Laplacian Matrices of Graphs: Algorithms and Applications · 2016-06-27 · Algorithms and...

Laplacian Matrices of Graphs:Algorithms and Applications

ICML, June 21, 2016

Daniel A. Spielman

LaplaciansInterpolation on graphsSpring networksClusteringIsotonic regression

Sparsification

Solving Laplacian EquationsBest resultsThe simplest algorithm

Outline

Interpolate values of a function at all verticesfrom given values at a few vertices.

Minimize

Subject to given values

1

0

ANAPC10

CDC27

ANAPC5 UBE2C

ANAPC2

CDC20

(Zhu,Ghahramani,Lafferty ’03)Interpolation on Graphs

X

(i,j)2E

(x(i)� x(j))2


Minimize


1

0

0.51

0.61

0.53

0.30

ANAPC10

CDC27

ANAPC5 UBE2C

ANAPC2

CDC20


X

(i,j)2E

(x(i)� x(j))2

X

(i,j)2E

(x(i)� x(j))2 = x

TLGx


Minimize


Takederivatives.MinimizebysolvingLaplacian

1

0

0.51

0.61

0.53

0.30

ANAPC10

CDC27

ANAPC5 UBE2C

ANAPC2

CDC20


X

(i,j)2E

(x(i)� x(j))2 = x

TLGx


Minimize


1

0

0.51

0.61

0.53

0.30

ANAPC10

CDC27

ANAPC5 UBE2C

ANAPC2

CDC20


X

(i,j)2E

(x(i)� x(j))2

The Laplacian Quadratic Form

x

TLGx =

X

(i,j)2E

(x(i)� x(j))2

The Laplacian Matrix of a Graph

Nail down some vertices, let rest settle

View edges as rubber bands or ideal linear springs

In equilibrium, nodes are averages of neighbors.

Spring Networks


View edges as rubber bands or ideal linear springs

Spring Networks

When stretched to length potential energy is

`

�2/2


Physics: position minimizes total potential energy

subject to boundary constraints (nails)

Spring Networks

1

2

X

(i,j)2E

(x(i)� x(j))2

X

(i,j)2E

(x(i)� x(j))2 = x

TLGx

0

ANAPC10

CDC27

ANAPC5 UBE2C

ANAPC2

CDC20

Spring Networks


Minimize

1

1

0

ANAPC10

CDC27

ANAPC5 UBE2C

ANAPC2

CDC20

Spring Networks


MinimizeX

(i,j)2E

(x(i)� x(j))2 = x

TLGx

X

(i,j)2E

(x(i)� x(j))2 = x

TLGx


Minimize

1

0

0.51

0.61

0.53

0.30

ANAPC10

CDC27

ANAPC5 UBE2C

ANAPC2

CDC20

Spring Networks

In the solution, variables are the average of their neighbors

(Tutte ’63)Drawing by Spring Networks

Drawing by Spring Networks (Tutte ’63)

If the graph is planar,then the spring drawinghas no crossing edges!


SS

Measuring boundaries of sets

Boundary: edges leaving a set


S

00

00

00

1

1 0

11

11 1

0

00

1

S

Characteristic Vector of S:

x(i) =

(1 i in S

0 i not in S



S

00

00

00

1

1 0

11

11 1

0

00

1

S

Characteristic Vector of S:

x(i) =

(1 i in S

0 i not in S


X

(i,j)2E

(x(i)� x(j))

2

= |boundary(S)|

Find large sets of small boundary

S

0.2

-0.200.4

-1.10.5

0.8

0.81.11.0

1.61.3

0.9

0.7-0.4

-0.3

1.3

S0.5

Heuristic to findx with small

Compute eigenvector

Consider the level sets

LGv2 = �2v2

x

TLGx

Spectral Clustering and Partitioning

2

1 43 5

6

0

BBBBBB@

3 �1 �1 �1 0 0�1 2 0 0 0 �1�1 0 3 �1 �1 0�1 0 �1 4 �1 �10 0 �1 �1 3 �10 �1 0 �1 �1 3

1

CCCCCCA

Symmetric

Non-positive off-diagonals

Diagonally dominant


x

TLGx =

X

(i,j)2E

(x(i)� x(j))2

x(i)� x(j) =�1 �1

�✓x(i)x(j)

