graph sparsification iii: ramanujan graphs, lifts, and ...graph sparsification iii: ramanujan...
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Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families
Nikhil Srivastava
Microsoft Research India
Simons Institute, August 27, 2014
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The Last Two Lectures
Lecture 1. Every weighted undirected πΊ has a
weighted subgraph π» with ππ log π
π2 edges
which satisfiesπΏπΊ βΌ πΏπ» βΌ 1 + π πΏπΊ
random sampling
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The Last Two Lectures
Lecture 1. Every weighted undirected πΊ has a
weighted subgraph π» with ππ log π
π2 edges
which satisfiesπΏπΊ βΌ πΏπ» βΌ 1 + π πΏπΊ
Lecture 2. Improved this to 4 π π2.
greedy
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The Last Two Lectures
Lecture 1. Every weighted undirected πΊ has a
weighted subgraph π» with ππ log π
π2 edges
which satisfiesπΏπΊ βΌ πΏπ» βΌ 1 + π πΏπΊ
Lecture 2. Improved this to 4 π π2.
Suboptimal for πΎπ in two ways: weights, and 2π/π2.
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|EH| = O(dn)
πΏπΎπ(1 β π) βΌ πΏπ» βΌ πΏπΎπ
1 + π
G=Kn H = random d-regular x (n/d)
|EG| = O(n2)
Good Sparsifiers of πΎπ
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|EH| = O(dn)
π(1 β π) βΌ πΏπ» βΌ π 1 + π
G=Kn H = random d-regular x (n/d)
|EG| = O(n2)
Good Sparsifiers of πΎπ
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G=Kn H = random d-regular
|EH| = dn/2
π(1 β π) βΌ πΏπ» βΌ π 1 + π
|EG| = O(n2)
Regular Unweighted Sparsifiers of πΎπ
Rescale weights back
to 1
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G=Kn H = random d-regular
|EH| = dn/2
π β 2 π β 1 βΌ πΏπ» βΌ π + 2 π β 1
|EG| = O(n2)
Regular Unweighted Sparsifiers of πΎπ
[Friedmanβ08]
+π(π)
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G=Kn H = random d-regular
|EH| = dn/2
π β 2 π β 1 βΌ πΏπ» βΌ π + 2 π β 1
|EG| = O(n2)
Regular Unweighted Sparsifiers of πΎπ
[Friedmanβ08]
+π(π)
will try to match this
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Why do we care so much about πΎπ?
Unweighted d-regular approximations of πΎπ are called expanders.
They behave like random graphs:
the right # edges across cuts
fast mixing of random walks
Prototypical βpseudorandom objectβ. Many uses in CS and math (Routing, Coding, Complexityβ¦)
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Switch to Adjacency Matrix
Let G be a graph and A be its adjacency matrix
eigenvalues π1 β₯ π2 β₯ β―ππ
a
c
d
eb
0 1 0 0 11 0 1 0 10 1 0 1 00 0 1 0 11 1 0 1 0
πΏ = ππΌ β π΄
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Switch to Adjacency Matrix
Let G be a graph and A be its adjacency matrix
eigenvalues π1 β₯ π2 β₯ β―ππIf d-regular, then π΄π = ππ so π1 = πIf bipartite then eigs are symmetric
about zero so ππ = βπ
a
c
d
eb
0 1 0 0 11 0 1 0 10 1 0 1 00 0 1 0 11 1 0 1 0
βtrivialβ
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Spectral Expanders
Definition: G is a good expander if all non-trivial eigenvalues are small
0[ ]-d d
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Spectral Expanders
Definition: G is a good expander if all non-trivial eigenvalues are small
0[ ]-d d
e.g. πΎπ and πΎπ,π have all nontrivial eigs 0.
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Spectral Expanders
Definition: G is a good expander if all non-trivial eigenvalues are small
0[ ]-d d
e.g. πΎπ and πΎπ,π have all nontrivial eigs 0.Challenge: construct infinite families.
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Spectral Expanders
Definition: G is a good expander if all non-trivial eigenvalues are small
0[ ]-d d
e.g. πΎπ and πΎπ,π have all nontrivial eigs 0.Challenge: construct infinite families.
Alon-Boppanaβ86: Canβt beat
[β2 π β 1, 2 π β 1]
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The meaning of 2 π β 1
The infinite d-ary tree
π π΄π = [β2 π β 1, 2 π β 1]
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The meaning of 2 π β 1
The infinite d-ary tree
π π΄π = [β2 π β 1, 2 π β 1]
Alon-Boppanaβ86: This is the best possible spectral expander.
