vectorized laplacians for dealing with high-dimensional...
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The Power Grid Graph drawn by Derrick Bezanson 2010
Vectorized Laplacians for dealing with high-dimensional data sets
Fan Chung Graham University of California, San Diego
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Singer + Wu 2011, C. + Zhao 2012
Vectorized Laplacians ----- Graph gauge theory
Earlier work (3 papers 1992-1994) with Shlomo Sternberg, Bert Kostanton the vibrational spectra and spins of the molecules.
New directions inorganizing/analyzinghigh-dimensional data sets,with applications in- vector diffusion maps, - graphics, imaging, sensing- resource allocation with dynamic demands
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=A5
B={(12345),(54321),(12)(34)}!A5
V a group H!
.u v u = gv for g B,a chosen subset of H! "!
Cayley graphs G = (V,E)Γ
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adjacency matrix !
0
0
irreduciblerepresentations
V a group H!
.u v u = gv for g B,a chosen subset of H! "!
Cayley graphs G = (V,E)
Frobenius 1897
Γ
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Eigenvalues of the buckyball
Roots of
• with multiplicity 5
(x2 + x − t 2 + t −1)(x3 − tx2 − x2 − t 2x − 3x + t 3 − t 2 + t + 2) = 0
• with multiplicity 4
(x2 + x − t 2 −1)(x2 + x − (t +1)2 ) = 0
• with multiplicity 3
(x2 + (2t +1)x + t 2 + t −1)(x4 − 3x3 + (−2t 2 + t −1)x2 +
with weight 1 for single bonds and t for double bonds
(3t 2 − 4t + 8)x + t 4 − t 3 + t 2 + 4t − 4) = 0
• with multiplicity 1x − t − 2 = 0
52 + 42 + 32 + 32 +1= 60Saturday, August 11, 2012
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The spectrum of a graph
1( ) ( ( ) ( ))y xx
f x f x f yd
! = "#!
Discrete Laplace operator
2
2 ( )jf xx!"! 1( ) ( ){ }j jf x f x
x x+! !" "! !
In a path Pn 1
1
1( ) ( ) ( )2
( ) ( )
{( )
( )}
j j j
j j
f x f x f x
f x f x
+
!
" = ! !
! !
For
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acts on Γ = (V ,E), ΔIn a graph
In general, we consider
acts on Δ (V ,X) = f :V → X{ }.
Δ = Δ180×180
Δ = Δnd×nd
V = n
F
F (V ,R) = { f :V → R}
Example 1: Vibrational spectra of molecules
Example 2: Vector Diffusion Maps
(V ,Rd ).
(V ,R3).F
F
geometric datatime based datastatistical data
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Vibrations and spins of the Buckyball Noise in the data sets, etc.
Connections: a classical framework for dealing with
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= {E Ex: x ∈V}Vector bundle
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= {E Ex: x ∈V}
Connection T
Vector bundle
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Vector bundle = {
Tx,e :
E
Connection Tfor x V, e =(x,y) E∈ ∈→Ey Ex
Tx,e Ty,e = id
Example 1:
Vibrational spectra of molecules
Spins of the Buckyball
Ex: x ∈V}
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Example 1: Vibrational spectra of molecules
= {h :V → R3}
uh(u)
h(v)
(V ,R3)F
the space of displacements
v
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Example 1: Vibrational spectra of molecules
Potential energy
Hooke’s law: h :V → R3
uvh(u)h(v)
= ku ,v
u + h(u)− v − h(v) − u − v( )u ,v∑ 2
≈ ku ,v wu ,v ⋅(h(u)− h(v))u ,v∑ 2
where wu ,v =
u − vu − v
,
= ku ,v < h(u)− h(v),Aeu ,v∑ (h(u)− h(v)) >
Ae = wu ,v ⊗wu ,vt
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The vibrational spectrum (infrared)of the Buckyball
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Vector bundle
Tx,e :
Selection sec s(u)∈ s∈forE
secEEu
Connection T e =(x,y) E
→Ey Ex
Tx,e Ty,e = id
Suppose a group G acts on V as graph automorphisms.
