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Graph Sparsification II:Rank one updates,
Interlacing, and Barriers
Nikhil Srivastava
Simons Institute August 26, 2014
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Definition. 𝐻 = (𝑉, 𝐹, 𝑢) is a 𝜅 −approximation of 𝐺 =𝑉, 𝐸, 𝑤 if:
Theorem. Every 𝐺 has a 1 + 𝜖 −approximation 𝐻with 𝑂(𝑛 log 𝑛 𝜖2) edges. There is a nearly linear time algorithm which finds it.
Previous Lecture
𝐿𝐻 ≼ 𝐿𝐺 ≼ 𝜅 ⋅ 𝐿𝐻
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|EH| = O(dn)
There is no log(n) here…
G=Kn H = random d-regular x (n/d)
|EG| = O(n2)
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Proof: Approximating the Identity
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Tool: The Matrix Chernoff Bound
Suppose 𝑋1, … , 𝑋𝑘 are i.i.d. random 𝑛 × 𝑛 matrices with
0 ≼ 𝑋𝑖 ≼ 𝑀 ⋅ 𝐼 and 𝔼𝑋𝑖 = 𝐼.
Then
ℙ1
𝑘
𝑖
𝑋𝑖 − 𝐼 ≥ 𝜖 ≤ 2𝒏 exp −𝑘𝜖2
4𝑀
Shows 𝑂𝑛 log 𝑛
𝜖2samples suffice in 𝑹𝑛.
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Tool: The Matrix Chernoff Bound
Suppose 𝑋1, … , 𝑋𝑘 are i.i.d. random 𝑛 × 𝑛 matrices with
0 ≼ 𝑋𝑖 ≼ 𝑛 ⋅ 𝐼 and 𝔼𝑋𝑖 = 𝐼.
then 𝑂𝑛 log 𝑛
𝜖2samples suffice in 𝑹𝑛.
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Tool: The Matrix Chernoff Bound
Suppose 𝑋1, … , 𝑋𝑘 are i.i.d. random 𝑛 × 𝑛 matrices with
0 ≼ 𝑋𝑖 ≼ 𝑛 ⋅ 𝐼 and 𝔼𝑋𝑖 = 𝐼.
then 𝑂𝑛 log 𝑛
𝜖2samples suffice in 𝑹𝑛.
Tight example:
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Tool: The Matrix Chernoff Bound
Suppose 𝑋1, … , 𝑋𝑘 are i.i.d. random 𝑛 × 𝑛 matrices with
0 ≼ 𝑋𝑖 ≼ 𝑛 ⋅ 𝐼 and 𝔼𝑋𝑖 = 𝐼.
then 𝑂𝑛 log 𝑛
𝜖2samples suffice in 𝑹𝑛.
Tight example:
Simple greedy algorithm gets 𝑂(𝑛): place next ball in emptiest bin
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This Lecture [Batson-Spielman-S’09]
Spectral Sparsification Theorem:
uses greedy approach.
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This is the goal
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Plan: Choose vectors one at a time
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Plan: Choose vectors one at a time
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Plan: Choose vectors one at a time
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Plan: Choose vectors one at a time
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Plan: Choose vectors one at a time
…..
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Plan: Choose vectors one at a time
…..
𝜆 ∈ [1, 𝜅]
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𝜆 ∈ [1, 𝜅]
Plan: Choose vectors one at a time
…..
Basic Question: What does a rank one update do to the eigenvalues?
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What happens when you add a vector?
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Interlacing (Cauchy, 1800s)
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The Characteristic Polynomial
Characteristic Polynomial:
where 𝜆1, … , 𝜆𝑛 = eigs(A).
𝑝𝐴 𝑥 =
𝑖
(𝑥 − 𝜆𝑖)
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Proof of Interlacing I
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Proof of Interlacing II
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Proof of Interlacing III
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The Characteristic Polynomial
Characteristic Polynomial:
Matrix-Determinant Lemma:
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The Characteristic Polynomial
Characteristic Polynomial:
Matrix-Determinant Lemma:are zeros of this.
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Physical model of interlacing
i = positive unit charges
resting at barriers on a slope
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Physical model of interlacing
<v,ui>2 =charges added
to barriers
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Physical model of interlacing
Barriers repel eigs.
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Physical model of interlacing
Barriers repel eigs.
gravity
Inverse law of repulsion
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Physical model of interlacing
Barriers repel eigs.
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Examples
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Ex1: All weight on u1
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Ex1: All weight on u1
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Ex1: All weight on u1
Pushed up against next barrier
resting due to gravity
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Ex2: Equal weight on u1 ,u2
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Ex2: Equal weight on u1 ,u2
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Ex2: Equal weight on u1 ,u2
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Ex3: Equal weight on all u1, u2 , …un
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Ex3: Equal weight on all u1, u2 , …un
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Adding a balanced vector
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Consider a random vector
If
thus a random vector has the same expected projection in every direction i :
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The Expected Characteristic Poly.
