graph sparsification by effective resistancescrs/icm2010conf/talks/srivastava-talk.pdf ·...
TRANSCRIPT
![Page 1: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/1.jpg)
Twice-Ramanujan Sparsifiers
Nikhil Srivastava
MSR India / Yale
with Josh Batson and Dan Spielman
![Page 2: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/2.jpg)
Sparsification
Approximate any graph G by a sparse graph H.
– Nontrivial statement about G
– H is faster to compute with than G
G H
![Page 3: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/3.jpg)
H approximates G if
for every cut S½V
sum of weights of edges leaving S is preserved
Can find H with O(nlogn/2) edges in time
Cut Sparsifiers [Benczur-Karger’96]
S S
![Page 4: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/4.jpg)
The Laplacian (quick review)
Quadratic form
Positive semidefinite
Ker(LG)=span(1) if G is connected
![Page 5: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/5.jpg)
Cuts and the Quadratic Form
For characteristic vector
So BK says:
![Page 6: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/6.jpg)
A Stronger Notion [ST’04]
For characteristic vector
So BK says:
![Page 7: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/7.jpg)
Why?
![Page 8: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/8.jpg)
1. All eigenvalues are preserved
By Courant-Fischer,
G and H have similar eigenvalues.
For spectral purposes, G and H are equivalent.
![Page 9: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/9.jpg)
(xTLx says a lot)
2. Behavior of electrical flows.
(xTLx = “energy” for potentials x:V->R)
3. Behavior of random walks: commute times, mixing time, etc.
4. ‘Relative condition number’ in lin-alg.
5. Fast linear system solvers.
strong notion of approximation.
![Page 10: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/10.jpg)
Examples
![Page 11: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/11.jpg)
Example: Sparsify Complete Graph by Ramanujan Expander [LPS,M]
G is complete on n vertices.
H is d-regular Ramanujan graph.
![Page 12: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/12.jpg)
Example: Sparsify Complete Graph by Ramanujan Expander [LPS,M]
G is complete on n vertices.
H is d-regular Ramanujan graph.
So, is a good sparsifier for G.
Each edge has weight (n/d)
![Page 13: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/13.jpg)
Example: Dumbell
Kn Kn
d-regularRamanujan,
times n/d
d-regularRamanujan,
times n/d
1
1
![Page 14: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/14.jpg)
Example: Dumbell
Kn Kn
d-regularRamanujan,
times n/d
d-regularRamanujan,
times n/d
1
1
G1
G2
G3
F1
F2
F3
![Page 15: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/15.jpg)
Results
![Page 16: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/16.jpg)
Results
We can do this well for every G.
(upto a factor of 2)
![Page 17: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/17.jpg)
Previously Known
Expanders/Ramanujan graphs exist:
“There are very sparse H that look like Kn”
![Page 18: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/18.jpg)
Previously Known
Expanders/Ramanujan graphs exist:
“There are very sparse H that look like Kn”
degree d
![Page 19: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/19.jpg)
New Result
Expanders/Ramanujan graphs exist:
“There are very sparse H that look like Kn”
Sparsifiers exist:
“There are very sparse H that look like any graph G.”
avg. degree 2d
degree d
![Page 20: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/20.jpg)
New Result
Expanders/Ramanujan graphs exist:
“There are very sparse H that look like Kn”
Sparsifiers exist:
“There are very sparse H that look like any graph G.”
avg. degree 2d
degree d
weightedsubgraph
![Page 21: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/21.jpg)
New Result
Expanders/Ramanujan graphs exist:
“There are very sparse H that look like Kn”
Sparsifiers exist:
“There are very sparse H that look like any graph G.”
avg. degree 2d
degree d
deterministic O(dmn3)algorithm
weightedsubgraph
![Page 22: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/22.jpg)
New Result
Expanders/Ramanujan graphs exist:
“There are very sparse H that look like Kn”
Sparsifiers exist:
“There are very sparse H that look like any graph G.”
avg. degree 2d
degree d
deterministic O(dmn3)algorithm
weightedsubgraph
1+²1¡²
8=²2
![Page 23: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/23.jpg)
The Method
![Page 24: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/24.jpg)
The Method
(13-approximation with 6n edges.)