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Ramanujan Graphs:
Definition: G is Ramanujan if all non-trivial eigs
have absolute value at most 2 π β 1
0[ ]-d d
[-2 π β 1
]2 π β 1
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Ramanujan Graphs:
Definition: G is Ramanujan if all non-trivial eigs
have absolute value at most 2 π β 1
0[ ]-d d
[-2 π β 1
]2 π β 1
Friedmanβ08: A random d-regular graph is almost
Ramanujan : 2 π β 1 + π(1)
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Ramanujan Graphs:
Definition: G is Ramanujan if all non-trivial eigs
have absolute value at most 2 π β 1
0[ ]-d d
[-2 π β 1
]2 π β 1
Friedmanβ08: A random d-regular graph is almost
Ramanujan : 2 π β 1 + π(1)
Margulis, Lubotzky-Phillips-Sarnakβ88: Infinite sequences of Ramanujan graphs exist for π = π + 1
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Ramanujan Graphs:
Definition: G is Ramanujan if all non-trivial eigs
have absolute value at most 2 π β 1
0[ ]-d d
[-2 π β 1
]2 π β 1
Friedmanβ08: A random d-regular graph is almost
Ramanujan : 2 π β 1 + π(1)
Margulis, Lubotzky-Phillips-Sarnakβ88: Infinite sequences of Ramanujan graphs exist for π = π + 1
What about π β π + 1?
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[Marcus-Spielman-Sβ13]
Theorem. Infinite families of bipartite Ramanujangraphs exist for every π β₯ 3.
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[Marcus-Spielman-Sβ13]
Theorem. Infinite families of bipartite Ramanujangraphs exist for every π β₯ 3.
Proof is elementary, doesnβt use number theory.
Not explicit.
Based on a new existence argument: method of interlacing families of polynomials.
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[Marcus-Spielman-Sβ13]
Theorem. Infinite families of bipartite Ramanujangraphs exist for every π β₯ 3.
Proof is elementary, doesnβt use number theory.
Not explicit.
Based on a new existence argument: method of interlacing families of polynomials.
πΌπ π = 1 β1
π
π
ππ₯
π
π₯π
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Bilu-Linialβ06 Approach
Find an operation which doubles the size of a graph without blowing up its eigenvalues.
0[ ]-d d
[-2 π β 1
]2 π β 1
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Bilu-Linialβ06 Approach
Find an operation which doubles the size of a graph without blowing up its eigenvalues.
0[ ]-d d
[-2 π β 1
]2 π β 1
![Page 28: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/28.jpg)
Bilu-Linialβ06 Approach
Find an operation which doubles the size of a graph without blowing up its eigenvalues.
0[ ]-d d
[-2 π β 1
]2 π β 1
β¦β
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2-lifts of graphs
a
c
d
e
b
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2-lifts of graphs
a
c
d
e
b
a
c
d
e
b
duplicate every vertex
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2-lifts of graphs
duplicate every vertex
a0
c0
d0
e0
b0
a1
c1
d1
e1
b1
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2-lifts of graphs
a0
c0
d0
e0
b0
a1
d1
e1
b1
c1
for every pair of edges:
leave on either side (parallel),
or make both cross
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2-lifts of graphs
for every pair of edges:
leave on either side (parallel),
or make both cross
a0
c0
d0
e0
b0
a1
d1
e1
b1
c1
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2-lifts of graphs
for every pair of edges:
leave on either side (parallel),
or make both cross
a0
c0
d0
e0
b0
a1
d1
e1
b1
c1
2π possibilities
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Eigenvalues of 2-lifts (Bilu-Linial)
Given a 2-lift of G,
create a signed adjacency matrix As
with a -1 for crossing edges
and a 1 for parallel edges
0 -1 0 0 1-1 0 1 0 10 1 0 -1 00 0 -1 0 11 1 0 1 0
a0
c0
d0
e0
b0
a1
d1
e1
b1
c1
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Eigenvalues of 2-lifts (Bilu-Linial)
Theorem:The eigenvalues of the 2-lift are:
π1, β¦ , ππ = ππππ π΄βͺ
π1β² β¦ππ
β² = ππππ (π΄π )
0 -1 0 0 1-1 0 1 0 10 1 0 -1 00 0 -1 0 11 1 0 1 0
π΄π =
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Eigenvalues of 2-lifts (Bilu-Linial)
Theorem:The eigenvalues of the 2-lift are the
union of the eigenvalues of A (old)and the eigenvalues of As (new)
Conjecture:Every d-regular graph has a 2-lift
in which all the new eigenvalueshave absolute value at most
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Eigenvalues of 2-lifts (Bilu-Linial)
Theorem:The eigenvalues of the 2-lift are the
union of the eigenvalues of A (old)and the eigenvalues of As (new)
Conjecture:
Every d-regular adjacency matrix A
has a signing π΄π with ||π΄π|| β€ 2 π β 1
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Eigenvalues of 2-lifts (Bilu-Linial)
Theorem:The eigenvalues of the 2-lift are the
union of the eigenvalues of A (old)and the eigenvalues of As (new)
Conjecture:
Every d-regular adjacency matrix A
has a signing π΄π with ||π΄π|| β€ 2 π β 1
Bilu-Linialβ06: This is true with π( π log3 π)
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Eigenvalues of 2-lifts (Bilu-Linial)
Conjecture:Every d-regular adjacency matrix A
has a signing π΄π with ||π΄π|| β€ 2 π β 1
We prove this in the bipartite case.
