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Matroid Bases andMatrix Concentration
Nick Harvey University of British Columbia
Joint work with Neil Olver (Vrije Universiteit)
Scalar concentration inequalities
Theorem: [Chernoff / Hoeffding Bound]Let Y1,…,Ym be independent, non-negative scalar random variables.Let Y=i Yi and ¹=E [ Y ]. Suppose Yi · 1 a.s. Then
Scalar concentration inequalities
Theorem: [Panconesi-Srinivasan ‘92, Dubhashi-Ranjan ‘96, etc.]Let Y1,…,Ym be negatively dependent, non-negative scalar rvs.Let Y=i Yi and ¹=E [ Y ]. Suppose Yi · 1 a.s. Then
Negative cylinder dependence: Yi 2 {0,1},
Stronger notions: negative association, determinantal distributions,strongly Rayleigh measures, etc.
Matrix concentration inequalities
Theorem: [Tropp ‘12, etc.]Let Y1,…,Ym be independent, PSD matrices of size nxn.Let Y=i Yi and M=E [ Y ]. Suppose ¹¢Yi ¹ M a.s. Then
Extensions of Chernoff Bounds
Independent Negative Dependent
Scalars Chernoff-Hoeffding
Panconesi-Srinivasan, etc.
Matrices Tropp, etc. ?
This talk: a special case of the missing common generalization, where the negatively dependent distribution is a certainrandom walk in a matroid base polytope.
Negative Dependence
Arises in many natural scenarios.
• Random spanning trees: Let Ye indicate if edge e is in tree.
Knowing that e2T decreases probability that f2T
e f
Negative Dependence
Arises in many natural scenarios.
• Random spanning trees: Let Ye indicate if edge e is in tree.
• Balls and bins: Let Yi be number of balls in bin i.
• Sampling without replacement, random permutations,random cluster models, etc.
Thin trees
A spanning tree T is ®-thin if |±T(S)| · ®¢|±G(S)| 8S
Global connectivity: K = min {|±G(S)| : ;(S(V }
Conjecture [Goddyn ’80s]: Every n-vertex graph has an®-thin tree with ®=O(1/K).
Would have deep consequences in graph theory.
S S
Cut ±(S) = { edge st : s2S, tS }
Thin trees
A spanning tree T is ®-thin if |±T(S)| · ®¢|±G(S)| 8S
Global connectivity: K = min {|±G(S)| : ;(S(V }
Theorem [Asadpour et al ‘10]: Every n-vertex graph has an®-thin spanning tree with ®= .
Uses negative dependence and Chernoff bounds.
S S
Cut ±(S) = { edge st : s2S, tS }
Asymmetric Traveling Salesman Problem[Julia Robinson, 1949]
• Let D=(V,E,w) be a weighted, directed graph.
• Goal: Find a tour sequence v1,v2,…,vk=v1 of vertices thatvisits every vertex in V at least once,has vivi+12E for every i,and minimizes total weight §1·i·k
w(vivi+1).
• Let D=(V,E,w) be a weighted, directed graph.
• Goal: Find a tour sequence v1,v2,…,vk=v1 of vertices thatvisits every vertex in V at least once,has vivi+12E for every i,and minimizes total weight §1·i·k w(vivi+1).
• Reduction [Oveis Gharan, Saberi ‘11]: If you can efficiently find an ®/K-thin spanning tree in any n-vertex graph, then you can find a tour whose weight is within O(®) of optimal.
Asymmetric Traveling Salesman Problem[Julia Robinson, 1949]
Graph Laplacians
Lbc =
0 0 0 0
0 1 -1 0
0 -1 1 0
0 0 0 0
a
b
c
d
a b c d
a
b
dc
Laplacian of edge bc
Graph Laplacians
LG = §e2E Le =
2 -1 -1
-1 2 -1
-1 -1 3 -1
-1 1
a
b
c
d
a b c d
degree of node
-1 for every edge
a
b
dc
Laplacian of graph G
Spectrally-thin trees
A spanning tree T is ®-spectrally-thin if LT ¹ ®¢LG
Effective Resistance from s to t: Rst = voltage difference when a 1-amp current source placed between s and t
Theorem [Harvey-Olver '14]: Every n-vertex graph has an®-spectrally-thin spanning tree with ®= .
Uses matrix concentration bounds. Algorithmic.