For a G and u V, ∈ ∈ →Eu Eaua :→Eu Eabu a b = ab :
(a ⋅T )x,e = aTa−1x,a−1ea−1
G acts as automorphisms on connections:
for x V,∈
∈
Define QT ,A(s) = (s(x)−Tx,ex,e∑ s(y)) ⋅Ax,e(s(x)−Tx,es(y))
= {E Ex: x ∈V}
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a G-invariant connection on defines an
Theorem: C.+Sternberg 1992
E
Γ =
Example 1: Vibrations and spins on the Buckyball
F = Av,beb∈B∑⎛⎝⎜
⎞⎠⎟⊗ I − Av,betb ⊗ r(b)
b∈B∑
tb ∈Auto(X)
QT ,A(s) = (s(x)−Tx,ex,e∑ s(y)) ⋅Ax,e(s(x)−Tx,es(y))
can be represented as
In a homogenous graph G/H with edge generating
= (Γ,X)E
isomorphism , such that the operator
set B,
For the Buckyball G = SL(2,5), PSL(2,5) A5,Isotropy subgroup Z2
G
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Examples of connection graphs
data point uOuv
Example 1: Vibrational spectra of molecules
Example 2: Vector Diffusion Maps
Spins of the buckyball
data point v
rotation
Singer + Wu 2011
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Examples of connection Laplacians
data point uOuv
Example 2: Vector Diffusion Maps
data point v
rotation
Singer + Wu 2011
DataAt a node u,
In a network,
local PCA vector in Rd
dimension reduction
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Consistency of connection Laplacians
For a connected connection graph G of dimension d,
For
< f ,L f >= f (u)Ouv − f (v)(i, j )∈E∑ 2
wuv
f :V → Rd
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Consistency of connection Laplacians
For a connected connection graph G of dimension d, the following statements are equivalent:
The 0 eigenvalue has multiplicity d.
All eigenvalues have multiplicity d.
For each vertex v, we haveOuv =Ou
−1Ov
Ov ∈SO(d),such that for all (u,v)∈E.
For any two vertices u and v, any two paths P, P’, Oxy
xy∈P∏ = Ozw
zw∈P '∏
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Consistency of connection Laplacians
For a connected connection graph G of dimension d,
if the 0 eigenvalue has multiplicity d, then
we can find
Theorem:
such that Ouv = Ou−1OvOv ∈SO(d),
for all (u,v)∈E.OuMethod 1: For a fixed vertex u, choose a frame
consisting of any d orthonormal vectors.For any v, define Ov = Ovivi+1
i∏ where
is any path joining u and v. u = v0,v1,...,vt
Method 2: Use eigenvectors. Singer+Spielman 2012Saturday, August 11, 2012
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Dealing with noise in the data
Random connection Laplacians
PageRank algorithms
Graph limits and graphlets
.........
Graph sparsification algorithms
Several combinatorial approaches
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Dealing with noise in the data
Random connection Laplacians
PageRank algorithms
Graph limits and graphlets
.........
Graph sparsification algorithms
Several combinatorial approaches
Saturday, August 11, 2012
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A random graph model for any given expected degree sequence
vj
Pr(Xij = 1) = Pr(vi vj ) = pij
viExample: The Erdos-Renyi model G(n, p)
with pij all equal to . p``
X = Xij( )
Xij = Xji ,1≤ i ≤ j ≤ n.
Independent random indicator variables
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A matrix concentration inequality
Theorem: C. + Radcliffe 2011
Xi : independent random Hermitian matrixn × n
Xi − E(Xi ) ≤ M
X = Xii=1
m
∑ satisfies
Pr X − E(X ) > a( ) ≤ 2n exp −a2
2v2 + 2Ma / 3⎛⎝⎜
⎞⎠⎟
where v2 = var(Xii=1
m
∑ )
with
Ahlswede-Winter,Cristofides-Markstrom, Oliveira, Gross, Recht, Tropp,....
Then
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Eigenvalues of random graphs with general distributions
Theorem: C. + Radcliffe 2011
G : a random graph, where Pr(vi vj ) = pij .
λi (A)− λi (A) ≤ 4dmax log(2n / ε )for all i, ,if the maximum degree satisfies
adjacency matrix of GA :
expected adjacency matrixA : Aij = pij
With probability , eigenvalues of and satisfyA1− ε
dmax > 49 log( 2nε ).
1≤ i ≤ n
A
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Eigenvalues of the normalized Laplacian for random graphs with general distributions
Theorem: C. + Radcliffe 2011
G : a random graph, where Pr(vi vj ) = pij .
λi (L)− λi (L) ≤3log(4n / ε )
δfor all i, ,and
adjacency matrix of GA :expected adjacency matrixA : Aij = pij
δ >10 logn.1≤ i ≤ n
diagonal degree matrix of GDL = I − D−1/2AD−1/2
if the minimum degree satisfiesL = I − D−1/2AD−1/2 ,
for all i, ,if the minimum degree satisfies
With probability , eigenvalues of 1− εsatisfy
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Dealing with noise in the data
Random connection Laplacians
PageRank algorithms
Graph limits and graphlets
.........
Graph sparsification algorithms
Several combinatorial approaches
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Graph sparsification
For a graph G=(V,E,W),
| hG (S) − hH (S) | ≤ εhG (S).
we say H is an ε-sparsifier if for all we have
S ⊆ V
O(n logn)Benczur and Karger 1994 --- sparsifier with size in running time
O(n2 )Spielman and Sriastava 2008 --- using effective resistance running time
O(n)
Batson,Spielman and Sriastava 2009 --- sparsifier running time
O(n)O(nm)
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Graph sparsification
Sparsification Sampling edges
| hG (S) − hH (S) | ≤ εhG (S)
e
The cut edges are more likely to be sampled in order to satisfy, for all S ⊆ V ,
We need to rank edges. Saturday, August 11, 2012
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Dealing with noise in the data
Random connection Laplacians
PageRank algorithms for ranking edges
Graph limits and graphlets
.........
