simplerelationship with k.

Hierarchical Graph Laplacian Eigen Transform (HGLET)

1 Aims & Objectives

7 References

Now we turn our focus to a novel transform that can be viewed as ageneralization of the block Discrete Cosine Transform. We refer to thistransform as the Hierarchical Graph Laplacian Eigen Transform (HGLET).

In order to utilize a hierarchical scheme, we will need to partition the graph.Therefore, we will now review some information about graph partitioning.

Now we turn our focus to a novel transform that can be viewed as ageneralization of the block Discrete Cosine Transform. We refer to thistransform as the Hierarchical Graph Laplacian Eigen Transform (HGLET).

In order to utilize a hierarchical scheme, we will need to partition the graph.Therefore, we will now review some information about graph partitioning.

Graph Partitioning via Spectral Clustering

Goal: split the vertices V into two subsets, X and X c .

Plan: minimize the RatioCut function1,

RatioCut(X , X c ) := cut(X , X c )

|X | + cut(X , X c )

|X c | ,

wherecut(X , X c ) := ∑

vi∈Xv j∈X c

Dividing by the number of nodes ensures that the partitions are of roughlythe same size ⇒ we do not simply cleave a small number of nodes

Dividing by the volume of nodes instead of the number of nodes leads to the

popular Normalized Cut (NCut) of Shi and Malik21L. Hagen and A. B. Kahng: �New spectral methods for ratio cut partitioning and

clustering,� IEEE Trans. Comput.-Aided Des., vol. 11, no. 9, pp. 1074-1085, 1992.2J. Shi & J. Malik: �Normalized cuts and image segmentation�, IEEE Trans. Pattern

Anal. Machine Intell., vol. 22, no. 8, pp. 888�905, 2000.saito@math.ucdavis.edu (UC Davis) Wavelet Packets on Graphs Sep. 11, 2013 14 / 56

|X | + cut(X , X c )

|X c | ,

vi∈Xv j∈X c

|X | + cut(X , X c )

|X c | ,

vi∈Xv j∈X c

Let us reformulate the RatioCut minimization problem.

1 De�ne f ∈RN as

√|X c ||X | if vi ∈ X

−√

|X ||X c | if vi ∈ X c

2 The RatioCut problem can be reformulated as

minX⊂V

f TL f s.t. f de�ned as above

√|X c ||X | if vi ∈ X

−√

minX⊂V

√|X c ||X | if vi ∈ X

−√

minX⊂V

f TL f = 1

N∑i , j=1

Wi j ( fi − f j )2

∑vi∈X

v j∈X c

(√|X c ||X | +

√|X ||X c |

∑vi∈X c

v j∈X

√|X c ||X | −

√|X ||X c |

= cut(X , X c )

( |X c ||X | + |X |

|X c | +2

)= cut(X , X c )

( |X |+ |X c ||X | + |X |+ |X c |

|X c |)

= |V |RatioCut(X , X c )

√|X c ||X | if vi ∈ X

−√

minX⊂V

f TL f , f de�ned as above

Unfortunately, this problem is NP hard...

√|X c ||X | if vi ∈ X

−√

minX⊂V

f TL f , f de�ned as above

Unfortunately, this problem is NP hard... Relax!

A couple things to note about f :f ⊥ 1 ⇔ ∑

fi = 0

N∑i=1

fi =∑

vi∈X

√|X c ||X | − ∑

vi∈X c

√|X ||X c |

= |X |√

|X c ||X | − |X c |

√|X ||X c | = 0

‖ f ‖ =pN

‖ f ‖2 =N∑

i=1f 2

= |X | |Xc |

|X | + |X c | |X ||X c |

= |X |+ |X c | = N

fi = 0

N∑i=1

fi =∑

vi∈X

√|X c ||X | − ∑

vi∈X c

√|X ||X c |

= |X |√

|X c ||X | − |X c |

√|X ||X c | = 0

‖ f ‖ =pN

‖ f ‖2 =N∑

i=1f 2

= |X | |Xc |

|X | + |X c | |X ||X c |

= |X |+ |X c | = N

fi = 0

N∑i=1

fi =∑

vi∈X

√|X c ||X | − ∑

vi∈X c

√|X ||X c |

= |X |√

|X c ||X | − |X c |

√|X ||X c | = 0

‖ f ‖ =pN

‖ f ‖2 =N∑

i=1f 2

= |X | |Xc |

|X | + |X c | |X ||X c |

= |X |+ |X c | = N

If we relax our previous de�nition of f and simply require that (i) f ⊥ 1and (ii) ‖ f ‖ =p

