graphs, vectors, and matrices daniel a. spielman yale ...cs- · yale university ams josiah willard...
TRANSCRIPT
![Page 1: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/1.jpg)
Graphs, Vectors, and Matrices Daniel A. Spielman
Yale University
AMS Josiah Willard Gibbs Lecture January 6, 2016
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From Applied to Pure Mathematics
Algebraic and Spectral Graph Theory Sparsification: approximating graphs by graphs with fewer edges The Kadison-Singer problem
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A Social Network Graph
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A Social Network Graph
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A Social Network Graph “vertex” or “node”
“edge” = pair of nodes
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“vertex” or “node”
“edge” = pair of nodes
A Social Network Graph
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A Big Social Network Graph
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A Graph
8
5
1
7 3 2
9 4 6
1012
11
G = (V,E)
= vertices, = edges, pairs of vertices V E
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The Graph of a Mesh
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Examples of Graphs
8
5
1
7 3 2
9 4 6
1012
11
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Examples of Graphs
8
5
1
7 3 2
9 4 6
1012
11
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How to understand large-scale structure Draw the graph Identify communities and hierarchical structure Use physical metaphors
Edges as resistors or rubber bands
Examine processes Diffusion of gas / Random Walks
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The Laplacian quadratic form of G = (V,E)X
(a,b)2E
(x(a)� x(b))2x : V ! R
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X
(a,b)2E
(x(a)� x(b))2x : V ! R
0
0.5
0.5
1
1
1.5
The Laplacian quadratic form of G = (V,E)
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X
(a,b)2E
(x(a)� x(b))2x : V ! R
0
0.5
0.5
1
1
1.5(0.5)2
(0.5)2
(0.5)2
(0.5)2
(0.5)2
(0.5)2
(0)2
The Laplacian quadratic form of G = (V,E)
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X
(a,b)2E
(x(a)� x(b))2x : V ! R
The Laplacian matrix of G = (V,E)
= x
TLx
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Graphs as Resistor Networks
View edges as resistors connecting vertices Apply voltages at some vertices. Measure induced voltages and current flow.
1V
0V
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Graphs as Resistor Networks
Induced voltages minimize , subject to constraints.
1V
0V
X
(a,b)2E
(x(a)� x(b))2
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0V
0.5V
0.5V
0.625V 0.375V
Graphs as Resistor Networks
Induced voltages minimize , subject to constraints.
1V
X
(a,b)2E
(x(a)� x(b))2
0V
1V
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1V
0V
0.5V
0.5V
0.625V 0.375V
(0.5)2
(0.5)2(0.
5)2
(0.5)2
(0.375) 2(0.125)2
(0.25)2
(0.125) 2
(0.375)2
Graphs as Resistor Networks
Induced voltages minimize , subject to constraints.
1V
X
(a,b)2E
(x(a)� x(b))2
0V
1V
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Graphs as Resistor Networks
Induced voltages minimize , subject to constraints.
X
(a,b)2E
(x(a)� x(b))2
1V
0V
0.5V
0.5V
0.625V 0.375V
(0.5)2
(0.5)2(0.5
)2
(0.5)2
(0.375) 2
(0.125
)2
(0.25)2
(0.125) 2
(0.375)2
1V
0V
1V
Effective conductance = current flow with one volt
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Weighted Graphs
Edge assigned a non-negative real weight measuring strength of connection 1/resistance
wa,b 2 R
x
TLx =
X
(a,b)2E
wa,b(x(a)� x(b))2
(a, b)
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Want to map with most edges short
Spectral Graph Drawing (Hall ’70)
V ! R
Edges are drawn as curves for visibility.
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Want to map with most edges short Minimize to fix scale, require
Spectral Graph Drawing (Hall ’70)
V ! R
x
TLx =
X
(a,b)2E
(x(a)� x(b))2
X
a
x(a)2 = 1
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Want to map with most edges short Minimize to fix scale, require
Spectral Graph Drawing (Hall ’70)
V ! R
x
TLx =
X
(a,b)2E
(x(a)� x(b))2
X
a
x(a)2 = 1
kxk = 1
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Courant-Fischer Theorem
Where is the smallest eigenvalue of and is the corresponding eigenvector.