![Page 25: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/25.jpg)
Step 1: Reduction to Linear Algebra
![Page 26: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/26.jpg)
Goal
![Page 27: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/27.jpg)
Outer Product Expansion
Recall:
![Page 28: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/28.jpg)
Outer Product Expansion
Recall:
For a weighted subgraph H:
![Page 29: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/29.jpg)
![Page 30: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/30.jpg)
![Page 31: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/31.jpg)
![Page 32: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/32.jpg)
![Page 33: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/33.jpg)
A closer look at ve
m vectors
in Rn-1
G
![Page 34: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/34.jpg)
A closer look at ve
m vectors
in Rn-1
G
![Page 35: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/35.jpg)
m vectors
in Rn-1
“decomposition of identity”
A closer look at ve
![Page 36: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/36.jpg)
Choosing a Subgraph
G H
![Page 37: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/37.jpg)
New Goal
G H
![Page 38: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/38.jpg)
New Goal
G H
G H
![Page 39: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/39.jpg)
Main theorem
If
then there are scalars with
and .
![Page 40: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/40.jpg)
Main theorem
If
then there are scalars with
and .
![Page 41: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/41.jpg)
Main theorem
If
then there are scalars with
and .
![Page 42: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/42.jpg)
Step 2: Intuition for the proof
![Page 43: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/43.jpg)
Main theorem
If
then there are scalars with
and .
![Page 44: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/44.jpg)
Main theorem
If
then there are scalars with
and . will build this one vector at a time.
![Page 45: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/45.jpg)
What happens when we add a vector?
![Page 46: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/46.jpg)
Interlacing
![Page 47: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/47.jpg)
More precisely
Characteristic Polynomial:
![Page 48: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/48.jpg)
More precisely
Characteristic Polynomial:
Matrix-Determinant Lemma:
![Page 49: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/49.jpg)
More precisely
Characteristic Polynomial:
Matrix-Determinant Lemma:are zeros of this.
![Page 50: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/50.jpg)
Physical model of interlacing
i = positive unit charges
resting at barriers on a slope
![Page 51: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/51.jpg)
Physical model of interlacing
<v,ui>2 =charges added
to barriers
![Page 52: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/52.jpg)
Physical model of interlacing
<v,ui>2 =charges added
to barriers
![Page 53: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/53.jpg)
Physical model of interlacing
Barriers repel eigs.
![Page 54: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/54.jpg)
Physical model of interlacing
Barriers repel eigs.
gravity
Inverse law repulsion
![Page 55: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/55.jpg)
Physical model of interlacing
Barriers repel eigs.
![Page 56: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/56.jpg)
Examples
![Page 57: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/57.jpg)
Ex1: All weight on u1
![Page 58: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/58.jpg)
Ex1: All weight on u1
![Page 59: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/59.jpg)
Ex1: All weight on u1
![Page 60: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/60.jpg)
Ex1: All weight on u1
![Page 61: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/61.jpg)
Ex2: Equal weight on u1 ,u2
![Page 62: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/62.jpg)
Ex2: Equal weight on u1 ,u2
![Page 63: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/63.jpg)
Ex2: Equal weight on u1 ,u2
![Page 64: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/64.jpg)
Ex3: Equal weight on all u1, u2 , …un
![Page 65: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/65.jpg)
Ex3: Equal weight on all u1, u2 , …un
![Page 66: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/66.jpg)
Adding a balanced vector
![Page 67: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/67.jpg)
Consider a random vector
If
![Page 68: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/68.jpg)
Consider a random vector
If
thus a ‘random’ vector has the same expected projection in every direction i :
![Page 69: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/69.jpg)
Ideal proof
![Page 70: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/70.jpg)
Ideal proof
![Page 71: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/71.jpg)
Ideal proof
![Page 72: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/72.jpg)
Ideal proof
![Page 73: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/73.jpg)
Ideal proof
![Page 74: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/74.jpg)
Ideal proof
![Page 75: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/75.jpg)
Ideal proof
![Page 76: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/76.jpg)
Ideal proof
![Page 77: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/77.jpg)
Ideal proof
![Page 78: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/78.jpg)
Ideal proof
![Page 79: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/79.jpg)
Ideal proof
![Page 80: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/80.jpg)
Ideal proof
![Page 81: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/81.jpg)
Ideal proof
![Page 82: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/82.jpg)
Ideal proof
![Page 83: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/83.jpg)
Punch Line
![Page 84: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/84.jpg)
Punch Line
![Page 85: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/85.jpg)
Punch Line find actual vectors that realize this ideal behavior.