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Eigenvalues of 2-lifts (Bilu-Linial)
Theorem:Every d-regular adjacency matrix A
has a signing π΄π with π1(π΄π) β€ 2 π β 1
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Eigenvalues of 2-lifts (Bilu-Linial)
Theorem:Every d-regular bipartite adjacency matrix A
has a signing π΄π with ||π΄π|| β€ 2 π β 1
Trick: eigenvalues of bipartite graphsare symmetric about 0, so only need to bound largest
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Random Signings
Idea 1: Choose π β β1,1 π randomly.
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Random Signings
Idea 1: Choose π β β1,1 π randomly.
Unfortunately,
(Bilu-Linial showed when
A is nearly Ramanujan )
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Random Signings
Idea 2: Observe thatwhere
Usually useless, but not here!
is an interlacing family.
such that
Consider
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Random Signings
Idea 2: Observe thatwhere
Usually useless, but not here!
is an interlacing family.
such that
Consider
![Page 47: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/47.jpg)
Random Signings
Idea 2: Observe thatwhere
Usually useless, but not here!
is an interlacing family.
such that
Consider
coefficient-wise average
![Page 48: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/48.jpg)
Random Signings
Idea 2: Observe thatwhere
Usually useless, but not here!
is an interlacing family.
such that
Consider
![Page 49: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/49.jpg)
3-Step Proof Strategy
1. Show that some poly does as well as the .
such that
![Page 50: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/50.jpg)
3-Step Proof Strategy
1. Show that some poly does as well as the .
such that
NOT ππππ₯ ππ΄π β€ πΌππππ₯(ππ΄π
)
![Page 51: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/51.jpg)
3-Step Proof Strategy
1. Show that some poly does as well as the .
2. Calculate the expected polynomial.
such that
![Page 52: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/52.jpg)
3-Step Proof Strategy
1. Show that some poly does as well as the .
2. Calculate the expected polynomial.
3. Bound the largest root of the expected poly.
such that
![Page 53: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/53.jpg)
3-Step Proof Strategy
1. Show that some poly does as well as the .
2. Calculate the expected polynomial.
3. Bound the largest root of the expected poly.
such that
![Page 54: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/54.jpg)
