Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok...
Transcript of Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok...
![Page 1: Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok ...cseweb.ucsd.edu/...theory-day-2014/trevisan-talk.pdf · U.C Berkeley Based on work withTsz Chiu Kwok,Lap Chi Lau,James](https://reader033.fdocuments.in/reader033/viewer/2022060416/5f13b13ebe7aa15e897b1630/html5/thumbnails/1.jpg)
Spectral Algorithms for Graph Partitioning
Luca TrevisanU.C Berkeley
Based on work with Tsz Chiu Kwok, Lap Chi Lau, James Lee,Yin Tat Lee, and Shayan Oveis Gharan
![Page 2: Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok ...cseweb.ucsd.edu/...theory-day-2014/trevisan-talk.pdf · U.C Berkeley Based on work withTsz Chiu Kwok,Lap Chi Lau,James](https://reader033.fdocuments.in/reader033/viewer/2022060416/5f13b13ebe7aa15e897b1630/html5/thumbnails/2.jpg)
spectral graph theory
Use linear algebra to
• understand/characterize graph properties
• design efficient graph algorithms
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linear algebra and graphs
Take an undirected graph G = (V ,E ), consider matrix
A[i , j ] := weight of edge (i , j)
eigenvalues −→ combinatorial propertieseigenvenctors −→ algorithms for combinatorial problems
![Page 4: Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok ...cseweb.ucsd.edu/...theory-day-2014/trevisan-talk.pdf · U.C Berkeley Based on work withTsz Chiu Kwok,Lap Chi Lau,James](https://reader033.fdocuments.in/reader033/viewer/2022060416/5f13b13ebe7aa15e897b1630/html5/thumbnails/4.jpg)
linear algebra and graphs
Take an undirected graph G = (V ,E ), consider matrix
A[i , j ] := weight of edge (i , j)
eigenvalues −→ combinatorial propertieseigenvenctors −→ algorithms for combinatorial problems
![Page 5: Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok ...cseweb.ucsd.edu/...theory-day-2014/trevisan-talk.pdf · U.C Berkeley Based on work withTsz Chiu Kwok,Lap Chi Lau,James](https://reader033.fdocuments.in/reader033/viewer/2022060416/5f13b13ebe7aa15e897b1630/html5/thumbnails/5.jpg)
linear algebra and graphs
Take an undirected graph G = (V ,E ), consider matrix
A[i , j ] := weight of edge (i , j)
eigenvalues −→ combinatorial propertieseigenvenctors −→ algorithms for combinatorial problems
![Page 6: Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok ...cseweb.ucsd.edu/...theory-day-2014/trevisan-talk.pdf · U.C Berkeley Based on work withTsz Chiu Kwok,Lap Chi Lau,James](https://reader033.fdocuments.in/reader033/viewer/2022060416/5f13b13ebe7aa15e897b1630/html5/thumbnails/6.jpg)
fundamental thm of spectral graph theory
G = (V ,E ) undirected, A adjacency matrix, dv degree of v , Ddiagonal matrix of degrees
L := I − D−12AD−
12
L is symmetric, all its eigenvalues are real and
• 0 = λ1 ≤ λ2 ≤ · · · ≤ λn ≤ 2
• λk = 0 iff G has ≥ k connected components
• λn = 2 iff G has a bipartite connected component
![Page 7: Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok ...cseweb.ucsd.edu/...theory-day-2014/trevisan-talk.pdf · U.C Berkeley Based on work withTsz Chiu Kwok,Lap Chi Lau,James](https://reader033.fdocuments.in/reader033/viewer/2022060416/5f13b13ebe7aa15e897b1630/html5/thumbnails/7.jpg)
fundamental thm of spectral graph theory
G = (V ,E ) undirected, A adjacency matrix, dv degree of v , Ddiagonal matrix of degrees
L := I − D−12AD−
12
L is symmetric, all its eigenvalues are real and
• 0 = λ1 ≤ λ2 ≤ · · · ≤ λn ≤ 2
• λk = 0 iff G has ≥ k connected components
• λn = 2 iff G has a bipartite connected component
![Page 8: Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok ...cseweb.ucsd.edu/...theory-day-2014/trevisan-talk.pdf · U.C Berkeley Based on work withTsz Chiu Kwok,Lap Chi Lau,James](https://reader033.fdocuments.in/reader033/viewer/2022060416/5f13b13ebe7aa15e897b1630/html5/thumbnails/8.jpg)
fundamental thm of spectral graph theory
G = (V ,E ) undirected, A adjacency matrix, dv degree of v , Ddiagonal matrix of degrees
L := I − D−12AD−
12
L is symmetric, all its eigenvalues are real and
• 0 = λ1 ≤ λ2 ≤ · · · ≤ λn ≤ 2
• λk = 0 iff G has ≥ k connected components
• λn = 2 iff G has a bipartite connected component