![Page 86: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/86.jpg)
Step 3: Actual Proof(for 6n vectors, 13-approx)
![Page 87: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/87.jpg)
Broad outline: moving barriers
0 n-n
![Page 88: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/88.jpg)
Step 1
0 n-n
![Page 89: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/89.jpg)
Step 1
0 n-n
0 n-n
![Page 90: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/90.jpg)
Step 1
0 n-n
0
+1/3 +2
-n+1/3 n+2
![Page 91: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/91.jpg)
Step 1
0 n-n
0
+1/3 +2
-n+1/3 n+2
tighter constraint
looser constraint
![Page 92: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/92.jpg)
Step i+1
0
![Page 93: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/93.jpg)
Step i+1
0
+1/3 +2
![Page 94: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/94.jpg)
Step i+1
0
![Page 95: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/95.jpg)
Step i+1
0
+1/3 +2
![Page 96: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/96.jpg)
Step i+1
0
![Page 97: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/97.jpg)
Step i+1
0
+1/3 +2
![Page 98: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/98.jpg)
Step i+1
0
![Page 99: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/99.jpg)
Step i+1
0
![Page 100: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/100.jpg)
Step i+1
0
![Page 101: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/101.jpg)
Step i+1
0
![Page 102: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/102.jpg)
Step 6n
0 … 13nn
![Page 103: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/103.jpg)
Step 6n
0 … n
13-approximation with 6n vectors.
13n
![Page 104: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/104.jpg)
Problem
need to show that an appropriate
always exists.
![Page 105: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/105.jpg)
Problem
is not strong enough to do the induction.
need to show that an appropriate
always exists.
![Page 106: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/106.jpg)
Problem
is not strong enough to do the induction.
need to show that an appropriate
always exists.need a better way to measure
quality of eigenvalues.
![Page 107: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/107.jpg)
The Upper Barrier
![Page 108: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/108.jpg)
The Upper Barrier
![Page 109: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/109.jpg)
No i within dist. 1No 2 i within dist. 2No 3 i within dist. 3
.
.No k i within dist. k
The Upper Barrier
![Page 110: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/110.jpg)
No i within dist. 1No 2 i within dist. 2No 3 i within dist. 3
.
.No k i within dist. k
The Upper Barrier‘Total repulsion’ in
physical model
![Page 111: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/111.jpg)
The Lower Barrier
![Page 112: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/112.jpg)
The Beginning
0 n-n
![Page 113: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/113.jpg)
The Beginning
0 n-n
![Page 114: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/114.jpg)
Step i+1
0
![Page 115: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/115.jpg)
Step i+1
0
+1/3 +2
Lemma.
can always choose
so that potentials do not increase
![Page 116: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/116.jpg)
Step i+1
0
![Page 117: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/117.jpg)
Step i+1
0
![Page 118: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/118.jpg)
Step i+1
0
![Page 119: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/119.jpg)
Step i+1
0
![Page 120: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/120.jpg)
Step 6n
0 … 13nn
![Page 121: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/121.jpg)
Step 6n
0 … n
13-approximation with 6n vectors.
13n
![Page 122: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/122.jpg)
Goal
+1/3 +2
Lemma.
can always choose so
that both potentials do not increase.
![Page 123: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/123.jpg)
The Right Question
“Which vector should we add?”
![Page 124: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/124.jpg)
The Right Question
“Which vector should we add?”
“Given a vector, how much of it can we add?”