3-Step Proof Strategy
1. Show that some poly does as well as the .
2. Calculate the expected polynomial.
3. Bound the largest root of the expected poly.
such that
![Page 55: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/55.jpg)
Step 2: The expected polynomial
Theorem [Godsil-Gutmanβ81]
For any graph G,
the matching polynomial of G
![Page 56: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/56.jpg)
The matching polynomial
(Heilmann-Lieb β72)
mi = the number of matchings with i edges
![Page 57: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/57.jpg)
![Page 58: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/58.jpg)
one matching with 0 edges
![Page 59: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/59.jpg)
7 matchings with 1 edge
![Page 60: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/60.jpg)
![Page 61: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/61.jpg)
![Page 62: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/62.jpg)
Proof that
x Β±1 0 0 Β±1 Β±1
Β±1 x Β±1 0 0 0
0 Β±1 x Β±1 0 0
0 0 Β±1 x Β±1 0
Β±1 0 0 Β±1 x Β±1
Β±1 0 0 0 Β±1 x
Expand using permutations
![Page 63: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/63.jpg)
Proof that
x Β±1 0 0 Β±1 Β±1
Β±1 x Β±1 0 0 0
0 Β±1 x Β±1 0 0
0 0 Β±1 x Β±1 0
Β±1 0 0 Β±1 x Β±1
Β±1 0 0 0 Β±1 x
Expand using permutations
same edge: same value
![Page 64: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/64.jpg)
Proof that
x Β±1 0 0 Β±1 Β±1
Β±1 x Β±1 0 0 0
0 Β±1 x Β±1 0 0
0 0 Β±1 x Β±1 0
Β±1 0 0 Β±1 x Β±1
Β±1 0 0 0 Β±1 x
Expand using permutations
same edge: same value
![Page 65: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/65.jpg)
Proof that
x Β±1 0 0 Β±1 Β±1
Β±1 x Β±1 0 0 0
0 Β±1 x Β±1 0 0
0 0 Β±1 x Β±1 0
Β±1 0 0 Β±1 x Β±1
Β±1 0 0 0 Β±1 x
Expand using permutations
Get 0 if hit any 0s
![Page 66: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/66.jpg)
Proof that
x Β±1 0 0 Β±1 Β±1
Β±1 x Β±1 0 0 0
0 Β±1 x Β±1 0 0
0 0 Β±1 x Β±1 0
Β±1 0 0 Β±1 x Β±1
Β±1 0 0 0 Β±1 x
Expand using permutations
Get 0 if take just one entry for any edge
![Page 67: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/67.jpg)
Proof that
x Β±1 0 0 Β±1 Β±1
Β±1 x Β±1 0 0 0
0 Β±1 x Β±1 0 0
0 0 Β±1 x Β±1 0
Β±1 0 0 Β±1 x Β±1
Β±1 0 0 0 Β±1 x
Expand using permutations
Only permutations that count are involutions
![Page 68: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/68.jpg)
Proof that
x Β±1 0 0 Β±1 Β±1
Β±1 x Β±1 0 0 0
0 Β±1 x Β±1 0 0
0 0 Β±1 x Β±1 0
Β±1 0 0 Β±1 x Β±1
Β±1 0 0 0 Β±1 x
Expand using permutations
Only permutations that count are involutions
![Page 69: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/69.jpg)
Proof that
x Β±1 0 0 Β±1 Β±1
Β±1 x Β±1 0 0 0
0 Β±1 x Β±1 0 0
0 0 Β±1 x Β±1 0
Β±1 0 0 Β±1 x Β±1
Β±1 0 0 0 Β±1 x
Expand using permutations
Only permutations that count are involutions
Correspond to matchings
![Page 70: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/70.jpg)
Proof that
x Β±1 0 0 Β±1 Β±1
Β±1 x Β±1 0 0 0
0 Β±1 x Β±1 0 0
0 0 Β±1 x Β±1 0
Β±1 0 0 Β±1 x Β±1
Β±1 0 0 0 Β±1 x
Expand using permutations
Only permutations that count are involutions
Correspond to matchings
![Page 71: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/71.jpg)
3-Step Proof Strategy
1. Show that some poly does as well as the .
2. Calculate the expected polynomial.
[Godsil-Gutmanβ81]
3. Bound the largest root of the expected poly.
such that
![Page 72: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/72.jpg)
3-Step Proof Strategy
1. Show that some poly does as well as the .
2. Calculate the expected polynomial.
[Godsil-Gutmanβ81]
3. Bound the largest root of the expected poly.
such that
![Page 73: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/73.jpg)
The matching polynomial
(Heilmann-Lieb β72)
Theorem (Heilmann-Lieb)
all the roots are real
![Page 74: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/74.jpg)
The matching polynomial
(Heilmann-Lieb β72)
Theorem (Heilmann-Lieb)
all the roots are real
and have absolute value at most
![Page 75: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/75.jpg)
The matching polynomial
(Heilmann-Lieb β72)
Theorem (Heilmann-Lieb)
all the roots are real
and have absolute value at most
The number 2 π β 1 comes by comparing
to an infinite π βary tree [Godsil].
![Page 76: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/76.jpg)
3-Step Proof Strategy
1. Show that some poly does as well as the .
2. Calculate the expected polynomial.
[Godsil-Gutmanβ81]
3. Bound the largest root of the expected poly.
[Heilmann-Liebβ72]
such that
![Page 77: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/77.jpg)
3-Step Proof Strategy
1. Show that some poly does as well as the .
2. Calculate the expected polynomial.
[Godsil-Gutmanβ81]
3. Bound the largest root of the expected poly.
[Heilmann-Liebβ72]
such that
![Page 78: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/78.jpg)
3-Step Proof Strategy
1. Show that some poly does as well as the .
Implied by:
β is an interlacing family.β
such that
![Page 79: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/79.jpg)
Averaging Polynomials
Basic Question: Given when are the roots of the related to roots of ?