◆

=

✓x(i)x(j)

◆T ✓1 �1�1 1

◆✓x(i)x(j)

◆


(x(i)� x(j))2 =

✓x(i)x(j)

◆T ✓1�1

◆✓1�1

◆T ✓x(i)x(j)

◆

x

TLGx =

X

(i,j)2E

wi,j(x(i)� x(j))2

Laplacian Matrices of Weighted Graphs

LG =X

(i,j)2E

wi,j(bi,jbTi,j) where bi,j = ei � ej

B is the signed edge-vertex adjacency matrixwith one row for each

W is the diagonal matrix of weights

LG =X

(i,j)2E

wi,j(bi,jbTi,j)

bi,j

wi,j

LG = BTWB


where bi,j = ei � ej

LG = BTWB


B =

0

BBBBBBBBBBBB@

1 �1 0 0 0 00 1 0 0 0 �11 0 0 �1 0 00 0 0 1 0 �11 0 �1 0 0 00 0 1 �1 0 00 0 1 0 �1 00 0 0 1 �1 00 0 0 0 1 �1

1

CCCCCCCCCCCCA

2

1 43 5

6

LG =X

(i,j)2E

wi,j(bi,jbTi,j)

Quickly Solving Laplacian Equations

Where m is number of non-zeros and n is dimension

O(m log

c n log ✏�1)

S,Teng ’04: Using low-stretch trees and sparsifiers


O(m log

c n log ✏�1)

eO(m log n log ✏�1)

Koutis, Miller, Peng ’11: Low-stretch trees and sampling




Cohen, Kyng, Pachocki, Peng, Rao ’14:

O(m log

c n log ✏�1)


eO(m log

1/2 n log ✏�1)




Cohen, Kyng, Pachocki, Peng, Rao ’14:

O(m log

c n log ✏�1)


eO(m log

1/2 n log ✏�1)




Good code:LAMG (lean algebraic multigrid) – Livne-BrandtCMG (combinatorial multigrid) – Koutis

An -accurate solution to is an x satisfying

where

LGx = b

✏

kvkLG=

pvTLGv = ||L1/2

G v||


O(m log

c n log ✏�1)


kx� x

⇤kLG ✏ kx⇤kLG

An -accurate solution to is an x satisfying

LGx = b

✏


O(m log

c n log ✏�1)


kx� x

⇤kLG ✏ kx⇤kLG

Allows fast computation of eigenvectorscorresponding to small eigenvalues.

Laplacians appear when solving Linear Programs onon graphs by Interior Point Methods

Lipschitz Learning : regularized interpolation on graphs(Kyng, Rao, Sachdeva,S ‘15)

Maximum and Min-Cost Flow (Daitch, S ’08, Mądry ‘13)

Shortest Paths (Cohen, Mądry, Sankowski, Vladu ‘16)

Isotonic Regression (Kyng, Rao, Sachdeva ‘15)

Laplacians in Linear Programming

(Ayer et. al. ‘55)Isotonic Regression

2.5

3.2

3.2

4.0

3.73.6

3.9

A function is isotonic with respect to a directed acyclic graph if x increases on edges.

x : V ! R

Isotonic Regression

2.5

3.2

3.2

4.0

3.73.6

3.9

College GPA

High-school GPA

SAT

(Ayer et. al. ‘55)

Isotonic Regression

2.5

3.2

3.2

4.0

3.73.6

3.9

College GPA

High-school GPA

SATEstimate bynearest neighbor?


Isotonic Regression

2.5

3.2

3.2

4.0

3.73.6

3.9

College GPA

High-school GPA

SATEstimate bynearest neighbor?