![Page 9: Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok ...cseweb.ucsd.edu/...theory-day-2014/trevisan-talk.pdf · U.C Berkeley Based on work withTsz Chiu Kwok,Lap Chi Lau,James](https://reader033.fdocuments.in/reader033/viewer/2022060416/5f13b13ebe7aa15e897b1630/html5/thumbnails/9.jpg)
fundamental thm of spectral graph theory
G = (V ,E ) undirected, A adjacency matrix, dv degree of v , Ddiagonal matrix of degrees
L := I − D−12AD−
12
L is symmetric, all its eigenvalues are real and
• 0 = λ1 ≤ λ2 ≤ · · · ≤ λn ≤ 2
• λk = 0 iff G has ≥ k connected components
• λn = 2 iff G has a bipartite connected component
![Page 10: Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok ...cseweb.ucsd.edu/...theory-day-2014/trevisan-talk.pdf · U.C Berkeley Based on work withTsz Chiu Kwok,Lap Chi Lau,James](https://reader033.fdocuments.in/reader033/viewer/2022060416/5f13b13ebe7aa15e897b1630/html5/thumbnails/10.jpg)
0,2
3,
2
3,
2
3,
2
3,
2
3,
5
3,
5
3,
5
3,
5
3
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0,2
3,
2
3,
2
3,
2
3,
2
3,
5
3,
5
3,
5
3,
5
3
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0, 0, .69, .69, 1.5, 1.5, 1.8, 1.8
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0, 0, .69, .69, 1.5, 1.5, 1.8, 1.8
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0,2
3,
2
3,
2
3,
4
3,
4
3,
4
3, 2
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0,2
3,
2
3,
2
3,
4
3,
4
3,
4
3, 2
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eigenvalues as solutions to optimizationproblems
If M ∈ Rn×n is symmetric, and
λ1 ≤ λ2 ≤ · · · ≤ λnare eigenvalues counted with multiplicities, then
λk = minA k−dim subspace of RV
maxx∈A
xTMx
xT x
and eigenvectors of λ1, . . . , λk are basis of opt A
![Page 17: Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok ...cseweb.ucsd.edu/...theory-day-2014/trevisan-talk.pdf · U.C Berkeley Based on work withTsz Chiu Kwok,Lap Chi Lau,James](https://reader033.fdocuments.in/reader033/viewer/2022060416/5f13b13ebe7aa15e897b1630/html5/thumbnails/17.jpg)
why eigenvalues relate to connectedcomponents
If G = (V ,E ) is undirected and L is Laplacian, then
xTLx
xT x=
∑(u,v)∈E |xu − xv |2∑
v dv · x2v
If λ1 ≤ λ2 ≤ · · ·λn are eigenvalues of L with multiplicities,
λk = minA k−dim subspace of RV
maxx∈A
R(x)
where
R(x) :=
∑(u,v)∈E |xu − xv |2∑
v dv · x2v
Note:∑
(u,v)∈E |xu − xv |2 = 0 iff x constant on connectedcomponents
![Page 18: Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok ...cseweb.ucsd.edu/...theory-day-2014/trevisan-talk.pdf · U.C Berkeley Based on work withTsz Chiu Kwok,Lap Chi Lau,James](https://reader033.fdocuments.in/reader033/viewer/2022060416/5f13b13ebe7aa15e897b1630/html5/thumbnails/18.jpg)
why eigenvalues relate to connectedcomponents
If G = (V ,E ) is undirected and L is Laplacian, then
xTLx
xT x=
∑(u,v)∈E |xu − xv |2∑
v dv · x2v
If λ1 ≤ λ2 ≤ · · ·λn are eigenvalues of L with multiplicities,
λk = minA k−dim subspace of RV
maxx∈A
R(x)
where
R(x) :=
∑(u,v)∈E |xu − xv |2∑
v dv · x2v
Note:∑
(u,v)∈E |xu − xv |2 = 0 iff x constant on connectedcomponents
![Page 19: Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok ...cseweb.ucsd.edu/...theory-day-2014/trevisan-talk.pdf · U.C Berkeley Based on work withTsz Chiu Kwok,Lap Chi Lau,James](https://reader033.fdocuments.in/reader033/viewer/2022060416/5f13b13ebe7aa15e897b1630/html5/thumbnails/19.jpg)
why eigenvalues relate to connectedcomponents
If G = (V ,E ) is undirected and L is Laplacian, then
xTLx
xT x=
∑(u,v)∈E |xu − xv |2∑
v dv · x2v
If λ1 ≤ λ2 ≤ · · ·λn are eigenvalues of L with multiplicities,
λk = minA k−dim subspace of RV
maxx∈A
R(x)
where
R(x) :=
∑(u,v)∈E |xu − xv |2∑
v dv · x2v
Note:∑
(u,v)∈E |xu − xv |2 = 0 iff x constant on connectedcomponents
![Page 20: Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok ...cseweb.ucsd.edu/...theory-day-2014/trevisan-talk.pdf · U.C Berkeley Based on work withTsz Chiu Kwok,Lap Chi Lau,James](https://reader033.fdocuments.in/reader033/viewer/2022060416/5f13b13ebe7aa15e897b1630/html5/thumbnails/20.jpg)
if G has ≥ k connected components
λk = minA k−dim subspace of RV
maxx∈A
∑(u,v)∈E |xu − xv |2∑
v dv · x2v
Consider the space of all vectors that are constant in eachconnected component; it has dimension at least k and so itwitnesses λk = 0.