𝐴(0) = 0
𝑝(0) = 𝑥𝑛
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The Expected Characteristic Poly.
𝐴(0) = 0
𝑝(0) = 𝑥𝑛
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The Expected Characteristic Poly.
𝐴(1) = 𝑣𝑣𝑇
𝔼𝑝(1) = 𝑝(0) −1
𝑚
𝜕
𝜕𝑥𝑝 0
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The Expected Characteristic Poly.
𝐴(1) = 𝑣𝑣𝑇
𝔼𝑝(1) = 𝑝(0) −1
𝑚
𝜕
𝜕𝑥𝑝 0 = 𝑥𝑛 −
𝑛
𝑚𝑥𝑛−1
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The Expected Characteristic Poly.
𝐴(2) = 𝐴(1) + 𝑣𝑣𝑇
𝔼𝑝(2) = 1 −1
𝑚
𝜕
𝜕𝑥𝑝 1 = 1 −
1
𝑚
𝜕
𝜕𝑥
2
𝑥𝑛
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The Expected Characteristic Poly.
𝐴(3) = 𝐴(2) + 𝑣𝑣𝑇
𝔼𝑝(3) = 1 −1
𝑚
𝜕
𝜕𝑥
3
𝑥𝑛
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Ideal proof
𝑟𝑜𝑜𝑡𝑠(𝔼𝑝 𝑘 )
𝔼𝑝 𝑘 = 1 −1
𝑚
𝜕
𝜕𝑥
𝑘
𝑥𝑛
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Ideal proof
𝑟𝑜𝑜𝑡𝑠(𝔼𝑝 𝑘 )
𝔼𝑝 𝑘 = 1 −1
𝑚
𝜕
𝜕𝑥
𝑘
𝑥𝑛
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Ideal proof
𝑟𝑜𝑜𝑡𝑠(𝔼𝑝 𝑘 )
𝔼𝑝 𝑘 = 1 −1
𝑚
𝜕
𝜕𝑥
𝑘
𝑥𝑛
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Ideal proof
𝑟𝑜𝑜𝑡𝑠(𝔼𝑝 𝑘 )
𝔼𝑝 𝑘 = 1 −1
𝑚
𝜕
𝜕𝑥
𝑘
𝑥𝑛
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Ideal proof
𝔼𝑝 𝑘 = 1 −1
𝑚
𝜕
𝜕𝑥
𝑘
𝑥𝑛
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Ideal proof
𝔼𝑝 𝑘 = 1 −1
𝑚
𝜕
𝜕𝑥
𝑘
𝑥𝑛
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Ideal proof
𝔼𝑝 𝑘 = 1 −1
𝑚
𝜕
𝜕𝑥
𝑘
𝑥𝑛
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Ideal proof
𝔼𝑝 𝑘 = 1 −1
𝑚
𝜕
𝜕𝑥
𝑘
𝑥𝑛
𝑝(𝑘)= Laguerre poly ℒ𝑘
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𝑝(𝑘)= Laguerre poly ℒ𝑘
Ideal proof
𝔼𝑝 𝑘 = 1 −1
𝑚
𝜕
𝜕𝑥
𝑘
𝑥𝑛
after 4𝑛/𝜖2 steps.
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This is not real
𝔼𝑝 𝑘 = 1 −1
𝑚
𝜕
𝜕𝑥
𝑘
𝑥𝑛
Problem: 𝑟𝑜𝑜𝑡𝑠 𝔼𝑝(𝑘) ≠ 𝔼𝑟𝑜𝑜𝑡𝑠(𝑝(𝑘))
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𝔼𝑝 𝑘 = 1 −1
𝑚
𝜕
𝜕𝑥
𝑘
𝑥𝑛 4𝑛/𝜖2
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𝔼𝑝 𝑘 = 1 −1
𝑚
𝜕
𝜕𝑥
𝑘
𝑥𝑛 4𝑛/𝜖2
Theorem. If we allow arbitrary scalings of the vectors, then
there is a greedy algorithm which matches what happens in the above dream.
…..
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𝔼𝑝 𝑘 = 1 −1
𝑚
𝜕
𝜕𝑥
𝑘
𝑥𝑛 4𝑛/𝜖2
Theorem. If we allow arbitrary scalings of the vectors, then
there is a greedy algorithm which matches what happens in the above dream.
…..
Keep track of total repulsion
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End Result [Batson-Spielman-S’09]
Spectral Sparsification Theorem:
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Actual Proof(for 6n vectors, 13-approx)
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Steady progress by moving barriers
0 n-n
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Step 1
0 n-n
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Step 1
0 n-n
0 n-n
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Step 1
0 n-n
0
+1/3 +2
-n+1/3 n+2
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Step 1
0 n-n
0
+1/3 +2
-n+1/3 n+2
tighter constraint
looser constraint
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Step i+1
0
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Step i+1
0
+1/3 +2
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Step i+1
0
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Step i+1
0
+1/3 +2
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Step i+1
0
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Step i+1
0
+1/3 +2
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Step i+1
0
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Step i+1
0
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Step i+1
0
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Step i+1
0
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Step 6n
0 … 13nn
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Step 6n
0 … n
13-approximation with 6n vectors.