We want the estimate to be monotonically increasing


Isotonic Regression

2.5

3.2

3.2

4.0

3.73.6

3.9

College GPA

High-school GPA

SAT

Given find the isotonic x minimizingy : V ! R kx� yk


Isotonic Regression

2.5

3.2

3.2

3.866

3.8663.6

3.866

College GPA

High-school GPA

SAT


Given find the isotonic x minimizingy : V ! R kx� yk

Given find the isotonic x minimizingy : V ! R kx� yk1

(Kyng, Rao, Sachdeva ’15)Fast IPM for Isotonic Regression

Given find the isotonic x minimizingy : V ! R kx� yk1

or for any

in time

kx� ykp p > 1

O(m3/2log

3 m)

(Kyng, Rao, Sachdeva ’15)Fast IPM for Isotonic Regression

Signed edge-vertex incidence matrix

x is isotonic if Bx 0

B =

0

BBBBBBBBBB@

1 0 0 �1 0 0 01 0 0 0 �1 0 00 1 �1 0 0 0 00 0 1 0 �1 0 00 0 1 0 0 �1 00 0 0 1 0 0 �10 0 0 0 1 0 �10 0 0 0 0 1 �1

1

CCCCCCCCCCA

Linear Program for Isotonic Regression

Given y, minimize

subject to

kx� yk1

Bx 0

B =

0

BBBBBBBBBB@

1 0 0 �1 0 0 01 0 0 0 �1 0 00 1 �1 0 0 0 00 0 1 0 �1 0 00 0 1 0 0 �1 00 0 0 1 0 0 �10 0 0 0 1 0 �10 0 0 0 0 1 �1

1

CCCCCCCCCCA


Bx 0

|xi � yi| = ri

Given y, minimize

subject to

Pi ri

B =

0

BBBBBBBBBB@

1 0 0 �1 0 0 01 0 0 0 �1 0 00 1 �1 0 0 0 00 0 1 0 �1 0 00 0 1 0 0 �1 00 0 0 1 0 0 �10 0 0 0 1 0 �10 0 0 0 0 1 �1

1

CCCCCCCCCCA


Given y, minimize

subject to

Pi ri

Bx 0

|xi � yi| ri

B =

0

BBBBBBBBBB@

1 0 0 �1 0 0 01 0 0 0 �1 0 00 1 �1 0 0 0 00 0 1 0 �1 0 00 0 1 0 0 �1 00 0 0 1 0 0 �10 0 0 0 1 0 �10 0 0 0 0 1 �1

1

CCCCCCCCCCA


Bx 0

xi � yi ri

�(xi � yi) ri

Given y, minimize

subject to

Pi ri

B =

0

BBBBBBBBBB@

1 0 0 �1 0 0 01 0 0 0 �1 0 00 1 �1 0 0 0 00 0 1 0 �1 0 00 0 1 0 0 �1 00 0 0 1 0 0 �10 0 0 0 1 0 �10 0 0 0 0 1 �1

1

CCCCCCCCCCA


Minimize

subject to

Pi ri

0

@0 B

�I I

�I �I

1

A✓r

x

◆

0

@0y

�y

1

A


Minimize

subject to

Pi ri

0

@0 B

�I I

�I �I

1

A✓r

x

◆

0

@0y

�y

1

A

IPM solves a sequence of equations of form0

@0 B�I I�I �I

1

AT 0

@S0 0 00 S1 00 0 S2

1

A

0

@0 B�I I�I �I

1

A

with positive diagonal matrices S0, S1, S2


0

@0 B�I I�I �I

1

AT 0

@S0 0 00 S1 00 0 S2

1

A

0

@0 B�I I�I �I

1

A

=

✓S1 + S2 S2 � S1

S2 � S1 BTS0B + S1 + S2

◆

S0, S1, S2 are positive diagonal

Laplacian!


0

@0 B�I I�I �I

1

AT 0

@S0 0 00 S1 00 0 S2

1

A

0

@0 B�I I�I �I

1

A

=

✓S1 + S2 S2 � S1

S2 � S1 BTS0B + S1 + S2

◆

S0, S1, S2 are positive diagonal

Laplacian!