![Page 80: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/80.jpg)
Averaging Polynomials
Basic Question: Given when are the roots of the related to roots of ?
Answer: Certainly not always
![Page 81: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/81.jpg)
Averaging Polynomials
Basic Question: Given when are the roots of the related to roots of ?
Answer: Certainly not alwaysβ¦
π π₯ = π₯ β 1 π₯ β 2 = π₯2 β 3π₯ + 2
π π₯ = π₯ β 3 π₯ β 4 = π₯2 β 7π₯ + 12
π₯ β 2.5 + 3π π₯ β 2.5 β 3π = π₯2 β 5π₯ + 7
1
2Γ
1
2Γ
![Page 82: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/82.jpg)
Averaging Polynomials
Basic Question: Given when are the roots of the related to roots of ?
But sometimes it works:
![Page 83: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/83.jpg)
A Sufficient Condition
Basic Question: Given when are the roots of the related to roots of ?
Answer: When they have a common interlacing.
Definition. interlaces
if
![Page 84: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/84.jpg)
Theorem. If have a common interlacing,
![Page 85: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/85.jpg)
Theorem. If have a common interlacing,
Proof.
![Page 86: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/86.jpg)
Theorem. If have a common interlacing,
Proof.
![Page 87: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/87.jpg)
Theorem. If have a common interlacing,
Proof.
![Page 88: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/88.jpg)
Theorem. If have a common interlacing,
Proof.
![Page 89: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/89.jpg)
Theorem. If have a common interlacing,
Proof.
![Page 90: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/90.jpg)
Theorem. If have a common interlacing,
Proof.
![Page 91: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/91.jpg)
Interlacing Family of Polynomials
Definition: is an interlacing family
if can be placed on the leaves of a tree so thatwhen every node is the sum of leaves below &sets of siblings have common interlacings
![Page 92: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/92.jpg)
Interlacing Family of Polynomials
Definition: is an interlacing family
if can be placed on the leaves of a tree so thatwhen every node is the sum of leaves below &sets of siblings have common interlacings
![Page 93: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/93.jpg)
Interlacing Family of Polynomials
Definition: is an interlacing family
if can be placed on the leaves of a tree so thatwhen every node is the sum of leaves below &sets of siblings have common interlacings
![Page 94: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/94.jpg)
Interlacing Family of Polynomials
Definition: is an interlacing family
if can be placed on the leaves of a tree so thatwhen every node is the sum of leaves below &sets of siblings have common interlacings
![Page 95: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/95.jpg)
Interlacing Family of Polynomials
Definition: is an interlacing family
if can be placed on the leaves of a tree so thatwhen every node is the sum of leaves below &sets of siblings have common interlacings
![Page 96: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/96.jpg)
Interlacing Family of Polynomials
Theorem: There is an s so that
![Page 97: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/97.jpg)
Proof: By common interlacing, one of ,has
Interlacing Family of Polynomials
Theorem: There is an s so that
![Page 98: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/98.jpg)
Proof: By common interlacing, one of ,has
Interlacing Family of Polynomials
Theorem: There is an s so that
![Page 99: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/99.jpg)
Proof: By common interlacing, one of ,has
Interlacing Family of Polynomials
Theorem: There is an s so that
![Page 100: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/100.jpg)
Proof: By common interlacing, one of ,has β¦.
Interlacing Family of Polynomials
Theorem: There is an s so that
![Page 101: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/101.jpg)
Proof: By common interlacing, one of ,has β¦.
Interlacing Family of Polynomials
Theorem: There is an s so that
![Page 102: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/102.jpg)
An interlacing family
Theorem:
Let
is an interlacing family
![Page 103: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/103.jpg)
To prove interlacing family
Let
Leaves of tree = signings π 1, β¦ , π π
Internal nodes = partial signings π 1, β¦ , π π
![Page 104: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/104.jpg)
To prove interlacing family
Let
Leaves of tree = signings π 1, β¦ , π π
Internal nodes = partial signings π 1, β¦ , π π
Need to find common interlacing for every
internal node
![Page 105: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/105.jpg)
How to Prove Common Interlacing
Lemma (Fiskβ08, folklore): Suppose π(π₯) and π(π₯) are monic and real-rooted. Then:
β a common interlacing π of π and π
β convex combinations,πΌπ + 1 β πΌ π
has real roots.