N , then we get the relaxed minimization problem1:

minX⊂V

f TL f s.t. f ⊥ 1, ‖ f ‖ =p

By the Rayleigh-Ritz Theorem, the solution is given by φ1 (scaled asnecessary), where φ1 is the eigenvector corresponding to the secondsmallest eigenvalue of L.

φ1 is known as the Fiedler vector and is often used to partition agraph into two subsets.

von Luxburg recommends the use of the random-walk version of theLaplacian matrix, Lrw :=I −D−1W , over the usual Laplacian matrix L,which leads to the NCut and the generalized eigenvalue problem:Lφ=λDφ.

1U. von Luxburg: �A tutorial on spectral clustering,� Statistics and Computing, vol.17, no. 4, pp.395-416, 2007.

minX⊂V

f TL f s.t. f ⊥ 1, ‖ f ‖ =p

minX⊂V

f TL f s.t. f ⊥ 1, ‖ f ‖ =p

minX⊂V

f TL f s.t. f ⊥ 1, ‖ f ‖ =p

The practice of using the Fiedler vector to partition a graph is supported bythe following theory.

De�nition (Weak Nodal Domain)

A positive (or negative) weak nodal domain of f on V (G) is a maximalconnected induced subgraph of G on vertices v ∈V with f (v) ≥ 0 (orf (v) ≤ 0) that contains at least one nonzero vertex. The number of weaknodal domains of f is denoted by W( f ).

Corollary (Fiedler (1975))

If G is connected, then W(φ1) = 2.

Example of Graph Partitioning

Figure: The MN road network

Example of Graph Partitioning

Figure: The MN road network partitioned into two regions via the Fiedler vector

And now, we present our Hierarchical Graph Laplacian Eigen Transform:1 Generate an orthonormal basis for the entire graph ⇒ Laplacian

eigenvectors (Notation is φjk,l with j = 0)