Kyng, Rao, Sachdeva ’15: Reduce to solving Laplacians to constant accuracy


Spectral Sparsification

Every graph can be approximated by a sparse graph with a similar Laplacian

for all x

A graph H is an -approximation of G if ✏

1

1 + � xTLHx

xTLGx 1 + �

Approximating Graphs

LH ⇡✏ LG

for all x


1

1 + � xTLHx

xTLGx 1 + �


Preserves boundaries of every set

Solutions to linear equations are similiar

for all x


1

1 + � xTLHx

xTLGx 1 + �


LH ⇡✏ LG () L�1H ⇡✏ L

�1G


Every graph G has an -approximation Hwith edges n(2 + ✏)2/✏2

✏

(Batson, S, Srivastava ’09)


Every graph G has an -approximation Hwith edges n(2 + ✏)2/✏2

✏

(Batson, S, Srivastava ’09)

Random regular graphs approximate complete graphs

Fast Spectral Sparsification

(S & Srivastava ‘08) If sample each edge with probability inversely proportional to its effective spring constant,only need samples

Takes time (Koutis, Levin, Peng ‘12)

O(n log n/✏2)

(Lee & Sun ‘15) Can find an -approximation with edges in time for every

O(n/✏2)✏O(n1+c) c > 0

O(m log

2 n)