![Page 106: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/106.jpg)
To prove interlacing family
Let
Need to prove that for all ,
is real rooted
![Page 107: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/107.jpg)
To prove interlacing family
Need to prove that for all ,
are fixed
is 1 with probability , -1 with
are uniformly
is real rooted
Let
![Page 108: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/108.jpg)
Generalization of Heilmann-Lieb
Suffices to prove that
is real rooted
for every product distributionon the entries of s
![Page 109: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/109.jpg)
Generalization of Heilmann-Lieb
Suffices to prove that
is real rooted
for every product distributionon the entries of s:
![Page 110: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/110.jpg)
Transformation to PSD Matrices
Suffices to show real rootedness of
![Page 111: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/111.jpg)
Transformation to PSD Matrices
Suffices to show real rootedness of
Why is this useful?
![Page 112: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/112.jpg)
Transformation to PSD Matrices
Suffices to show real rootedness of
Why is this useful?
![Page 113: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/113.jpg)
Transformation to PSD Matrices
![Page 114: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/114.jpg)
π£ππ = πΏπ β πΏπ with probability Ξ»ππ
πΏπ + πΏπ with probability (1βπππ)
Transformation to PSD Matrices
πΌπ det π₯πΌ β ππΌ β π΄π = πΌdet π₯πΌ β
ππβπΈ
π£πππ£πππ
where
![Page 115: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/115.jpg)
Master Real-Rootedness Theorem
Given any independent random vectors π£1, β¦ , π£π β βπ, their expected characteristic polymomial
has real roots.
πΌdet π₯πΌ β
π
π£ππ£ππ
![Page 116: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/116.jpg)
Master Real-Rootedness Theorem
Given any independent random vectors π£1, β¦ , π£π β βπ, their expected characteristic polymomial
has real roots.
πΌdet π₯πΌ β
π
π£ππ£ππ
How to prove this?
![Page 117: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/117.jpg)
The Multivariate Method
A. Sokal, 90βs-2005:ββ¦it is often useful to consider the multivariate polynomial β¦ even if one is ultimately interested in a particular one-variable specializationβ
Borcea-Branden 2007+: prove that univariatepolynomials are real-rooted by showing that they are nice transformations of real-rootedmultivariate polynomials.
![Page 118: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/118.jpg)
Real Stable Polynomials
Definition. π β β π₯1, β¦ , π₯π is real stable if every univariate restriction in the strictly positive orthant:
π π‘ β π π₯ + π‘ π¦ π¦ > 0
is real-rooted.
If it has real coefficients, it is called real stable.
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Definition. π β β π₯1, β¦ , π₯π is real stable if every univariate restriction in the strictly positive orthant:
π π‘ β π π₯ + π‘ π¦ π¦ > 0
is real-rooted.
Real Stable Polynomials
![Page 120: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/120.jpg)
Definition. π β β π₯1, β¦ , π₯π is real stable if every univariate restriction in the strictly positive orthant:
π π‘ β π π₯ + π‘ π¦ π¦ > 0
is real-rooted.
Real Stable Polynomials
![Page 121: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/121.jpg)
Definition. π β β π₯1, β¦ , π₯π is real stable if every univariate restriction in the strictly positive orthant:
π π‘ β π π₯ + π‘ π¦ π¦ > 0
is real-rooted.
Real Stable Polynomials
![Page 122: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/122.jpg)
Definition. π β β π₯1, β¦ , π₯π is real stable if every univariate restriction in the strictly positive orthant:
π π‘ β π π₯ + π‘ π¦ π¦ > 0
is real-rooted.
Real Stable Polynomials
![Page 123: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/123.jpg)
Definition. π β β π₯1, β¦ , π₯π is real stable if every univariate restriction in the strictly positive orthant:
π π‘ β π π₯ + π‘ π¦ π¦ > 0
is real-rooted.
Not positive
Real Stable Polynomials
![Page 124: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/124.jpg)
A Useful Real Stable Poly
Borcea-BrΓ€ndΓ©n β08:
For PSD matrices
is real stable
![Page 125: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/125.jpg)
A Useful Real Stable Poly
Borcea-BrΓ€ndΓ©n β08:
For PSD matrices
is real stable
Proof: Every positive univariate restriction is the characteristic polynomial of a symmetric matrix.
det
π
π₯ππ΄π + π‘
π
π¦ππ΄π = det(π‘πΌ + π)
![Page 126: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/126.jpg)
If is real stable, then so is
1. π(πΌ, π§2, β¦ , π§π) for any πΌ β β
2. 1 β ππ§ππ(π§1, β¦ π§π) [Lieb-Sokalβ81]
Excellent Closure Properties
is real stable if for all i
Implies .