![Page 21: Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok ...cseweb.ucsd.edu/...theory-day-2014/trevisan-talk.pdf · U.C Berkeley Based on work withTsz Chiu Kwok,Lap Chi Lau,James](https://reader033.fdocuments.in/reader033/viewer/2022060416/5f13b13ebe7aa15e897b1630/html5/thumbnails/21.jpg)
if G has ≥ k connected components
λk = minA k−dim subspace of RV
maxx∈A
∑(u,v)∈E |xu − xv |2∑
v dv · x2v
Consider the space of all vectors that are constant in eachconnected component; it has dimension at least k and so itwitnesses λk = 0.
![Page 22: Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok ...cseweb.ucsd.edu/...theory-day-2014/trevisan-talk.pdf · U.C Berkeley Based on work withTsz Chiu Kwok,Lap Chi Lau,James](https://reader033.fdocuments.in/reader033/viewer/2022060416/5f13b13ebe7aa15e897b1630/html5/thumbnails/22.jpg)
if λk = 0
λk = minA k−dim subspace of RV
maxx∈A
∑(u,v)∈E |xu − xv |2∑
v dv · x2v
If λk = 0 there is a k-dimensional space A of vectors, eachconstant on connected components
The graph must have ≥ k connected components
![Page 23: Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok ...cseweb.ucsd.edu/...theory-day-2014/trevisan-talk.pdf · U.C Berkeley Based on work withTsz Chiu Kwok,Lap Chi Lau,James](https://reader033.fdocuments.in/reader033/viewer/2022060416/5f13b13ebe7aa15e897b1630/html5/thumbnails/23.jpg)
if λk = 0
λk = minA k−dim subspace of RV
maxx∈A
∑(u,v)∈E |xu − xv |2∑
v dv · x2v
If λk = 0 there is a k-dimensional space A of vectors, eachconstant on connected components
The graph must have ≥ k connected components
![Page 24: Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok ...cseweb.ucsd.edu/...theory-day-2014/trevisan-talk.pdf · U.C Berkeley Based on work withTsz Chiu Kwok,Lap Chi Lau,James](https://reader033.fdocuments.in/reader033/viewer/2022060416/5f13b13ebe7aa15e897b1630/html5/thumbnails/24.jpg)
if λk = 0
λk = minA k−dim subspace of RV
maxx∈A
∑(u,v)∈E |xu − xv |2∑
v dv · x2v
If λk = 0 there is a k-dimensional space A of vectors, eachconstant on connected components
The graph must have ≥ k connected components
![Page 25: Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok ...cseweb.ucsd.edu/...theory-day-2014/trevisan-talk.pdf · U.C Berkeley Based on work withTsz Chiu Kwok,Lap Chi Lau,James](https://reader033.fdocuments.in/reader033/viewer/2022060416/5f13b13ebe7aa15e897b1630/html5/thumbnails/25.jpg)
conductance
vol(S) :=∑
v∈S dv
φ(S) :=E (S , S)
vol(S)
φ(G ) := minS⊆V :0<vol(S)≤ 1
2vol(V )
φ(S)
In regular graphs, same as edge expansion and, up to factor of 2non-uniform sparsest cut
![Page 26: Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok ...cseweb.ucsd.edu/...theory-day-2014/trevisan-talk.pdf · U.C Berkeley Based on work withTsz Chiu Kwok,Lap Chi Lau,James](https://reader033.fdocuments.in/reader033/viewer/2022060416/5f13b13ebe7aa15e897b1630/html5/thumbnails/26.jpg)
conductance
0, 0, .69, .69, 1.5, 1.5, 1.8, 1.8
φ(G ) = φ(0, 1, 2) =0
6= 0
![Page 27: Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok ...cseweb.ucsd.edu/...theory-day-2014/trevisan-talk.pdf · U.C Berkeley Based on work withTsz Chiu Kwok,Lap Chi Lau,James](https://reader033.fdocuments.in/reader033/viewer/2022060416/5f13b13ebe7aa15e897b1630/html5/thumbnails/27.jpg)
conductance
0, .055, .055, .211, . . .
