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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
Saturday, August 11, 2012
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
Saturday, August 11, 2012
=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
Saturday, August 11, 2012
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
Saturday, August 11, 2012
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
Saturday, August 11, 2012
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
Saturday, August 11, 2012
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
Saturday, August 11, 2012
Vibrations and spins of the Buckyball Noise in the data sets, etc.
Connections: a classical framework for dealing with
Saturday, August 11, 2012
= {E Ex: x ∈V}Vector bundle
Saturday, August 11, 2012
= {E Ex: x ∈V}
Connection T
Vector bundle
Saturday, August 11, 2012
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}
Saturday, August 11, 2012
Example 1: Vibrational spectra of molecules
= {h :V → R3}
uh(u)
h(v)
(V ,R3)F
the space of displacements
v
Saturday, August 11, 2012
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
Saturday, August 11, 2012
The vibrational spectrum (infrared)of the Buckyball
Saturday, August 11, 2012
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}
Saturday, August 11, 2012
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
Saturday, August 11, 2012
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
Saturday, August 11, 2012
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
Saturday, August 11, 2012
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
Saturday, August 11, 2012
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 '∏
Saturday, August 11, 2012
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
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
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
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
Saturday, August 11, 2012
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
Saturday, August 11, 2012
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
Saturday, August 11, 2012
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
Saturday, August 11, 2012
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
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)
Saturday, August 11, 2012
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
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
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
Saturday, August 11, 2012
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
Saturday, August 11, 2012
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
Saturday, August 11, 2012
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
Saturday, August 11, 2012
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)
Saturday, August 11, 2012
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
Saturday, August 11, 2012
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
Saturday, August 11, 2012
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.
Saturday, August 11, 2012
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
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, ...
Saturday, August 11, 2012
Millions of photos in a box!
Navigate using connection graphs.
- finding consistent subgraphs, paths ...
Saturday, August 11, 2012
Feature detectionDetect features using SIFT [Lowe, IJCV 2004]
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Feature matchingMatch features between each pair of images
Saturday, August 11, 2012
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
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
Saturday, August 11, 2012
Thank you.
Saturday, August 11, 2012
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
An example of connection graphsBuilding Rome in a day
Agarwal, Snavely, Simon, Seitz, Szeliski 2009
Saturday, August 11, 2012