Definition:
![Page 127: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/127.jpg)
A Useful Real Stable Poly
Borcea-BrΓ€ndΓ©n β08:
For PSD matrices
is real stable
Plan: apply closure properties to this
to show that πΌdet π₯πΌ β π π£ππ£ππ is real stable.
![Page 128: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/128.jpg)
Central Identity
Suppose π£1, β¦ , π£π are independent random vectors with π΄π β πΌπ£ππ£π
π. Then
πΌdet π₯πΌ β
π
π£ππ£ππ
=
π=1
π
1 βπ
ππ§πdet π₯πΌ +
π
π§ππ΄π
π§1=β―=π§π=0
![Page 129: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/129.jpg)
Central Identity
Suppose π£1, β¦ , π£π are independent random vectors with π΄π β πΌπ£ππ£π
π. Then
πΌdet π₯πΌ β
π
π£ππ£ππ
=
π=1
π
1 βπ
ππ§πdet π₯πΌ +
π
π§ππ΄π
π§1=β―=π§π=0
Key Principle: random rank one updates β‘ (1 β ππ§) operators.
![Page 130: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/130.jpg)
Proof of Master Real-Rootedness Theorem
Suppose π£1, β¦ , π£π are independent random vectors with π΄π β πΌπ£ππ£π
π. Then
πΌdet π₯πΌ β
π
π£ππ£ππ
=
π=1
π
1 βπ
ππ§πdet π₯πΌ +
π
π§ππ΄π
π§1=β―=π§π=0
![Page 131: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/131.jpg)
Proof of Master Real-Rootedness Theorem
Suppose π£1, β¦ , π£π are independent random vectors with π΄π β πΌπ£ππ£π
π. Then
πΌdet π₯πΌ β
π
π£ππ£ππ
=
π=1
π
1 βπ
ππ§πdet π₯πΌ +
π
π§ππ΄π
π§1=β―=π§π=0
![Page 132: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/132.jpg)
Proof of Master Real-Rootedness Theorem
Suppose π£1, β¦ , π£π are independent random vectors with π΄π β πΌπ£ππ£π
π. Then
πΌdet π₯πΌ β
π
π£ππ£ππ
=
π=1
π
1 βπ
ππ§πdet π₯πΌ +
π
π§ππ΄π
π§1=β―=π§π=0
![Page 133: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/133.jpg)
Proof of Master Real-Rootedness Theorem
Suppose π£1, β¦ , π£π are independent random vectors with π΄π β πΌπ£ππ£π
π. Then
πΌdet π₯πΌ β
π
π£ππ£ππ
=
π=1
π
1 βπ
ππ§πdet π₯πΌ +
π
π§ππ΄π
π§1=β―=π§π=0
![Page 134: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/134.jpg)
Proof of Master Real-Rootedness Theorem
Suppose π£1, β¦ , π£π are independent random vectors with π΄π β πΌπ£ππ£π
π. Then
πΌdet π₯πΌ β
π
π£ππ£ππ
=
π=1
π
1 βπ
ππ§πdet π₯πΌ +
π
π§ππ΄π
π§1=β―=π§π=0
![Page 135: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/135.jpg)
The Whole Proof
πΌdet π₯πΌ β π π£ππ£ππ is real-rooted for all indep. π£π.
![Page 136: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/136.jpg)
The Whole Proof
πΌdet π₯πΌ β π π£ππ£ππ is real-rooted for all indep. π£π.
πΌππ΄π (π β π₯) is real-rooted for all product
distributions on signings.
rank one structure naturally reveals interlacing.
![Page 137: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/137.jpg)
The Whole Proof
πΌdet π₯πΌ β π π£ππ£ππ is real-rooted for all indep. π£π.
πΌππ΄π (π₯) is real-rooted for all product
distributions on signings.
![Page 138: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/138.jpg)
The Whole Proof
πΌdet π₯πΌ β π π£ππ£ππ is real-rooted for all indep. π£π.
πΌππ΄π (π₯) is real-rooted for all product
distributions on signings.
ππ΄π π₯
π₯β Β±1 π is an interlacing family
![Page 139: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/139.jpg)
The Whole Proof
πΌdet π₯πΌ β π π£ππ£ππ is real-rooted for all indep. π£π.
such that
πΌππ΄π (π₯) is real-rooted for all product
distributions on signings.