φ(G ) =2
18
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conductance
0,2
3,
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3,
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3,
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3,
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3,
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3
φ(G ) =5
15
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finding S of small conductance
• G is a social network
• S is a set of users more likely to be friends with each otherthan anyone else
• G is a similarity graph on items of a data sets• S are items more similar to each other than to other items• [Shi, Malik]: image segmentation
• G is an input of a NP-hard optimization problem• Recurse on S , V − S , use dynamic programming to combine in
time 2O(|E(S,S)|)
![Page 30: Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok ...cseweb.ucsd.edu/...theory-day-2014/trevisan-talk.pdf · U.C Berkeley Based on work withTsz Chiu Kwok,Lap Chi Lau,James](https://reader033.fdocuments.in/reader033/viewer/2022060416/5f13b13ebe7aa15e897b1630/html5/thumbnails/30.jpg)
finding S of small conductance
• G is a social network• S is a set of users more likely to be friends with each other
than anyone else
• G is a similarity graph on items of a data sets• S are items more similar to each other than to other items• [Shi, Malik]: image segmentation
• G is an input of a NP-hard optimization problem• Recurse on S , V − S , use dynamic programming to combine in
time 2O(|E(S,S)|)
![Page 31: Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok ...cseweb.ucsd.edu/...theory-day-2014/trevisan-talk.pdf · U.C Berkeley Based on work withTsz Chiu Kwok,Lap Chi Lau,James](https://reader033.fdocuments.in/reader033/viewer/2022060416/5f13b13ebe7aa15e897b1630/html5/thumbnails/31.jpg)
finding S of small conductance
• G is a social network• S is a set of users more likely to be friends with each other
than anyone else
• G is a similarity graph on items of a data sets
• S are items more similar to each other than to other items• [Shi, Malik]: image segmentation
• G is an input of a NP-hard optimization problem• Recurse on S , V − S , use dynamic programming to combine in
time 2O(|E(S,S)|)
![Page 32: Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok ...cseweb.ucsd.edu/...theory-day-2014/trevisan-talk.pdf · U.C Berkeley Based on work withTsz Chiu Kwok,Lap Chi Lau,James](https://reader033.fdocuments.in/reader033/viewer/2022060416/5f13b13ebe7aa15e897b1630/html5/thumbnails/32.jpg)
finding S of small conductance
• G is a social network• S is a set of users more likely to be friends with each other
than anyone else
• G is a similarity graph on items of a data sets• S are items more similar to each other than to other items• [Shi, Malik]: image segmentation
• G is an input of a NP-hard optimization problem• Recurse on S , V − S , use dynamic programming to combine in
time 2O(|E(S,S)|)
![Page 33: Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok ...cseweb.ucsd.edu/...theory-day-2014/trevisan-talk.pdf · U.C Berkeley Based on work withTsz Chiu Kwok,Lap Chi Lau,James](https://reader033.fdocuments.in/reader033/viewer/2022060416/5f13b13ebe7aa15e897b1630/html5/thumbnails/33.jpg)
finding S of small conductance
• G is a social network• S is a set of users more likely to be friends with each other
than anyone else
• G is a similarity graph on items of a data sets• S are items more similar to each other than to other items• [Shi, Malik]: image segmentation
• G is an input of a NP-hard optimization problem• Recurse on S , V − S , use dynamic programming to combine in
time 2O(|E(S,S)|)
![Page 34: Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok ...cseweb.ucsd.edu/...theory-day-2014/trevisan-talk.pdf · U.C Berkeley Based on work withTsz Chiu Kwok,Lap Chi Lau,James](https://reader033.fdocuments.in/reader033/viewer/2022060416/5f13b13ebe7aa15e897b1630/html5/thumbnails/34.jpg)
conductance versus λ2
Recall:λ2 = 0⇔ G disconnected
⇔ φ(G ) = 0
Cheeger inequality (Alon, Milman, Dodzuik, mid-1980s):
λ2
2≤ φ(G ) ≤
√2λ2
![Page 35: Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok ...cseweb.ucsd.edu/...theory-day-2014/trevisan-talk.pdf · U.C Berkeley Based on work withTsz Chiu Kwok,Lap Chi Lau,James](https://reader033.fdocuments.in/reader033/viewer/2022060416/5f13b13ebe7aa15e897b1630/html5/thumbnails/35.jpg)
conductance versus λ2
Recall:λ2 = 0⇔ G disconnected⇔ φ(G ) = 0
Cheeger inequality (Alon, Milman, Dodzuik, mid-1980s):
λ2
2≤ φ(G ) ≤
√2λ2
![Page 36: Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok ...cseweb.ucsd.edu/...theory-day-2014/trevisan-talk.pdf · U.C Berkeley Based on work withTsz Chiu Kwok,Lap Chi Lau,James](https://reader033.fdocuments.in/reader033/viewer/2022060416/5f13b13ebe7aa15e897b1630/html5/thumbnails/36.jpg)
conductance versus λ2
Recall:λ2 = 0⇔ G disconnected⇔ φ(G ) = 0
Cheeger inequality (Alon, Milman, Dodzuik, mid-1980s):
λ2
2≤ φ(G ) ≤
√2λ2
![Page 37: Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok ...cseweb.ucsd.edu/...theory-day-2014/trevisan-talk.pdf · U.C Berkeley Based on work withTsz Chiu Kwok,Lap Chi Lau,James](https://reader033.fdocuments.in/reader033/viewer/2022060416/5f13b13ebe7aa15e897b1630/html5/thumbnails/37.jpg)
Cheeger inequality
λ2
2≤ φ(G ) ≤
√2λ2
Constructive proof of φ(G ) ≤√
2λ2:
• given eigenvector x of λ2.