�1
v1L
v1 = arg minx 6=0
kxk=1
x
T
Lx�1 = minx 6=0
kxk=1
x
T
Lx
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For
�1 = 0 and is a constant vector v1
Where is the smallest eigenvalue of and is the corresponding eigenvector.
�1
v1L
v1 = arg minx 6=0
kxk=1
x
T
Lx�1 = minx 6=0
kxk=1
x
T
Lx
Courant-Fischer Theorem
x
TLx =
X
(a,b)2E
(x(a)� x(b))2
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Want to map with most edges short Minimize Such that and
Spectral Graph Drawing (Hall ’70)
V ! R
x
TLx =
X
(a,b)2E
(x(a)� x(b))2
X
a
x(a) = 0kxk = 1
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Want to map with most edges short Minimize Such that and
Spectral Graph Drawing (Hall ’70)
V ! R
x
TLx =
X
(a,b)2E
(x(a)� x(b))2
X
a
x(a) = 0
Courant-Fischer Theorem: solution is , the eigenvector of , the second-smallest eigenvalue
v2 �2
kxk = 1
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Spectral Graph Drawing (Hall ’70)
X
(a,b)2E
(x(a)� x(b))2 = area under blue curves
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Spectral Graph Drawing (Hall ’70)
X
(a,b)2E
(x(a)� x(b))2 = area under blue curves
0 =X
a
x(a)kxk = 1
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Space the points evenly
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And, move them to the circle
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Finish by putting me back in the center
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Want to map with most edges short Such that and
Spectral Graph Drawing (Hall ’70)
X
a
x(a) = 0
V ! R2
X
(a,b)2E
����
✓x(a)y(a)
◆�✓x(b)y(b)
◆����2
X
a
y(a) = 0and
Minimize
kxk = 1
kyk = 1
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Want to map with most edges short Such that and
Spectral Graph Drawing (Hall ’70)
V ! R2
X
(a,b)2E
����
✓x(a)y(a)
◆�✓x(b)y(b)
◆����2
Minimize
kxk = 1 1Tx = 0
and kyk = 1 1T y = 0
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Want to map with most edges short Such that and
Spectral Graph Drawing (Hall ’70)
V ! R2
X
(a,b)2E
����
✓x(a)y(a)
◆�✓x(b)y(b)
◆����2
and
Minimize
kxk = 1
kyk = 1
1Tx = 0
1T y = 0
and , to prevent x
Ty = 0
x = y
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Such that
Spectral Graph Drawing (Hall ’70) X
(a,b)2E
����
✓x(a)y(a)
◆�✓x(b)y(b)
◆����2
Minimize
Courant-Fischer Theorem: solution is , up to rotation
x = v2, y = v3
and
kxk = 1 kyk = 1
1Tx = 0 1T y = 0 x
Ty = 0
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Spectral Graph Drawing (Hall ’70)
31 2
4
56 7
8 9
1 2
4
5
6
9
3
8
7
Arbitrary Drawing
Spectral Drawing
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Spectral Graph Drawing (Hall ’70)
Original Drawing
Spectral Drawing
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Spectral Graph Drawing (Hall ’70)
Original Drawing
Spectral Drawing
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Dodecahedron
Best embedded by first three eigenvectors
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Spectral drawing of Erdos graph: edge between co-authors of papers
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When there is a “nice” drawing:
Most edges are short Vertices are spread out and don’t clump too much
is close to 0 �2
When is big, say there is no nice picture of the graph
�2 > 10/ |V |1/2
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Expanders: when is big �2