ππ΄π π₯
π₯β Β±1 π is an interlacing family
![Page 140: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/140.jpg)
3-Step Proof Strategy
1. Show that some poly does as well as the .
2. Calculate the expected polynomial.
3. Bound the largest root of the expected poly.
such that
![Page 141: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/141.jpg)
Infinite Sequences of Bipartite Ramanujan Graphs
Find an operation which doubles the size of a graph without blowing up its eigenvalues.
0[ ]-d d
[-2 π β 1
]2 π β 1
β¦β
![Page 142: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/142.jpg)
Main Theme
Reduced the existence of a good matrix to:
1. Proving real-rootedness of an
expected polynomial.
2. Bounding roots of the expected
polynomial.
![Page 143: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/143.jpg)
Main Theme
Reduced the existence of a good matrix to:1. Proving real-rootedness of an
expected polynomial. (rank-1 structure + real stability)
2. Bounding roots of the expectedpolynomial.
(matching poly + combinatorics)
![Page 144: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/144.jpg)
Beyond complete graphs
Unweighted sparsifiers of general graphs?
![Page 145: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/145.jpg)
G
H
Beyond complete graphs
1O(n)
![Page 146: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/146.jpg)
G
H
Weights are Required in General
1O(n)
This edge has high resistance.
![Page 147: Graph Sparsification III: Ramanujan Graphs, Lifts, and ...Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons](https://reader030.vdocument.in/reader030/viewer/2022040520/5e798e692a40a01aa05d4193/html5/thumbnails/147.jpg)
What if all edges are equally important?
?G
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Unweighted Decomposition Thm.
G H1
H2
Theorem [MSSβ13]: If all edges have resistance π(π/π), there is a partition of G into unweighted 1 + π-
sparsifiers, each with ππ
π2 edges.
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Unweighted Decomposition Thm.
G H1
H2
Theorem [MSSβ13]: If all edges have resistance β€ πΌ, there is a partition of G into unweighted π(1)-sparsifiers, each with π ππΌ edges.
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Unweighted Decomposition Thm.
G H1
H2
Theorem [MSSβ13]: If all edges have resistance πΌ, there is a partition of G into two unweighted 1 + πΌ-
approximations, each with half as many edges.
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Unweighted Decomposition Thm.
Theorem [MSSβ13]: Given any vectors π π£ππ£ππ = πΌ and
π£π β€ π, there is a partition into approximately
1/2-spherical quadratic forms, each πΌ
2Β± π π .
π
π£ππ£ππ = πΌ
π
π£ππ£ππ β
πΌ
2
π
π£ππ£ππ β
πΌ
2
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Unweighted Decomposition Thm.
Theorem [MSSβ13]: Given any vectors π π£ππ£ππ = πΌ and
π£π β€ π, there is a partition into approximately
1/2-spherical quadratic forms, each πΌ
2Β± π π .
π
π£ππ£ππ = πΌ
π
π£ππ£ππ β
πΌ
2
π
π£ππ£ππ β
πΌ
2
Proof: Analyze expected charpoly of a random partition:
πΌdet π₯πΌ β π π£ππ£ππ det π₯πΌ β π π£ππ£π
π
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Unweighted Decomposition Thm.
Theorem [MSSβ13]: Given any vectors π π£ππ£ππ = πΌ and
π£π β€ π, there is a partition into approximately
1/2-spherical quadratic forms, each πΌ
2Β± π π .
π
π£ππ£ππ = πΌ
π
π£ππ£ππ β
πΌ
2
π
π£ππ£ππ β
πΌ
2
Other applications:Kadison-Singer ProblemUncertainty principles.
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Summary of Algorithms
Result Edges Weights Time
Spielman-Sβ08 O(nlogn) Yes O~(m)
Random Sampling with Effective Resistances
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Summary of Algorithms
Result Edges Weights Time
Spielman-Sβ08 O(nlogn) Yes O~(m)
Batson-Spielman-Sβ09 O(n) Yes π(π4)
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Summary of Algorithms
Result Edges Weights Time
Spielman-Sβ08 O(nlogn) Yes O~(m)
Batson-Spielman-Sβ09 O(n) Yes π(π4)
Marcus-Spielman-Sβ13 O(n) No π(2π)
πΌdet π₯πΌ β
π
π£ππ£ππ det π₯πΌ β
π
π£ππ£ππ
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Open Questions
Non-bipartite graphs
Algorithmic construction
(computing generalized ππΊ is hard)
More general uses of interlacing families
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Open Questions
Nearly linear time algorithm for 4π/π2 size sparsifiers
Improve to 2π/π2 for graphs?
Fast combinatorial algorithm for approximating resistances