• sort vertices v according to xv
• a set of vertices S occurring as initial segment or finalsegment of sorted order has
φ(S) ≤√
2λ2 ≤ 2√φ(G )
Fiedler’s sweep algorithm can be implemented in O(|V |+ |E |) time
![Page 38: Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok ...cseweb.ucsd.edu/...theory-day-2014/trevisan-talk.pdf · U.C Berkeley Based on work withTsz Chiu Kwok,Lap Chi Lau,James](https://reader033.fdocuments.in/reader033/viewer/2022060416/5f13b13ebe7aa15e897b1630/html5/thumbnails/38.jpg)
Cheeger inequality
λ2
2≤ φ(G ) ≤
√2λ2
Constructive proof of φ(G ) ≤√
2λ2:
• given eigenvector x of λ2.
• sort vertices v according to xv
• a set of vertices S occurring as initial segment or finalsegment of sorted order has
φ(S) ≤√
2λ2
≤ 2√φ(G )
Fiedler’s sweep algorithm can be implemented in O(|V |+ |E |) time
![Page 39: Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok ...cseweb.ucsd.edu/...theory-day-2014/trevisan-talk.pdf · U.C Berkeley Based on work withTsz Chiu Kwok,Lap Chi Lau,James](https://reader033.fdocuments.in/reader033/viewer/2022060416/5f13b13ebe7aa15e897b1630/html5/thumbnails/39.jpg)
Cheeger inequality
λ2
2≤ φ(G ) ≤
√2λ2
Constructive proof of φ(G ) ≤√
2λ2:
• given eigenvector x of λ2.
• sort vertices v according to xv
• a set of vertices S occurring as initial segment or finalsegment of sorted order has
φ(S) ≤√
2λ2 ≤ 2√φ(G )
Fiedler’s sweep algorithm can be implemented in O(|V |+ |E |) time
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applications
λ2
2≤ φ(G ) ≤
√2λ2
• rigorous analysis of sweep algorithm(practical performance usually better than worst-case analysis)
• to certify φ ≥ Ω(1), enough to certify λ2 ≥ Ω(1)
• mixing time of random walk is O(
1λ2
log |V |)
and so also
O
(1
φ2(G )log |V |
)
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applications
λ2
2≤ φ(G ) ≤
√2λ2
• rigorous analysis of sweep algorithm(practical performance usually better than worst-case analysis)
• to certify φ ≥ Ω(1), enough to certify λ2 ≥ Ω(1)
• mixing time of random walk is O(
1λ2
log |V |)
and so also
O
(1
φ2(G )log |V |
)
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applications
λ2
2≤ φ(G ) ≤
√2λ2
• rigorous analysis of sweep algorithm(practical performance usually better than worst-case analysis)
• to certify φ ≥ Ω(1), enough to certify λ2 ≥ Ω(1)
• mixing time of random walk is O(
1λ2
log |V |)
and so also
O
(1
φ2(G )log |V |
)
![Page 43: Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok ...cseweb.ucsd.edu/...theory-day-2014/trevisan-talk.pdf · U.C Berkeley Based on work withTsz Chiu Kwok,Lap Chi Lau,James](https://reader033.fdocuments.in/reader033/viewer/2022060416/5f13b13ebe7aa15e897b1630/html5/thumbnails/43.jpg)
applications
λ2
2≤ φ(G ) ≤
√2λ2
• rigorous analysis of sweep algorithm(practical performance usually better than worst-case analysis)
• to certify φ ≥ Ω(1), enough to certify λ2 ≥ Ω(1)
• mixing time of random walk is O(
1λ2
log |V |)
and so also
O
(1
φ2(G )log |V |
)
![Page 44: Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok ...cseweb.ucsd.edu/...theory-day-2014/trevisan-talk.pdf · U.C Berkeley Based on work withTsz Chiu Kwok,Lap Chi Lau,James](https://reader033.fdocuments.in/reader033/viewer/2022060416/5f13b13ebe7aa15e897b1630/html5/thumbnails/44.jpg)
goal
A similar theory for other eigenvectors, with algorithmicapplications
One intended application: Like Cheeger’s inequality is worst caseanalysis of Fiedler’s algorithm, develop worst-case analysis ofspectral clustering
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spectral clustering
The following algorithm works well in practice. (But to solve whatproblem?)