Formally: infinite families of graphs of constant degree d and large Examples: random d-regular graphs Ramanujan graphs Have no communities or clusters. Incredibly useful in Computer Science:
Act like random graphs (pseudo-random) Used in many important theorems and algorithms
�2
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-regular graphs with �2, ...,�n ⇡ d
Courant-Fischer: for all
d
x
TLGx ⇡ d
1Tx = 0
kxk = 1
Good Expander Graphs
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-regular graphs with
Courant-Fischer: for all
For , the complete graph on vertices Kn n
�2, ...,�n = n , so for
LKn ⇡ n
dLG
d
x
TLGx ⇡ d
1Tx = 0
kxk = 1
Good Expander Graphs
1Tx = 0
kxk = 1
�2, ...,�n ⇡ d
x
TLKnx = n
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LKn ⇡ n
dLG
Good Expander Graphs
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Sparse Approximations of Graphs
A graph is a sparse approximation of if has few edges and
H G
H LH ⇡ LG
few: the number of edges in is H
O(n) O(n log n) n = |V |or , where
LH ⇡✏ LG if for all 1
1 + � xTLHx
xTLGx 1 + �
x
(S-Teng ‘04)
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Sparse Approximations of Graphs
A graph is a sparse approximation of if has few edges and
H G
H LH ⇡ LG
few: the number of edges in is H
O(n) O(n log n) n = |V |or , where
LH ⇡✏ LG if for all 1
1 + � xTLHx
xTLGx 1 + �
x
(S-Teng ‘04)
Where M 4 fMx
TMx x
T fMxif for all x
1
1 + ✏LG 4 LH 4 (1 + ✏)LG
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Sparse Approximations of Graphs
A graph is a sparse approximation of if has few edges and
H G
H LH ⇡ LG
few: the number of edges in is H
O(n) O(n log n) n = |V |or , where
LH ⇡✏ LG if for all 1
1 + � xTLHx
xTLGx 1 + �
x
(S-Teng ‘04)
Where M 4 fMx
TMx x
T fMxif for all x
1
1 + ✏LG 4 LH 4 (1 + ✏)LG
![Page 52: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/52.jpg)
Sparse Approximations of Graphs
The number of edges in is H
O(n) O(n log n) n = |V |or , where
(S-Teng ‘04)
Where M 4 fMx
TMx x
T fMxif for all x
1
1 + ✏LG 4 LH 4 (1 + ✏)LG
![Page 53: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/53.jpg)
Why we sparsify graphs
To save memory when storing graphs. To speed up algorithms:
flow problems in graphs (Benczur-Karger ‘96)
linear equations in Laplacians (S-Teng ‘04)
![Page 54: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/54.jpg)
Graph Sparsification Theorems
For every , there is a s.t.
G = (V,E,w) H = (V, F, z)
and
(Batson-S-Srivastava ‘09)
|F | (2 + ✏)2n/✏2LG ⇡✏ LH
![Page 55: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/55.jpg)
Graph Sparsification Theorems
For every , there is a s.t.
G = (V,E,w) H = (V, F, z)
and
(Batson-S-Srivastava ‘09)
By careful random sampling, can quickly get
(S-Srivastava ‘08)
|F | O(n log n/✏2)
|F | (2 + ✏)2n/✏2LG ⇡✏ LH
![Page 56: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/56.jpg)
L1,2 =
✓1 �1�1 1
◆
=
✓1�1
◆�1 �1
�
x
TLGx =
X
(a,b)2E
(x(a)� x(b))2
LG =X
(a,b)2E
La,b
Laplacian Matrices
![Page 57: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/57.jpg)
x
TLGx =
X
(a,b)2E
(x(a)� x(b))2
LG =X
(a,b)2E
La,b
=X
(a,b)2E
ua,buTa,b ua,b = �a � �b
Laplacian Matrices
![Page 58: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/58.jpg)
=� �✓ ◆
x
TLGx =
X
(a,b)2E
(x(a)� x(b))2
LG =X
(a,b)2E
La,b
=X
(a,b)2E
ua,buTa,b
U UT
ua,b
Laplacian Matrices
ua,b = �a � �b
![Page 59: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/59.jpg)
Matrix Sparsification
� �=