![Page 125: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/125.jpg)
Upper Barrier Update
Add & set
+2
![Page 126: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/126.jpg)
Upper Barrier Update
Add & set
+2
![Page 127: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/127.jpg)
Upper Barrier Update
Add & set
Sherman-Morrisson
+2
![Page 128: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/128.jpg)
Upper Barrier Update
Add & set
+2
![Page 129: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/129.jpg)
Upper Barrier Update
Add & set
+2
![Page 130: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/130.jpg)
How much of can we add?
Rearranging:
![Page 131: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/131.jpg)
How much of can we add?
Rearranging:
![Page 132: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/132.jpg)
The Lower Barrier
Similarly:
![Page 133: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/133.jpg)
Goal
Show that we can always add some vector while respecting both barriers.
+1/3 +2
![Page 134: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/134.jpg)
There is always a vector with
Both Barriers
![Page 135: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/135.jpg)
There is always a vector with
Both Barrierscan add must add
![Page 136: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/136.jpg)
There is always a vector with
Both Barriers
Then, can squeeze scaling factor in between:
can add must add
![Page 137: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/137.jpg)
Taking Averages
![Page 138: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/138.jpg)
Taking Averages
![Page 139: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/139.jpg)
Bounding Tr(UA)
![Page 140: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/140.jpg)
Bounding Tr(UA)
![Page 141: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/141.jpg)
Bounding Tr(UA)
![Page 142: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/142.jpg)
Bounding Tr(UA)
induction
![Page 143: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/143.jpg)
Bounding Tr(UA)
induction
![Page 144: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/144.jpg)
Bounding Tr(UA)
induction
convexity
![Page 145: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/145.jpg)
Bounding Tr(UA)
induction
convexity
![Page 146: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/146.jpg)
Taking Averages
![Page 147: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/147.jpg)
Taking Averages
![Page 148: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/148.jpg)
Taking Averages
=3/22
![Page 149: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/149.jpg)
Taking Averages
=3/2
2
2
1/3
![Page 150: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/150.jpg)
Taking Averages
=3/2
2
2
1/3
![Page 151: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/151.jpg)
Step i+1
0
+1/3 +2
Lemma.
can always choose
so that potentials do not increase.
![Page 152: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/152.jpg)
Step i+1
0
![Page 153: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/153.jpg)
Step i+1
0
![Page 154: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/154.jpg)
Step i+1
0
![Page 155: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/155.jpg)
Step i+1
0
![Page 156: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/156.jpg)
Step 6n
0 … 13nn
![Page 157: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/157.jpg)
Step 6n
0 … n
13-approximation with 6n vectors.
13n
![Page 158: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/158.jpg)
Main Sparsification Theorem
If
then there are scalars with
and .
![Page 159: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/159.jpg)
Sparsification of Graphs
G H
G H
![Page 160: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/160.jpg)
Twice-Ramanujan
Fixing steps and tightening parameters gives
(zeros of Laguerre polynomials).
![Page 161: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/161.jpg)
Major Themes
• Electrical model of interlacing is useful
• Can use barrier potential to iterativelyconstruct matrices with desired spectra
• Analysis of progress is greedy / local
• Requires fractional weights on vectors
![Page 162: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/162.jpg)
Sparsification of PSD Matrices
Theorem. If
then there are scalars for which
and at most are nonzero.
V =Pi viv
Ti
si ¸ 0
V ¹Pi sivivTi ¹ (1+ ²)V
n=²2
![Page 163: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/163.jpg)
Given:
Spectral Theorem: If rank(V)=n then
for eigenvectors ui.
To put this in context…
V =Pi·n¸iuiu
Ti
V =Pi·m viv
Ti
n terms!