Compute k eigenvectors x(1), . . . , x(k) ofk smallest eigenvalues of L
Define mapping F : V → Rk
F (v) := (x(1)v , x
(2)v , . . . , x
(k)v )
Apply k-means to the points that vertices are mapped to
![Page 46: Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok ...cseweb.ucsd.edu/...theory-day-2014/trevisan-talk.pdf · U.C Berkeley Based on work withTsz Chiu Kwok,Lap Chi Lau,James](https://reader033.fdocuments.in/reader033/viewer/2022060416/5f13b13ebe7aa15e897b1630/html5/thumbnails/46.jpg)
spectral embedding of a random graphk = 3
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When λk is small
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Lee, Oveis-Gharan, T, 2012
From fundamental theorem of spectral graph theory:
• λk = 0⇔ G has ≥ k connected components
We prove:
• λk small ⇔ G has ≥ k disjoint sets of small conductance
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Lee, Oveis-Gharan, T, 2012
From fundamental theorem of spectral graph theory:
• λk = 0⇔ G has ≥ k connected components
We prove:
• λk small ⇔ G has ≥ k disjoint sets of small conductance
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order-k conductance
Define order-k conductance
φk(G ) := minS1,...,Sk disjoint
maxi=1,...,k
φ(Si )
Note:
• φ(G ) = φ2(G )
• φk(G ) = 0⇔ G has ≥ k connected components
![Page 51: Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok ...cseweb.ucsd.edu/...theory-day-2014/trevisan-talk.pdf · U.C Berkeley Based on work withTsz Chiu Kwok,Lap Chi Lau,James](https://reader033.fdocuments.in/reader033/viewer/2022060416/5f13b13ebe7aa15e897b1630/html5/thumbnails/51.jpg)
0, 0, .69, .69, 1.5, 1.5, 1.8, 1.8
φ3(G ) = max
0,
2
6,
2
4
=
1
2
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0, 0, .69, .69, 1.5, 1.5, 1.8, 1.8
φ3(G ) = max
0,
2
6,
2
4
=
1
2
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0, .055, .055, .211, . . .
φ3(G ) = max
2
12,
2
12,
2
14
=
1
6
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λk
λk = 0⇔ φk = 0
λk2≤ φk(G ) ≤ O(k2) ·
√λk
(Lee,Oveis Gharan, T, 2012)
Upper bound is constructive: in nearly-linear time we can find kdisjoint sets each of expansion at most O(k2)
√λk .
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λk
λk = 0⇔ φk = 0
λk2≤ φk(G ) ≤ O(k2) ·
√λk
(Lee,Oveis Gharan, T, 2012)
Upper bound is constructive: in nearly-linear time we can find kdisjoint sets each of expansion at most O(k2)
√λk .
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λk
λk2≤ φk(G ) ≤ O(k2) ·
√λk
φk(G ) ≤ O(√
(log k) · λ1.1k)
(Lee,Oveis Gharan, T, 2012)
φk(G ) ≤ O(√
(log k) · λO(k))
(Louis, Raghavendra, Tetali, Vempala, 2012)
Factor O(√
log k) is tight
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λk
λk2≤ φk(G ) ≤ O(k2) ·
√λk
φk(G ) ≤ O(√
(log k) · λ1.1k)
(Lee,Oveis Gharan, T, 2012)
φk(G ) ≤ O(√
(log k) · λO(k))
(Louis, Raghavendra, Tetali, Vempala, 2012)
Factor O(√
log k) is tight
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λk
φk(G ) ≤ O(√
(log k) · λO(k))
(Louis, Raghavendra, Tetali, Vempala, 2012)(Lee,Oveis Gharan, T, 2012)
Miclo 2013:
• Analog for infinite-dimensional Markov process.
• Application: every hyper-bounded operator has a spectral gap.