� �✓ ◆M
� �=
� �✓ ◆fM
1
(1 + ✏)M 4 fM 4 (1 + ✏)M
U UT
![Page 60: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/60.jpg)
� �=
� �✓ ◆M
� �=
� �✓ ◆fM
1
(1 + ✏)M 4 fM 4 (1 + ✏)M
subset of vectors, scaled up
Matrix Sparsification
U UT
![Page 61: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/61.jpg)
� �=
� �✓ ◆M
� �=
� �✓ ◆fM
1
(1 + ✏)M 4 fM 4 (1 + ✏)M
subset of vectors, scaled up
Matrix Sparsification
U UT
![Page 62: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/62.jpg)
� �=
� �✓ ◆M
1
(1 + ✏)M 4 fM 4 (1 + ✏)M
� �=
� �✓ ◆fM
most si = 0
Matrix Sparsification
U UT =X
i
uiuTi
=X
i
siuiuTi
![Page 63: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/63.jpg)
Simplification of Matrix Sparsification
1
(1 + ✏)M 4 fM 4 (1 + ✏)M
1
(1 + ✏)I 4 M�1/2fMM�1/2 4 (1 + ✏)I
is equivalent to
![Page 64: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/64.jpg)
1
(1 + ✏)I 4 M�1/2fMM�1/2 4 (1 + ✏)I
Set
We need X
i
sivivTi ⇡✏ I
Simplification of Matrix Sparsification
X
i
vivTi = Ivi = M�1/2ui
![Page 65: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/65.jpg)
1
(1 + ✏)I 4 M�1/2fMM�1/2 4 (1 + ✏)I
“Decomposition of the identity”
“Parseval frame” “Isotropic Position”
X
i
(vTi t)2 = ktk2
Set vi = M�1/2ui
Simplification of Matrix Sparsification
X
i
vivTi = I
![Page 66: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/66.jpg)
Matrix Sparsification by Sampling
For with X
i
vivTi = Iv1, ..., vm 2 Rn
si =
(1/pi with probability pi0 with probability 1� pi
(Rudelson ‘99, Ahlswede-Winter ‘02, Tropp ’11)
E"X
i
sivivTi
#=
X
i
vivTi
Choose with probability If choose , set
visi = 1/pivi
pi ⇠ kvik2
![Page 67: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/67.jpg)
Matrix Sparsification by Sampling
For with X
i
vivTi = Iv1, ..., vm 2 Rn
si =
(1/pi with probability pi0 with probability 1� pi
(Rudelson ‘99, Ahlswede-Winter ‘02, Tropp ’11)
E"X
i
sivivTi
#=
X
i
vivTi
Choose with probability If choose , set
visi = 1/pivi
pi ⇠ kvik2
(effective conductance)
![Page 68: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/68.jpg)
Matrix Sparsification by Sampling
For with X
i
vivTi = Iv1, ..., vm 2 Rn
si =
(1/pi with probability pi0 with probability 1� pi
(Rudelson ‘99, Ahlswede-Winter ‘02, Tropp ’11)
E"X
i
sivivTi
#=
X
i
vivTi
Choose with probability If choose , set
visi = 1/pivi
pi = C(log n) kvik2 /✏2
![Page 69: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/69.jpg)
Matrix Sparsification by Sampling
For with X
i
vivTi = Iv1, ..., vm 2 Rn
(Rudelson ‘99, Ahlswede-Winter ‘02, Tropp ’11)
Choose with probability If choose , set
visi = 1/pivi
pi = C(log n) kvik2 /✏2
X
i
sivivTi ⇡✏ I
With high probability, choose vectors
and
O(n log n/✏2)
![Page 70: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/70.jpg)
Optimal (?) Matrix Sparsification
For with X
i
vivTi = Iv1, ..., vm 2 Rn
Can choose vectors and nonzero values for the so that
X
i
sivivTi ⇡✏ I
(Batson-S-Srivastava ‘09)
si(2 + ✏)2n/✏2
![Page 71: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/71.jpg)
Optimal (?) Matrix Sparsification
For with X
i
vivTi = Iv1, ..., vm 2 Rn
Can choose vectors and nonzero values for the so that
X
i
sivivTi ⇡✏ I
(Batson-S-Srivastava ‘09)
si
si
(2 + ✏)2n/✏2
![Page 72: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/72.jpg)
Optimal (?) Matrix Sparsification
For with X
i
vivTi = Iv1, ..., vm 2 Rn
Can choose vectors and nonzero values for the so that
X
i