2 Partition the graph using the Fiedler vector φjk,1

3 Generate an orthonormal basis for each of the partitions ⇒ Laplacianeigenvectors

4 Repeat...5 Select an orthonormal basis from this collection of orthonormal bases

0,0 φ00,1 φ0

0,2 · · · φ00,N−1

0,0 φ10,1 φ1

0,2 · · · φ10,N0−1

] [φ1

1,0 φ11,1 φ1

1,2 · · · φ11,N1−1

0,0φ20,1 · · ·φ2

0,N0−1

] [φ2

1,0φ21,1 · · ·φ2

1,N1−1

] [φ2

2,0φ22,1 · · ·φ2

2,N2−1

] [φ2

3,0φ23,1 · · ·φ2

3,N3−1

0,0 φ00,1 φ0

0,2 · · · φ00,N−1

0,0 φ10,1 φ1

0,2 · · · φ10,N0−1

] [φ1

1,0 φ11,1 φ1

1,2 · · · φ11,N1−1

0,0φ20,1 · · ·φ2

0,N0−1

] [φ2

1,0φ21,1 · · ·φ2

1,N1−1

] [φ2

2,0φ22,1 · · ·φ2

2,N2−1

] [φ2

3,0φ23,1 · · ·φ2

3,N3−1

0,0 φ00,1 φ0

0,2 · · · φ00,N−1

0,0 φ10,1 φ1

0,2 · · · φ10,N0−1

] [φ1

1,0 φ11,1 φ1

1,2 · · · φ11,N1−1

0,0φ20,1 · · ·φ2

0,N0−1

] [φ2

1,0φ21,1 · · ·φ2

1,N1−1

] [φ2

2,0φ22,1 · · ·φ2

2,N2−1

] [φ2

3,0φ23,1 · · ·φ2

3,N3−1

0,0 φ00,1 φ0

0,2 · · · φ00,N−1

0,0 φ10,1 φ1

0,2 · · · φ10,N0−1

] [φ1

1,0 φ11,1 φ1

1,2 · · · φ11,N1−1

0,0φ20,1 · · ·φ2

0,N0−1

] [φ2

1,0φ21,1 · · ·φ2

1,N1−1

] [φ2

2,0φ22,1 · · ·φ2

2,N2−1

] [φ2

3,0φ23,1 · · ·φ2

3,N3−1

0,0 φ00,1 φ0

0,2 · · · φ00,N−1

0,0 φ10,1 φ1

0,2 · · · φ10,N0−1

] [φ1

1,0 φ11,1 φ1

1,2 · · · φ11,N1−1

0,0φ20,1 · · ·φ2

0,N0−1

] [φ2

1,0φ21,1 · · ·φ2

1,N1−1

] [φ2

2,0φ22,1 · · ·φ2

2,N2−1

] [φ2

3,0φ23,1 · · ·φ2

3,N3−1

0,0 φ00,1 φ0

0,2 · · · φ00,N−1

0,0 φ10,1 φ1

0,2 · · · φ10,N0−1

] [φ1

1,0 φ11,1 φ1

1,2 · · · φ11,N1−1

0,0φ20,1 · · ·φ2

0,N0−1

] [φ2

1,0φ21,1 · · ·φ2

1,N1−1

] [φ2

2,0φ22,1 · · ·φ2

2,N2−1

] [φ2

3,0φ23,1 · · ·φ2

3,N3−1

0,0 φ00,1 φ0

0,2 · · · φ00,N−1

0,0 φ10,1 φ1

0,2 · · · φ10,N0−1

] [φ1

1,0 φ11,1 φ1

1,2 · · · φ11,N1−1

0,0φ20,1 · · ·φ2

0,N0−1

] [φ2

1,0φ21,1 · · ·φ2

1,N1−1

] [φ2

2,0φ22,1 · · ·φ2

2,N2−1

] [φ2

3,0φ23,1 · · ·φ2

3,N3−1

0,0 φ00,1 φ0

0,2 · · · φ00,N−1

0,0 φ10,1 φ1

0,2 · · · φ10,N0−1

] [φ1

1,0 φ11,1 φ1

1,2 · · · φ11,N1−1

0,0φ20,1 · · ·φ2

0,N0−1

] [φ2

1,0φ21,1 · · ·φ2

1,N1−1

] [φ2

2,0φ22,1 · · ·φ2

2,N2−1

] [φ2

3,0φ23,1 · · ·φ2

3,N3−1

0,0 φ00,1 φ0

0,2 · · · φ00,N−1

0,0 φ10,1 φ1

0,2 · · · φ10,N0−1

] [φ1

1,0 φ11,1 φ1

1,2 · · · φ11,N1−1

0,0φ20,1 · · ·φ2

0,N0−1

] [φ2

1,0φ21,1 · · ·φ2

1,N1−1

] [φ2

2,0φ22,1 · · ·φ2

2,N2−1

] [φ2

3,0φ23,1 · · ·φ2

3,N3−1

0,0 φ00,1 φ0

0,2 · · · φ00,N−1

0,0 φ10,1 φ1

0,2 · · · φ10,N0−1

] [φ1

1,0 φ11,1 φ1

1,2 · · · φ11,N1−1

0,0φ20,1 · · ·φ2

0,N0−1

] [φ2

1,0φ21,1 · · ·φ2

1,N1−1

] [φ2

2,0φ22,1 · · ·φ2

2,N2−1

] [φ2

3,0φ23,1 · · ·φ2

3,N3−1

Observations

For an unweighted path graph, this yields a dictionary of the blockDCT-II

Similar to wavelet packet or local cosine dictionaries in that itgenerates an overcomplete basis from which we can select a basisuseful for the task at hand ⇒ best-basis algorithm, local discriminantbasis algorithm, . . .