• Solves 40+ year old open problem
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spectral clustering
Recall spectral clustering algorithm
1 Compute k eigenvectors x(1), . . . , x(k) ofk smallest eigenvalues of L
2 Define mapping F : V → Rk
F (v) := (x(1)v , x
(2)v , . . . , x
(k)v )
3 Apply k-means to the points that vertices are mapped to
LOT algorithms and LRTV algorithm find k low-conductance setsif they exist
First step is spectral embedding
Then geometric partitioning but not k-means
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spectral clustering
Recall spectral clustering algorithm
1 Compute k eigenvectors x(1), . . . , x(k) ofk smallest eigenvalues of L
2 Define mapping F : V → Rk
F (v) := (x(1)v , x
(2)v , . . . , x
(k)v )
3 Apply k-means to the points that vertices are mapped to
LOT algorithms and LRTV algorithm find k low-conductance setsif they exist
First step is spectral embedding
Then geometric partitioning but not k-means
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When λk is large
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Cheeger inequality
λ2
2≤ φ(G ) ≤
√2λ2
φ(G ) ≤ O(k) · λ2√λk
(Kwok, Lau, Lee, Oveis Gharan, T, 2013)
and the upper bound holds for the sweep algorithm
Nearly-linear time sweep algorithm finds S such that, for every k ,
φ(S) ≤ φ(G ) · O(
k√λk
)
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Cheeger inequality
λ2
2≤ φ(G ) ≤
√2λ2
φ(G ) ≤ O(k) · λ2√λk
(Kwok, Lau, Lee, Oveis Gharan, T, 2013)
and the upper bound holds for the sweep algorithm
Nearly-linear time sweep algorithm finds S such that, for every k ,
φ(S) ≤ φ(G ) · O(
k√λk
)
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Cheeger inequality
λ2
2≤ φ(G ) ≤
√2λ2
φ(G ) ≤ O(k) · λ2√λk
(Kwok, Lau, Lee, Oveis Gharan, T, 2013)
and the upper bound holds for the sweep algorithm
Nearly-linear time sweep algorithm finds S such that, for every k ,
φ(S) ≤ φ(G ) · O(
k√λk
)
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Cheeger inequality
λ2
2≤ φ(G ) ≤
√2λ2
φ(G ) ≤ O(k) · λ2√λk
(Kwok, Lau, Lee, Oveis Gharan, T, 2013)
Liu 2014: analog for Riemann manifolds. Applications to boundingeigenvalue gaps
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Cheeger inequality
λ2
2≤ φ(G ) ≤
√2λ2
φ(G ) ≤ O(k) · λ2√λk
(Kwok, Lau, Lee, Oveis Gharan, T, 2013)
Liu 2014: analog for Riemann manifolds. Applications to boundingeigenvalue gaps
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proof structure
1 We find a 2k-valued vector y such that
||x − y ||2 ≤ O
(λ2
λk
)where x is eigenvector of λ2
2 For every k-valued vector y , applying the sweep algorithm tox finds a set S such that
φ(S) ≤ O(k) ·(λ2 +
√λ2 · ||x − y ||
)
So the sweep algorithm finds S
φ(S) ≤ O(k) · λ2 ·1√λk
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proof structure
1 We find a 2k-valued vector y such that
||x − y ||2 ≤ O
(λ2
λk
)where x is eigenvector of λ2
2 For every k-valued vector y , applying the sweep algorithm tox finds a set S such that
φ(S) ≤ O(k) ·(λ2 +
√λ2 · ||x − y ||
)
So the sweep algorithm finds S
φ(S) ≤ O(k) · λ2 ·1√λk
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proof structure
1 We find a 2k-valued vector y such that
||x − y ||2 ≤ O
(λ2
λk
)where x is eigenvector of λ2
2 For every k-valued vector y , applying the sweep algorithm tox finds a set S such that
φ(S) ≤ O(k) ·(λ2 +
√λ2 · ||x − y ||
)
So the sweep algorithm finds S
φ(S) ≤ O(k) · λ2 ·1√λk
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bounded threshold rank
[Arora, Barak, Steurer], [Barak, Raghavendra, Steurer],[Guruswami, Sinop], ...
If λk is large,then a cut of approximately minimal conductance
can be found in time exponential in kusing spectral methods [ABS]or convex relaxations [BRS, GS]
[Kwok, Lau, Lee, Oveis-Gharan, T]: the nearly-linear time sweepalgorithm finds good approximation if λk is large
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bounded threshold rank
[Arora, Barak, Steurer], [Barak, Raghavendra, Steurer],[Guruswami, Sinop], ...