sivivTi ⇡✏ I
(Batson-S-Srivastava ‘09)
si
si ⇠ 1/ kvik2
(2 + ✏)2n/✏2
![Page 73: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/73.jpg)
The Kadison-Singer Problem ‘59
Equivalent to: Anderson’s Paving Conjectures (‘79, ‘81) Bourgain-Tzafriri Conjecture (‘91) Feichtinger Conjecture (‘05) Many others
Implied by:
Weaver’s KS2 conjecture (‘04)
![Page 74: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/74.jpg)
v1
�v1�v2
�v3
�v4
v4
v3
v2
for every unit vector
Weaver’s Conjecture: Isotropic vectors X
i
vivTi = I
X
i
(vTi t)2 = 1
t
t
![Page 75: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/75.jpg)
Partition into approximately ½-Isotropic Sets S1 S2
![Page 76: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/76.jpg)
S1 S2
1/4 P
i2Sj(vTi t)
2 3/4
Partition into approximately ½-Isotropic Sets
![Page 77: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/77.jpg)
1/4 eigs(P
i2SjvivTi ) 3/4
S1 S2
1/4 P
i2Sj(vTi t)
2 3/4
Partition into approximately ½-Isotropic Sets
![Page 78: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/78.jpg)
1/4 eigs(P
i2SjvivTi ) 3/4
S1 S2
eigs(P
i2SjvivTi ) 3/4
X
i2S1
vivTi = I �
X
i2S2
vivTibecause
()
1/4 P
i2Sj(vTi t)
2 3/4
Partition into approximately ½-Isotropic Sets
![Page 79: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/79.jpg)
S1
S2
Big vectors make this difficult
![Page 80: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/80.jpg)
S1 S2
Big vectors make this difficult
![Page 81: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/81.jpg)
Weaver’s Conjecture KS2
There exist positive constants and so that if all and then exists a partition into S1 and S2 with
↵ ✏
PvivTi = I
eigs(P
i2SjvivTi ) 1� ✏
kvik2 ↵
![Page 82: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/82.jpg)
For all if all and then exists a partition into S1 and S2 with
↵ > 0
Theorem (Marcus-S-Srivastava ‘15)
eigs(P
i2SjvivTi ) 1
2 + 3↵
PvivTi = Ikvik2 ↵
![Page 83: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/83.jpg)
We want
eigs
0
BB@
X
i2S1
vivTi 0
0X
i2S2
vivTi
1
CCA 12 + 3↵
![Page 84: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/84.jpg)
We want
roots
0
BB@poly
0
BB@
X
i2S1
vivTi 0
0
X
i2S2
vivTi
1
CCA
1
CCA 12 + 3↵
![Page 85: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/85.jpg)
Consider expected polynomial of a random partition.
We want
roots
0
BB@poly
0
BB@
X
i2S1
vivTi 0
0
X
i2S2
vivTi
1
CCA
1
CCA 12 + 3↵
![Page 86: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/86.jpg)
Proof Outline
1. Prove expected characteristic polynomial has real roots
2. Prove its largest root is at most
3. Prove is an interlacing family, so exists a partition whose polynomial
has largest root at most
1/2 + 3↵
1/2 + 3↵
![Page 87: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/87.jpg)
Interlacing
Polynomial
interlaces
if
q(x) =Qd�1
i=1 (x� �i)
p(x) =Qd
i=1(x� ↵i)
↵1 �1 ↵2 · · ·↵d�1 �d�1 ↵d
Example: q(x) =d
dx
p(x)
![Page 88: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/88.jpg)
p1(x)
Common Interlacing
and have a common interlacing if can partition the line into intervals so that each contains one root from each polynomial
p2(x)
�1
�2
�3
) ) ) ) ( ( ( ( �d�1
![Page 89: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/89.jpg)
�1
�2
�3
) ) ) ) ( ( ( (
max-root (pi) max-root (Ei [ pi ])
Largest root of average
Common Interlacing
�d�1
If p1 and p2 have a common interlacing,
for some i.