13n
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Problem
need to show that an appropriate
always exists.
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Problem
need to show that an appropriate
always exists.
Hope: vectors are well-spread: there must be
one which is well-behaved.
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Bad: Accumulation of Eigenvalues
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Bad: Accumulation of Eigenvalues
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Bad: Accumulation of Eigenvalues
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Bad: Accumulation of Eigenvalues
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Bad: Accumulation of Eigenvalues
is not strong enough to do the induction.
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Bad: Accumulation of Eigenvalues
is not strong enough to do the induction.
need a better way to measure
quality of eigenvalues.
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The Upper Barrier
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The Upper Barrier
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No i within dist. 1No 2 i within dist. 2No 3 i within dist. 3
.
.No k i within dist. k
The Upper Barrier
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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
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The Lower Barrier
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The Beginning
0 n-n
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The Beginning
0 n-n
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Step i+1
0
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Step i+1
0
+1/3 +2
Lemma.
can always choose
so that potentials do not increase
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Step i+1
0
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Step i+1
0
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Step i+1
0
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Step i+1
0
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Step 6n
0 … 13nn
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Step 6n
0 … n
13-approximation with 6n vectors.
13n
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Goal
+1/3 +2
Lemma.
can always choose so
that both potentials do not increase.
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The Right Question
“Which vector should we add?”
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The Right Question
“Which vector should we add?”
“Given a vector, how much of it can we add?”
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Upper Barrier Update
Add & set
+2
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Upper Barrier Update
Add & set
+2
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Upper Barrier Update
Add & set
Sherman-Morrisson
+2
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Upper Barrier Update
Add & set
+2
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Upper Barrier Update
Add & set
+2
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Upper feasibility condition
Rearranging:
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Rearranging:
Upper feasibility condition
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Rearranging:
s=0 always feasible
Upper feasibility condition
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Lower Feasibility
Similarly:
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Goal
Show that we can always add some vector while respecting both barriers.
+1/3 +2
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There is always a vector with
Both BarriersGoal
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There is always a vector with
Both Barrierscan add must add
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There is always a vector with
Both Barriers
Then, can squeeze scaling factor in between:
can add must add
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Average over all veGoal
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Average over all veGoal
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Average over all veGoal
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Average over all veGoal
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Bounding Tr(UA)
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Bounding Tr(UA)
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Bounding Tr(UA)
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Bounding Tr(UA)
induction
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Bounding Tr(UA)
induction
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Bounding Tr(UA)
induction
convexity
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Bounding Tr(UA)
induction
convexity
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Taking AveragesGoal
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Taking Averages
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Taking Averages
=3/22
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Taking Averages
=3/2
2
2
1/3
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Taking Averages
=3/2
2
2
1/3
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Step i+1
0
+1/3 +2
Lemma.
can always choose
so that potentials do not increase.
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Step i+1
0
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Step i+1
0
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Step i+1
0
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Step i+1
0
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Step 6n
0 … 13nn
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Step 6n
0 … n
13-approximation with 6n vectors.
13n
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Done!
Spectral Sparsification Theorem:
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Fixing steps and tightening parameters gives
(zeros of Laguerre polynomials).
This is within a factor of 2 of the optimal
Ramanujan Bound [LPS, Alon-Boppana].
Nearly Optimal bound
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Why does this work?
Efficiently bounds
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Why does this work?
Efficiently bounds
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guarantees good vector
Why does this work?
Efficiently bounds
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guarantees good vector
Why does this work?
Efficiently bounds
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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
Instead of directly reasoning about 𝜆𝑖(𝐴), reason about 𝑧𝐼 − 𝐴 −1.
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Open Questions
Fast algorithm currently O(n^4)
Optimization proof?
More applications
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|EH| = O(dn)
There are no weights here…
G=Kn H = random d-regular x (n/d)
|EG| = O(n2)
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|EH| = O(dn)
And off by a factor of 2
G=Kn H = random d-regular x (n/d)
|EG| = O(n2)
We get 4𝑛/𝜖2
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Tomorrow
2/𝜖2 degree unweighted approximations for 𝐾𝑛“Ramanujan Graphs”
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Tomorrow
2/𝜖2 degree unweighted approximations for 𝐾𝑛“Ramanujan Graphs”
𝔼𝑝 𝑘 = 1 −1
𝑚
𝜕
𝜕𝑥
𝑘
𝑥𝑛 4𝑛/𝜖2
This is not a dream.