If λk is large,then a cut of approximately minimal conductance
can be found in time exponential in kusing spectral methods [ABS]or convex relaxations [BRS, GS]
[Kwok, Lau, Lee, Oveis-Gharan, T]: the nearly-linear time sweepalgorithm finds good approximation if λk is large
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bounded threshold rank
If λk is large,
[Kwok, Lau, Lee, Oveis-Gharan, T]: the nearly-linear time sweepalgorithm finds good approximation
[Oveis-Gharan, T]: the Goemans-Linial relaxation (solvable inpolynomial time independent of k) can be rounded better than inthe Arora-Rao-Vazirani analysis
[Oveis-Gharan, T]: the graph satisfies a weak regularity lemma inthe sense of Frieze and Kannan
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bounded threshold rank
If λk is large,
[Kwok, Lau, Lee, Oveis-Gharan, T]: the nearly-linear time sweepalgorithm finds good approximation
[Oveis-Gharan, T]: the Goemans-Linial relaxation (solvable inpolynomial time independent of k) can be rounded better than inthe Arora-Rao-Vazirani analysis
[Oveis-Gharan, T]: the graph satisfies a weak regularity lemma inthe sense of Frieze and Kannan
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bounded threshold rank
If λk is large,
[Kwok, Lau, Lee, Oveis-Gharan, T]: the nearly-linear time sweepalgorithm finds good approximation
[Oveis-Gharan, T]: the Goemans-Linial relaxation (solvable inpolynomial time independent of k) can be rounded better than inthe Arora-Rao-Vazirani analysis
[Oveis-Gharan, T]: the graph satisfies a weak regularity lemma inthe sense of Frieze and Kannan
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bounded threshold rank
If λk is large, the graph is an easy instance for several algorithms.Two types of results:
• There is a near-optimal combinatorial solution with a simplestructure
Find it by brute force
• The optimum of a relaxation has a special structure
Improved rounding algorithm that uses the special structure
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When λk+1 is large and λk is small
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eigenvalue gap
• if λk = 0 and λk+1 > 0 then G has exactly k connectedcomponents;
• [Tanaka 2012] If λk+1 > 5k√λk , then vertices of G can be
partitioned into k sets of small conductance, each inducing asubgraph of large conductance. [Non-algorithmic proof]
• [Oveis-Gharan, T 2013] Sufficient φk+1(G ) > (1 + ε)φk(G ); ifλk+1 > O(k2)
√λk , then partition can be found
algorithmically
![Page 78: Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok ...cseweb.ucsd.edu/...theory-day-2014/trevisan-talk.pdf · U.C Berkeley Based on work withTsz Chiu Kwok,Lap Chi Lau,James](https://reader033.fdocuments.in/reader033/viewer/2022060416/5f13b13ebe7aa15e897b1630/html5/thumbnails/78.jpg)
eigenvalue gap
• if λk = 0 and λk+1 > 0 then G has exactly k connectedcomponents;
• [Tanaka 2012] If λk+1 > 5k√λk , then vertices of G can be
partitioned into k sets of small conductance, each inducing asubgraph of large conductance. [Non-algorithmic proof]
• [Oveis-Gharan, T 2013] Sufficient φk+1(G ) > (1 + ε)φk(G ); ifλk+1 > O(k2)
√λk , then partition can be found
algorithmically
![Page 79: Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok ...cseweb.ucsd.edu/...theory-day-2014/trevisan-talk.pdf · U.C Berkeley Based on work withTsz Chiu Kwok,Lap Chi Lau,James](https://reader033.fdocuments.in/reader033/viewer/2022060416/5f13b13ebe7aa15e897b1630/html5/thumbnails/79.jpg)
eigenvalue gap
• if λk = 0 and λk+1 > 0 then G has exactly k connectedcomponents;
• [Tanaka 2012] If λk+1 > 5k√λk , then vertices of G can be
partitioned into k sets of small conductance, each inducing asubgraph of large conductance. [Non-algorithmic proof]
• [Oveis-Gharan, T 2013] Sufficient φk+1(G ) > (1 + ε)φk(G ); ifλk+1 > O(k2)
√λk , then partition can be found
algorithmically
![Page 80: Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok ...cseweb.ucsd.edu/...theory-day-2014/trevisan-talk.pdf · U.C Berkeley Based on work withTsz Chiu Kwok,Lap Chi Lau,James](https://reader033.fdocuments.in/reader033/viewer/2022060416/5f13b13ebe7aa15e897b1630/html5/thumbnails/80.jpg)
In Summary
λk is small: There are k disjoint non-expanding sets; they can befound efficiently
λk is large: Easy instance for various algorithms.
λk is small and λk+1 is large: Nodes can be partitioned into ksets, each with high internal expansion and small externalexpansion
![Page 81: Luca Trevisan U.C Berkeley Based on work withTsz Chiu Kwok ...cseweb.ucsd.edu/...theory-day-2014/trevisan-talk.pdf · U.C Berkeley Based on work withTsz Chiu Kwok,Lap Chi Lau,James](https://reader033.fdocuments.in/reader033/viewer/2022060416/5f13b13ebe7aa15e897b1630/html5/thumbnails/81.jpg)
lucatrevisan.wordpress.com/tag/expanders/