![Page 90: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/90.jpg)
�1
�2
�3
) ) ) ) ( ( ( (
If p1 and p2 have a common interlacing,
for some i.
max-root (pi) max-root (Ei [ pi ])
Largest root of average
Common Interlacing
�d�1
![Page 91: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/91.jpg)
Without a common interlacing
(x+ 1)(x+ 2) (x� 1)(x� 2)
![Page 92: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/92.jpg)
(x+ 1)(x+ 2) (x� 1)(x� 2)
x
2 + 4
Without a common interlacing
![Page 93: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/93.jpg)
(x+ 4)(x� 1)(x� 8)
(x+ 3)(x� 9)(x� 10.3)
Without a common interlacing
![Page 94: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/94.jpg)
(x+ 4)(x� 1)(x� 8)
(x+ 3)(x� 9)(x� 10.3)
(x+ 3.2)(x� 6.8)(x� 7)
Without a common interlacing
![Page 95: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/95.jpg)
�1
�2
�3
) ) ) ) ( ( ( (
Largest root of average
�d�1
max-root (pi) max-root (Ei [ pi ])
Common Interlacing
If p1 and p2 have a common interlacing,
for some i.
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p1(x) and have a common interlacing iff p2(x)
�p1(x) + (1� �)p2(x) is real rooted for all 0 � 1
�1
�2
�3
�d�1
) ) ) ) ( ( ( (
Common Interlacing
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Ei [ p2,i ]Ei [ p1,i ]
p2,2p2,1p1,1 p1,2
is an interlacing family {p�}�2{1,2}n
if its members can be placed on the leaves of a tree so that when every node is labeled with the average of leaves below, siblings have common interlacings
Interlacing Family of Polynomials
Ei,j [ pi,j ]
![Page 98: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/98.jpg)
Ei [ p2,i ]Ei [ p1,i ]
p2,2p2,1p1,1 p1,2
have a common interlacing
is an interlacing family {p�}�2{1,2}n
if its members can be placed on the leaves of a tree so that when every node is labeled with the average of leaves below, siblings have common interlacings
Interlacing Family of Polynomials
Ei,j [ pi,j ]
![Page 99: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/99.jpg)
Ei [ p2,i ]Ei [ p1,i ]
p2,2p2,1p1,1 p1,2
Ei,j [ pi,j ]
have a common interlacing
is an interlacing family {p�}�2{1,2}n
if its members can be placed on the leaves of a tree so that when every node is labeled with the average of leaves below, siblings have common interlacings
Interlacing Family of Polynomials
![Page 100: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/100.jpg)
p2,2p2,1p1,1 p1,2
Theorem: There is a so that �
Interlacing Family of Polynomials
max-root(p�) max-root(E�p�)
Ei [ p2,i ]Ei [ p1,i ]
Ei,j [ pi,j ]
![Page 101: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/101.jpg)
p2,2p2,1p1,1 p1,2
Theorem: There is a so that �
Interlacing Family of Polynomials
have a common interlacing
Ei [ p2,i ]Ei [ p1,i ]
Ei,j [ pi,j ]
max-root(p�) max-root(E�p�)
![Page 102: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/102.jpg)
Theorem: There is a so that �
p2,2p2,1p1,1 p1,2
Interlacing Family of Polynomials
have a common interlacing
Ei [ p2,i ]Ei [ p1,i ]
Ei,j [ pi,j ]
max-root(p�) max-root(E�p�)
![Page 103: Graphs, Vectors, and Matrices Daniel A. Spielman Yale ...cs- · Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . From Applied to Pure Mathematics Algebraic and Spectral](https://reader036.vdocuments.site/reader036/viewer/2022071020/5fd4850a690a645496637f4e/html5/thumbnails/103.jpg)
Our family is interlacing
Form other polynomials in the tree by fixing the choices of where some vectors go
ES1,S2
2
664 poly
0
BB@
X
i2S1
vivTi 0
0
X
i2S2
vivTi
1
CCA
3
775
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Summary
1. Prove expected characteristic polynomial has real roots
2. Prove its largest root is at most
3. Prove is an interlacing family, so exists a partition whose polynomial
has largest root at most
1/2 + 3↵
1/2 + 3↵
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To learn more about Laplacians, see
My web page on Laplacian linear equations, sparsification, etc.
My class notes from “Spectral Graph Theory” and “Graphs and Networks”
Papers in Annals of Mathematics and survey from ICM. Available on arXiv and my web page
To learn more about Kadison-Singer