A union of bases on disjoint subsets is obviously orthonormal[φ0

0,0 φ00,1 φ0

0,2 · · · φ00,N−1

0,0 φ10,1 φ1

0,2 · · · φ10,N0−1

1,0 φ11,1 φ1

1,2 · · · φ11,N1−1

0,0 · · · φ20,N0−1

1,0 · · · φ21,N1−1

2,0 · · · φ22,N2−1

3,0 · · · φ23,N3−1

]saito@math.ucdavis.edu (UC Davis) Wavelet Packets on Graphs Sep. 11, 2013 23 / 56

Observations

0,0 φ00,1 φ0

0,2 · · · φ00,N−1

0,0 φ10,1 φ1

0,2 · · · φ10,N0−1

1,0 φ11,1 φ1

1,2 · · · φ11,N1−1

0,0 · · · φ20,N0−1

1,0 · · · φ21,N1−1

2,0 · · · φ22,N2−1

3,0 · · · φ23,N3−1

Observations

0,0 φ00,1 φ0

0,2 · · · φ00,N−1

0,0 φ10,1 φ1

0,2 · · · φ10,N0−1

1,0 φ11,1 φ1

1,2 · · · φ11,N1−1

0,0 · · · φ20,N0−1

1,0 · · · φ21,N1−1

2,0 · · · φ22,N2−1

3,0 · · · φ23,N3−1

Observations

0,0 φ00,1 φ0

0,2 · · · φ00,N−1

0,0 φ10,1 φ1

0,2 · · · φ10,N0−1

1,0 φ11,1 φ1

1,2 · · · φ11,N1−1

0,0 · · · φ20,N0−1

1,0 · · · φ21,N1−1

2,0 · · · φ22,N2−1

3,0 · · · φ23,N3−1

Observations

0,0 φ00,1 φ0

0,2 · · · φ00,N−1

0,0 φ10,1 φ1

0,2 · · · φ10,N0−1

1,0 φ11,1 φ1

1,2 · · · φ11,N1−1

0,0 · · · φ20,N0−1

1,0 · · · φ21,N1−1

2,0 · · · φ22,N2−1

3,0 · · · φ23,N3−1

Observations

0,0 φ00,1 φ0

0,2 · · · φ00,N−1

0,0 φ10,1 φ1

0,2 · · · φ10,N0−1

1,0 φ11,1 φ1

1,2 · · · φ11,N1−1

0,0 · · · φ20,N0−1

1,0 · · · φ21,N1−1

2,0 · · · φ22,N2−1

3,0 · · · φ23,N3−1

Observations

0,0 φ00,1 φ0

0,2 · · · φ00,N−1

0,0 φ10,1 φ1

0,2 · · · φ10,N0−1

1,0 φ11,1 φ1

1,2 · · · φ11,N1−1

0,0 · · · φ20,N0−1

1,0 · · · φ21,N1−1

2,0 · · · φ22,N2−1

3,0 · · · φ23,N3−1

Observations

0,0 φ00,1 φ0

0,2 · · · φ00,N−1

0,0 φ10,1 φ1

0,2 · · · φ10,N0−1

1,0 φ11,1 φ1

1,2 · · · φ11,N1−1

0,0 · · · φ20,N0−1

1,0 · · · φ21,N1−1

2,0 · · · φ22,N2−1

3,0 · · · φ23,N3−1

HGLET Basis Vectors on MN

Here we display some of the basis vectors generated by our HGLET scheme onthe MN road network. (Note: j = 0 is the coarsest scale, j = 14 is the �nest.)

Level j = 14, Region k = 0, φ1

Level j = 0, Region k = 0, φ00,1

Computational Complexity: HGLET

Computational Run Time

Complexity for MN1

HGLET (redundant) O(N 3) 83 sec

1Computations performed on a personal laptop (4.00 GB RAM, 2.26 GHz), N = 2640 and

nnz(W)= 6604.saito@math.ucdavis.edu (UC Davis) Wavelet Packets on Graphs Sep. 11, 2013 25 / 56

Hierarchical Graph Laplacian Eigen Transform (HGLET) HGLET Variation 1: Haar-like Basis

1 Aims & Objectives

7 References

Now we present a Haar-like modi�cation of our scheme:

1 Starting with the entire graph (i.e., level j = 0), compute the Fiedler vectorφ1 (φ0 is trivially known, and we denote it by ϕ0,0). Convert φ1 to aHaar-like vector:1