(Kyng & Sachdeva ‘16)Approximate Gaussian Elimination

Gaussian Elimination:compute upper triangular U so that

LG = UTU

Approximate Gaussian Elimination:compute sparse upper triangular U so that

LG ⇡ UTU

Gaussian Elimination

1. Find the rank-1 matrix that agrees on the first row and column

2. Subtract it

0

BB@

16 �4 �8 �4�4 5 0 �1�8 0 14 0�4 �1 0 7

1

CCA

0

BB@

16 �4 �8 �4�4 1 2 1�8 2 4 2�4 1 2 1

1

CCA =

0

BB@

4�1�2�1

1

CCA�4 �1 �2 �1

�

Gaussian Elimination1. Find the rank-1 matrix that agrees

on the first row and column

2. Subtract it

0

BB@

16 �4 �8 �4�4 1 2 1�8 2 4 2�4 1 2 1

1

CCA =

0

BB@

4�1�2�1

1

CCA�4 �1 �2 �1

�

0

BB@

16 �4 �8 �4�4 5 0 �1�8 0 14 0�4 �1 0 7

1

CCA�

0

BB@

16 �4 �8 �4�4 1 2 1�8 2 4 2�4 1 2 1

1

CCA =

0

BB@

0 0 0 00 4 �2 �20 �2 10 �20 �2 �2 6

1

CCA

3. Repeat


1. Find the rank-1 matrix that agrees on the next row and column

2. Subtract it0

BB@

16 �4 �8 �4�4 5 0 �1�8 0 14 0�4 �1 0 7

1

CCA�

0

BB@

16 �4 �8 �4�4 1 2 1�8 2 4 2�4 1 2 1

1

CCA =

0

BB@

0 0 0 00 4 �2 �20 �2 10 �20 �2 �2 6

1

CCA

0

BB@

0 0 0 00 4 �2 �20 �2 1 10 �2 1 1

1

CCA =

0

BB@

02�1�1

1

CCA�0 2 �1 �1

�

Gaussian Elimination1. Find the rank-1 matrix that agrees

on the next row and column

2. Subtract it

0

BB@

0 0 0 00 4 �2 �20 �2 1 10 �2 1 1

1

CCA =

0

BB@

02�1�1

1

CCA�0 2 �1 �1

�

0

BB@

0 0 0 00 4 �2 �20 �2 10 �20 �2 �2 6

1

CCA�

0

BB@

0 0 0 00 4 �2 �20 �2 1 10 �2 1 1

1

CCA =

0

BB@

0 0 0 00 0 0 00 0 9 �30 0 �3 5

1

CCA


=

0

BB@

4�1�2�1

1

CCA

0

BB@

4�1�2�1

1

CCA

T

+

0

BB@

02�1�1

1

CCA

0

BB@

02�1�1

1

CCA

T

+

0

BB@

003�1

1

CCA

0

BB@

003�1

1

CCA

T

+

0

BB@

0002

1

CCA

0

BB@

0002

1

CCA

T

0

BB@

16 �4 �8 �4�4 5 0 �1�8 0 14 0�4 �1 0 7

1

CCA


=

0

BB@

4�1�2�1

1

CCA

0

BB@

4�1�2�1

1

CCA

T

+

0

BB@

02�1�1

1

CCA

0

BB@

02�1�1

1

CCA

T

+

0

BB@

003�1

1

CCA

0

BB@

003�1

1

CCA

T

+

0

BB@

0002

1

CCA

0

BB@

0002

1

CCA

T

0

BB@

16 �4 �8 �4�4 5 0 �1�8 0 14 0�4 �1 0 7

1

CCA

=

0

BB@

4 0 0 0�1 2 0 0�2 �1 3 0�1 �1 �1 2

1

CCA

0

BB@

4 �1 �2 �10 2 �1 �10 0 3 �10 0 0 2

1

CCA


=

0

BB@

4�1�2�1

1

CCA

0

BB@

4�1�2�1

1

CCA

T

+

0

BB@

02�1�1

1

CCA

0

BB@

02�1�1

1

CCA

T

+

0

BB@

003�1

1

CCA

0

BB@

003�1

1

CCA

T

+

0

BB@

0002

1

CCA

0

BB@

0002

1

CCA

T

0

BB@

16 �4 �8 �4�4 5 0 �1�8 0 14 0�4 �1 0 7

1

CCA

=

0

BB@

4 �1 �2 �10 2 �1 �10 0 3 �10 0 0 2

1

CCA

T 0

BB@

4 �1 �2 �10 2 �1 �10 0 3 �10 0 0 2

1

CCA


=

0

BB@

4�1�2�1

1

CCA

0

BB@

4�1�2�1

1

CCA

T

+

0

BB@

02�1�1

1

CCA

0

BB@

02�1�1

1

CCA

T

+

0

BB@

003�1

1

CCA

0

BB@

003�1

1

CCA

T

+

0

BB@

0002

1

CCA

0

BB@

0002

1

CCA

T

0

BB@

16 �4 �8 �4�4 5 0 �1�8 0 14 0�4 �1 0 7

1

CCA

Computation time proportional to the sum of the squares of the number of nonzeros

in these vectors

Gaussian Elimination of Laplacians

0

BB@

16 �4 �8 �4�4 5 0 �1�8 0 14 0�4 �1 0 7

1

CCA�

0

BB@

4�1�2�1

1

CCA

0

BB@

4�1�2�1

1

CCA

T

=

0

BB@

0 0 0 00 4 �2 �20 �2 10 �20 �2 �2 6

1

CCA

If this is a Laplacian, then so is this

0

BB@

16 �4 �8 �4�4 5 0 �1�8 0 14 0�4 �1 0 7

1

CCA�

0

BB@

4�1�2�1

1

CCA

0

BB@

4�1�2�1

1

CCA

T

=

0

BB@

0 0 0 00 4 �2 �20 �2 10 �20 �2 �2 6

1

CCA

Gaussian Elimination of Laplacians

If this is a Laplacian, then so is this

2

1 4

3 6

54

8

4 12

4

3 6

5

222

When eliminate a node, add a clique on its neighbors

Approximate Gaussian Elimination

2

1 4

3 6

54

8

4 12

4

3 6

5

222

1. when eliminate a node, add a clique on its neighbors

2. Sparsify that clique, without ever constructing it

(Kyng & Sachdeva ‘16)


1. When eliminate a node of degree d,add d edges at random between its neighbors, sampled with probability proportional to the weight of the edge to the eliminated node

1


0. Initialize by randomly permuting vertices, and making copies of every edgeO(log

2 n)


Total time is O(m log

3 n)


0. Initialize by randomly permuting vertices, and making copies of every edgeO(log

2 n)

Total time is O(m log

3 n)


Can be improved by sacrificing some simplicity


Analysis by Random Matrix Theory:

Write UTU as a sum of random matrices.

Random permutation and copying control the variances of the random matrices

Apply Matrix Freedman inequality (Tropp ‘11)

E⇥UTU

⇤= LG

Other families of linear systems

complex-weighted Laplacians

connection Laplacians

✓1 ei✓

e�i✓ 1

◆

✓I QQT I

◆

Recent Developments

(Kyng, Lee, Peng, Sachdeva, S ‘16)

Laplacians.jl

To learn more

My web page on:Laplacian linear equations, sparsification, local graph clustering, low-stretch spanning trees, and so on.

My class notes from “Graphs and Networks” and “Spectral Graph Theory”

Lx = b, by Nisheeth Vishnoi

Laplacian Matrices of Graphs: Algorithms and Applications · 2016-06-27 · Algorithms and...

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