5 -1 -1 -1 -1 -14 -1 -1 -1 -1
-1 -1 6 -1 -1 -1 -1-1 5 -1 -1 -1 -1
-1 -1 -1 7 -1 -1 -1 -1-1 -1 -1 5 -1 -1-1 -1 -1 5 -1 -1
-1 -1 -1 -1 6 -1 -1-1 -1 -1 -1 -1 5
-1 -1 -1 -1 -1 -1 6
6 -1 -55 -1 -3 -1
-1 2 -18 -8
-1 2 -11 -1
-3 -1 5 -12 -1 -1
-5 -1 -1 -1 8-1 -8 -1 10
Spectrally-thin trees
A spanning tree T is ®-spectrally-thin if LT ¹ ®¢LG
Effective Resistance from s to t: Rst = voltage difference when a 1-amp current source placed between s and t
Theorem: Every n-vertex graph has an®-spectrally-thin spanning tree with ®= .
Follows from Kadison-Singer solution of MSS'13. Not algorithmic.
5 -1 -1 -1 -1 -14 -1 -1 -1 -1
-1 -1 6 -1 -1 -1 -1-1 5 -1 -1 -1 -1
-1 -1 -1 7 -1 -1 -1 -1-1 -1 -1 5 -1 -1-1 -1 -1 5 -1 -1
-1 -1 -1 -1 6 -1 -1-1 -1 -1 -1 -1 5
-1 -1 -1 -1 -1 -1 6
6 -1 -55 -1 -3 -1
-1 2 -18 -8
-1 2 -11 -1
-3 -1 5 -12 -1 -1
-5 -1 -1 -1 8-1 -8 -1 10
Asymmetric Traveling Salesman Problem
• Recent breakthrough: [Ansari, Oveis-Gharan Dec 2014]Show how to build on the O(1)-spectrally-thin tree resultto approximate optimal weight of an ATSP solution to within poly(log log n) of optimal.
• But, no algorithm to find the actual sequence of vertices!
Our Main ResultLet P½[0,1]m be a matroid base
polytope (e.g., convex hull of characteristic
vectors of spanning trees)Let A1,…, Am be PSD matrices of size
nxn.Define and Q;.There is an extreme point Â(S) of P with
Our Main ResultLet P½[0,1]m be a matroid base polytope.Let A1,…, Am be PSD matrices of size nxn.
Define and Q;.There is an extreme point Â(S) of P with
What is dependence on ®?• Easy: ® ¸ 1.5, even with n=2.• Standard random matrix theory: ® = O(log
n).• Our result: • Ideally: ®<2. This would solve Kadison-
Singer problem.• MSS ‘13: Solved Kadison-Singer, achieve ®
= O(1).
Our Main ResultLet P½[0,1]m be a matroid base polytope.Let A1,…, Am be PSD matrices of size nxn.
Define and Q;.There is an extreme point Â(S) of P with ,
Furthermore,• there is a random process that starts at any
x02Q and terminates after m steps at such a point Â(S), whp.
• each step of this process can be performed algorithmically.
• the entire process can be derandomized.
Pipage rounding[Ageev-Svirideno ‘04, Srinivasan ‘01, Calinescu et al. ‘07, Chekuri et al. ‘09]
Let P be any matroid polytope.
Given fractional xFind coordinates a and b s.t. linez x + z ( ea – eb ) stays in current face
Find two points where line leaves P
Randomly choose one of thosepoints s.t. expectation is x
Repeat until x = ÂT is integral
x is a martingale: expectation of final ÂT is original fractional x.
ÂT1
ÂT2
ÂT3
ÂT4
ÂT5
ÂT6
x
Definition: “Pessimistic Estimator”Let E µ {0,1}m be an event. Let D(x) be the product distribution on {0,1}m with expectation x.Then g : [0,1]m ! R is a pessimistic estimator for E if
Example: If E is the event { x : wT x>t }
then Chernoff bounds give the pessimistic estimator
Pessimistic estimators
Definition:A functionf : Rm ! R is concave under swaps if z ! f( x + z(ea-eb) ) is concave 8x2P, 8a, b2[m].
Example: is concave under swaps.
Pipage Rounding:Let X0 be initial point and ÂT be final point visited by pipage rounding.
Claim: If f concave under swaps then E[f(ÂT)] · f(X0). [by Jensen]
Pessimistic Estimators:
Let E be an event and g a pessimistic estimator for E.Claim: Suppose g is concave under swaps. Then Pr[ ÂT 2
E ] · g(X0).
Concavity under swaps
Matrix Pessimistic Estimators
Main Technical Result: gt,µ is concave under swaps.
Special case of Tropp ‘12: Let A1,…,Am be nxn PSD
matrices.
Let D(x) be the product distribution on {0,1}m with expectation x.Let Suppose ¹¢Ai ¹ M.
Let
Then
and .
) Tropp’s bound for independent sampling also achieved by pipage rounding
Pessimistic estimator
Our Variant of Lieb’s Theorem:
PD
Questions
• Does Tropp’s matrix concentration bound hold in a negatively dependent scenario?
• Does our variant of Lieb’s theorem have other uses?
• O(maxe Re)-spectrally thin trees exist by MSS’13.Can they be constructed algorithmically?