ψ0,0(i ) :={

1 if φ1(i ) ≥ 0

− # nonnegative# negative if φ1(i ) < 0

and then `2-normalize it

2 Partition the graph ⇒ Fiedler vector

3 Compute the Fiedler vector for each partition and convert it to a Haar-likevector on its respective partition1 ⇒ ψ j ,k

4 Repeat...

This yields an orthonormal basis: ϕ0,0 ∪ {ψ j ,k }0≤ j<J , k

1As with the HGLET, we could generate a full orthonormal basis by converting all theLaplacian eigenvectors into piecewise-constant orthonormal vectors according to their sign,similar to the Walsh-Hadamard transform.

ψ0,0(i ) :={

1 if φ1(i ) ≥ 0

4 Repeat...

ψ0,0(i ) :={

1 if φ1(i ) ≥ 0

4 Repeat...

ψ0,0(i ) :={

1 if φ1(i ) ≥ 0

4 Repeat...

ψ0,0(i ) :={

1 if φ1(i ) ≥ 0

4 Repeat...

ψ0,0(i ) :={

1 if φ1(i ) ≥ 0

4 Repeat...

HGLET Haar-like Basis Example

1 2 3 4 5 610 10 10 1 10

[ϕ0,0 ψ0,0 φ0,2 φ0,3 φ0,4 φ0,5

][φ1,0 ψ1,0 φ1,2 φ1,3

] [φ1,1 ψ1,1

][φ2,0 ψ2,0

] [φ2,1 ψ2,1

] [φ1,1 ψ1,1

ϕ0,0 ψ0,0 ψ1,0 ψ1,1 ψ2,0 ψ2,1

1 1 1 0 1 01 1 1 0 −1 01 1 −1 0 0 11 1 −1 0 0 −11 −2 0 1 0 01 −2 0 −1 0 0

1 2 3 4 5 610 10 10 1 10

[ϕ0,0 ψ0,0 φ0,2 φ0,3 φ0,4 φ0,5

][φ1,0 ψ1,0 φ1,2 φ1,3

] [φ1,1 ψ1,1

][φ2,0 ψ2,0

] [φ2,1 ψ2,1

] [φ1,1 ψ1,1

ϕ0,0 ψ0,0 ψ1,0 ψ1,1 ψ2,0 ψ2,1

1 1 1 0 1 01 1 1 0 −1 01 1 −1 0 0 11 1 −1 0 0 −11 −2 0 1 0 01 −2 0 −1 0 0

1 2 3 4 5 610 10 10 1 10

[ϕ0,0 ψ0,0 φ0,2 φ0,3 φ0,4 φ0,5

][φ1,0 ψ1,0 φ1,2 φ1,3

] [φ1,1 ψ1,1

][φ2,0 ψ2,0

] [φ2,1 ψ2,1

] [φ1,1 ψ1,1

ϕ0,0 ψ0,0 ψ1,0 ψ1,1 ψ2,0 ψ2,1

1 1 1 0 1 01 1 1 0 −1 01 1 −1 0 0 11 1 −1 0 0 −11 −2 0 1 0 01 −2 0 −1 0 0

1 2 3 4 5 610 10 10 1 10

[ϕ0,0 ψ0,0 φ0,2 φ0,3 φ0,4 φ0,5

][φ1,0 ψ1,0 φ1,2 φ1,3

] [φ1,1 ψ1,1

][φ2,0 ψ2,0

] [φ2,1 ψ2,1

] [φ1,1 ψ1,1

ϕ0,0 ψ0,0 ψ1,0 ψ1,1 ψ2,0 ψ2,1

1 1 1 0 1 01 1 1 0 −1 01 1 −1 0 0 11 1 −1 0 0 −11 −2 0 1 0 01 −2 0 −1 0 0

1 2 3 4 5 610 10 10 1 10

[ϕ0,0 ψ0,0 φ0,2 φ0,3 φ0,4 φ0,5

][φ1,0 ψ1,0 φ1,2 φ1,3

] [φ1,1 ψ1,1

][φ2,0 ψ2,0

] [φ2,1 ψ2,1

] [φ1,1 ψ1,1

]Thus, we generate a matrix whose columns (after `2-normalization) forman orthonormal basis:

ϕ0,0 ψ0,0 ψ1,0 ψ1,1 ψ2,0 ψ2,1

1 1 1 0 1 01 1 1 0 −1 01 1 −1 0 0 11 1 −1 0 0 −11 −2 0 1 0 01 −2 0 −1 0 0

Computational Complexity: Haar-like HGLET

Complexity for MN1

Haar-like HGLET O(N log N ) 5 sec

Hierarchical Graph Laplacian Eigen Transform (HGLET) HGLET Variation 2: OHFT

1 Aims & Objectives

7 References

We have also developed and implemented a modi�cation that is similar tothe Haar-like HGLET, but yields a smoother set of orthonormal basisfunctions. We call this the Orthonormalized Hierarchical Fiedler Transform(OHFT).

1 Starting with the entire graph (i.e., level j = 0), compute the Fiedlervector φ1 and denote it as ψ0,0 (φ0 is trivially known, and we denoteit by ϕ0,0)

3 Compute the Fiedler vector for each partition and orthonormalize itagainst all ψ j ,k 's computed thus far (it is already orthogonal to ϕ0,0)

⇒ ψ j ,k

4 Repeat...

1As with the HGLET, we could generate a full orthonormal basis. However, thiswould require computing all of the eigenvectors, and so we do not perform this step.But we point this out to show consistency with the HGLET.

⇒ ψ j ,k

4 Repeat...

⇒ ψ j ,k

4 Repeat...

⇒ ψ j ,k

4 Repeat...

⇒ ψ j ,k

4 Repeat...

⇒ ψ j ,k

4 Repeat...

Haar-like HGLET vs. OHFT

1 2 3 4 5 610 10 10 1 10

(a) (b)

1 2 3 4 5 610 10 10 1 10

ϕ0,0 is the same in both cases: a global constant vector.

(a) Haar-like ϕ0,0 (b) OHFT ϕ0,0

1 2 3 4 5 610 10 10 1 10

(a) Haar-like ψ0,0 (b) OHFT ψ0,0

1 2 3 4 5 610 10 10 1 10

(These vectors look the same, but they are not.)

1 2 3 4 5 610 10 10 1 10

Now we compare the basis functions they generate on the MN roadnetwork.

Computational Complexity: OHFT

Complexity for MN1

Haar-like HGLET O(N log N ) 5 sec

OHFT O(N 3) 8 sec

Approximation Experiments

1 Aims & Objectives

7 References

We have performed some preliminary approximation experiments on thefollowing datasets...

(a) Thickness data on dendritic tree#100

(b) The pixels of the Barbara imagemapped to the MN road network

(c) A Gaussian on the MN roadnetwork

(d) A mutilated Gaussian on the MNroad network

(a) Thickness data on dendritic tree#100

(b) The pixels of the Barbara imagemapped to the MN road network

(c) A Gaussian on the MN roadnetwork

(d) A mutilated Gaussian on the MNroad network

Explanation of Barbara on MN Road Network

The Barbara image (512×512) and the MN road network (2640nodes)

1 Stretch the MN road network so that it is on a [1,512]×[1,512] grid

2 Superimpose the stretched MN road network onto Barbara

3 Set each node value to be the nearest pixel value

Barbara on the original MN road network

Approximation Results for Dendrite #100

Approximation Results for MN Barbara

Approximation Results for MN Gaussian

Approximation Results for MN Mutilated Gaussian

Approximation Experiments Discussions

1 Aims & Objectives

7 References

Discussion of Approximation Results

Overall, the Haar-like HGLET variation was the best performer, followed bythe OHFT. This makes a strong case for using localized basis functions onmultiple scales.

Level 5 of the HGLET outperforms Level 3. Both outperform Laplacianeigenvectors (i.e., HGLET Level 0). Again, this demonstrates the merit ofusing localized basis vectors. Future work will investigate the advantages ofusing a basis comprised of HGLET vectors from multiple levels.