![Page 164: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/164.jpg)
ui need not be ‘meaningful’ directions…
(e.g., vi = edges of graph)
Given:
Spectral Theorem: If rank(V)=n then
To put this in context…
V =Pi·n¸iuiu
Ti
V =Pi·m viv
Ti
n terms
![Page 165: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/165.jpg)
Given:
Spectral Theorem: If rank(V)=n then
This Theorem. Can find scalars si so that:
To put this in context…
V =Pi·n¸iuiu
Ti
V =Pi·m viv
Ti
n terms
V »²Pi siviv
Ti
terms!n=²2
![Page 166: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/166.jpg)
Open Questions
• The Ramanujan bound
• Unweighted sparsifiers for Kn
• A faster algorithm
• Directed graphs? (must be weaker notion)
• The Kadison-Singer Conjecture
![Page 167: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/167.jpg)
Open Questions
• The Ramanujan bound
• Unweighted sparsifiers for Kn
• A faster algorithm
• Directed graphs? (must be weaker notion)
• The Kadison-Singer Conjecture
![Page 168: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/168.jpg)
Sums of Outer Products
A=P
i·m vivTi vi 2Rn;mÀ n
![Page 169: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/169.jpg)
Sums of Outer Products
A=P
i·m vivTi vi 2Rn;mÀ n
•elementary / “meaningful” directions
![Page 170: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/170.jpg)
Sums of Outer Products
A=P
i·m vivTi vi 2Rn;mÀ n
•Q: Can we write A as a sparse sum?
![Page 171: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/171.jpg)
Sums of Outer Products
A=P
i·m vivTi vi 2Rn;mÀ n
•Q: Can we write A as a sparse sum?
•A: Yes, Spectral Theorem:
A=Pi·n¸iuiu
Ti for eigvecs ui
![Page 172: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/172.jpg)
Sums of Outer Products
A=P
i·m vivTi vi 2Rn;mÀ n
•Q: Can we write A as a sparse sum?
•A: Yes, Spectral Theorem:
•Good: only n terms = optimal!
•Bad: ui hard to interpret in terms of vi
A=Pi·n¸iuiu
Ti for eigvecs ui
![Page 173: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/173.jpg)
Example: Graphs
Undirected graph G(V,E)
Laplacian Matrix
LG =P
ij2E(ei ¡ ej)(ei ¡ ej)T
![Page 174: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/174.jpg)
Example: Graphs
Undirected graph G(V,E)
Laplacian Matrix
Interested in `sparse approximation’: write LG as sum of a small number of edges.
LG =P
ij2E(ei ¡ ej)(ei ¡ ej)T
![Page 175: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/175.jpg)
Example: Graphs
Undirected graph G(V,E)
Laplacian Matrix
Interested in `sparse approximation’: write LG as sum of a small number of edges.
Spectral Thm: LG = i=1:n ¸i ui uiT
Sparse, but ui do not correspond to edges… : (
LG =P
ij2E(ei ¡ ej)(ei ¡ ej)T
![Page 176: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/176.jpg)
Spectral Sparsification [BSS’09]
Theorem. Given
there are nonnegative weights si¸ 0 s.t.
and at most 1.1n of the si are nonzero.
A=P
i·m vivTi
A» ~A=P
i sivivTi
![Page 177: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/177.jpg)
Spectral Sparsification [BSS’09]
Theorem. Given
there are nonnegative weights si¸ 0 s.t.
and at most 1.1n of the si are nonzero.
•cf. n terms for ¸i uiuiT
A» ~A=P
i sivivTi
A=P
i·m vivTi
•same vi
![Page 178: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/178.jpg)
Back to Graphs
Undirected graph G(V,E)
Laplacian Matrix
LG =P
ij2E(ei ¡ ej)(ei ¡ ej)T
![Page 179: Graph Sparsification by Effective Resistancescrs/icm2010conf/talks/Srivastava-talk.pdf · Expanders/Ramanujan graphs exist: “There are very sparse H that look like K n” Sparsifiers](https://reader036.vdocument.in/reader036/viewer/2022071010/5fc7e0106228dc4a642852bd/html5/thumbnails/179.jpg)
Back to Graphs
Undirected graph G(V,E)
Laplacian Matrix
Apply Theorem:
LG =P
ij2E(ei ¡ ej)(ei ¡ ej)T
~LG =P
ij2E si(ei ¡ ej)(ei ¡ ej)T
L » ~L
•· 1.1n edges !