Graph sparsification algorithms
Several combinatorial approaches
Saturday, August 11, 2012
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Two equivalent ways to define PageRank p=pr(!,s)
(1) (1 )p s pW! != + "
0
(1 ) ( )t t
t
p sW! !"
=
= #$(2)
s = 1 1 1( , ,...., )n n n
Definition of PageRank
the (original) PageRank
some “seed”, e.g.,s = (1,0,....,0)
personalized PageRank
PageRank for ranking vertices
jumping constant
seed
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PageRank as generalized Green’s function
Discrete Green’s function Chung+Yau 2000
restricted to
L = D1/2ΔD−1/2 = λkPk
k=1
n−1
∑Laplacian
(ker L )⊥
Green’s function G =
1λk
Pkk=1
n−1
∑
LG = I
The general Green’s function Gβ =
1β + λk
Pkk=1
n−1
∑
PageRank pr(α, s) = βsD−1/2GβD1/2 , β = 2α 1−α
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Ranking pages using Green’s function and electrical network theory
: a weighted graphG = (V ,E,W ) wu ,v
vu
injected current functionconductance
1
iV :V →
−1 iE :E→ induced current function
B(e,v) =
1 if v is e 's head
−1 if v is e 's tail0 otherwise
⎧
⎨⎪⎪
⎩⎪⎪
Kirchoff’s current law: iV = iEB
Ohm’s current law: iE = f BTWvoltage potantial function
L = BTWB
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Ranking pages using Green’s function and electrical network theory
iV = fV BTW B = fV L
Kirchoff’s current law: iV = iEB
Ohm’s current law: iE = f BTW
fV = iVD−1/2G D−1/2
R(u,v) = fV (χu − χv )T
= iVD−1/2G D−1/2 (χu − χv )
T
= (χu − χv ) D−1/2G D−1/2 (χu − χv )
T
B ' =D1/2
B⎡
⎣⎢
⎤
⎦⎥
Wβ =βI 0 0 W⎡
⎣⎢
⎤
⎦⎥
Lβ = βI + L
β β
β
LβGβ = Iβ
β
β
prα (s) = sD−1/2GβD
1/2 where β =2α1−α
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Ranking pages using Green’s function and electrical network theory
Rα (u,v) = hα (u,v) + hα (v,u)
=prα ,u (u)du
−prα ,u (v)dv
+prα ,v (v)dv
−prα ,v (u)du
The Green value of (u,v)
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Vectorized Laplacian and vectorized PageRank
A connection graph G = (V ,E,O,w)
A connection matrix
A =
wuvOuv if (u,v)∈E,0d×d otherwise.
⎧⎨⎩
fLf T = wuv f (u)Ouv − f (v)
(u ,v)∈E∑ 2
L =D − A D(v,v) = duId×d
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Vectorized Laplacian and vectorized PageRank
A connection graph G = (V ,E,O,w)
The transition probability matrix
P =D−1A
p(α,v) = α s + (1−α ) pZ
Z =
I + P2
The vectorized PageRank
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Ranking pages using Green’s function and electrical network theory
Rα (u,v) =pα ,u (u)du
−pα ,u (v)dv
+pα ,v (v)dv
−pα ,v (u)du
The Green value of (u,v)
which the sampling subroutine is based upon.
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Dealing with noise in the data
Random connection Laplacians
PageRank algorithms for ranking edges
Graph limits and graphlets
.........
Graph sparsification algorithms
Several combinatorial approaches
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Examples of connection Laplacians
Example 1: Vibrational spectra of molecules
Example 2: Electron microscopy Vector Diffusion Maps
Spins of the buckyball
Managing millions of images, photos.Example 3: Graphics, vision, ...
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Millions of photos in a box!
Navigate using connection graphs.
- finding consistent subgraphs, paths ...
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Feature detectionDetect features using SIFT [Lowe, IJCV 2004]
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Feature matchingMatch features between each pair of images
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Image connectivity graph
(graph layout produced using the Graphviz toolkit: http://www.graphviz.org/)Part of an image connection graphBuilding Rome in a day
Agarwal, Snavely, Simon, Seitz, Szeliski 2009Saturday, August 11, 2012
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The Power Grid Graph drawn by Derrick Bezanson 2010
Vectorized Laplacians for dealing with high-dimensional data sets
Fan Chung Graham University of California, San Diego
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Thank you.
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Image connectivity graph
(graph layout produced using the Graphviz toolkit: http://www.graphviz.org/)Part of an image connection graphBuilding Rome in a day
Agarwal, Snavely, Simon, Seitz, Szeliski 2009Saturday, August 11, 2012
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An example of connection graphsBuilding Rome in a day
Agarwal, Snavely, Simon, Seitz, Szeliski 2009
Saturday, August 11, 2012