The basis vectors for both are derived from the same Fiedler vectors ⇒convert to a Haar-like vector vs. orthonormalize against pre-existingbasis vectorsThe OHFT o�ers a compromise between the localization of theHaar-like HGLET and the smoothness of the HGLET (includingLaplacian eigenvectors)

This explains why the Haar-like HGLET performs better for thedendrite #100 data (piecewise constant), while the OHFT performsbetter for < 50% coe�cients kept on the MN Gaussian data (smooth)

Bonus: Simultaneous Signal Segmentation & Compression

1 Aims & Objectives

7 References

As a bonus, we can apply the HGLET for simultaneously segmentingand compressing a given nonstationary regularly-sampled signal.

Our proposed procedure is:1 Form a graph of a given signal by associating each vertex (i.e., the

signal sample location) with a set of signal amplitude at that vertexand those of its local neighbors (e.g., 3 or 5 points around it);

2 Compute the graph Laplacian matrix and the Fiedler vector;3 Segment the signal based on the polarity of the Fiedler vector;4 In each segment, apply the standard DCT;5 Store the compressed coe�cients and the segment location info.

Of course, one can use more sophisticated feature vectors instead ofthe local samples at each vertex; also can use a few more eigenvectorsfor the segmentation above.

Preliminary Result

Figure: Noisy `Piece-Regular' Signal from WaveLab

Preliminary Result

Figure: Segmentation intervals using the Fiedler vector

Preliminary Result

Figure: Approximation comparison

Preliminary Result

Figure: More concise approximations

Preliminary Result

Figure: Segmentation using φ2

Summary and Future Work

1 Aims & Objectives

7 References

Summary

We developed a set of multiscale transforms on graphs and networks:HGLET; Haar-like HGLET; OHFT.

They are direct generalizations of Hierarchical Block Discrete Cosine

Transforms originally developed for regularly-sampled signals andimages.

They allow us to choose an orthonormal basis most suitable for one'stask at hand, e.g., approximation, classi�cation, regression, . . .

They may also be useful for regularly-sampled signals.

Developing a true generalization of wavelet and wavelet packettransforms is more challenging due to the di�culty of the notion ofthe frequency domain of a given graph.

Summary

Future Work

Implement basis selection algorithms to be used in conjunction withthe HGLET

Approximation/Denoising ⇒ the best-basis algorithm of Coifmanand Wickerhauser (1992)Classi�cation ⇒ the local discriminant basis algorithms of Saito,Coifman, Geshwind, Warner, Marchand (1995, 2002, 2013)

Perform classi�cation experiments and compare the results using eachof the 3 schemes presented herein

Explore other methods for graph partitioning

Allow for splitting of a region into an arbitrary number of subregionsConsider a bottom-up clustering method, rather than a top-downpartitioning method

Future Work

References

http://www.math.ucdavis.edu/�saito/courses/HarmGraph/ contains my course

slides and useful information on �Harmonic Analysis on Graphs and Networks�

http://www.math.ucdavis.edu/�saito/confs/ICIAM11/ contains talk slides of the

minisymposium on Harmonic Analysis on Graphs and Networks, ICIAM 2011,Zürich (Organizers: NS, Mauro Maggioni)

Also visit http://www.math.ucdavis.edu/�saito/publications/ for various related

publications including:

N. Saito: �Data analysis and representation using eigenfunctions of Laplacianon a general domain,� Applied & Computational Harmonic Analysis, vol. 25,no. 1, pp. 68�97, 2008.N. Saito & E. Woei: �Analysis of neuronal dendrite patterns using eigenvaluesof graph Laplacians,� Japan SIAM Letters, vol. 1, pp. 13�16, 2009.N. Saito & E. Woei: �On the phase transition phenomenon of graphLaplacian eigenfunctions on trees,� RIMS Kôkyûroku, vol. 1743, pp. 77�90,2011.

Y. Nakatsukasa, N. Saito, & E. Woei: �Mysteries around graph Laplacian

eigenvalue 4,� Linear Algebra & Its Applications, vol. 438, no. 8, pp.

3231�3246, 2013.

Acknowledgment

This research was partially supported by the grants receivedfrom the O�ce of Naval Research and the National DefenseScience and Engineering Graduate Fellowship.

Thank you very much for your attention!

Any Questions?

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