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**Matroid Bases and Matrix Concentration**

Nick Harvey University of British Columbia Joint work with Neil Olver (Vrije Universiteit) TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAA

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**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

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**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.

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**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

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**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 certain random walk in a matroid base polytope.

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**Knowing that e2T decreases probability that f2T**

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

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**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.

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Thin trees S S Cut ±(S) = { edge st : s2S, tS } 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. Let me begin the talk by motivating what sparsification is and why it’s important. Sparsification is a concept that’s familiar from our daily lives: you want to take some object that’s dense, and replace by an object that’s sparser, but almost as good as the original object. Here’s an example I learned about when I bought my first house a few years ago: floor joists used to be made of solid wood, but now they’re “engineered”, often with a truss structure, often with other materials. Another familiar example is image compression: we express the image in a particular basis (perhaps a wavelet basis), then keep the most important components and throw away the less important ones.

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Thin trees S S Cut ±(S) = { edge st : s2S, tS } 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. Let me begin the talk by motivating what sparsification is and why it’s important. Sparsification is a concept that’s familiar from our daily lives: you want to take some object that’s dense, and replace by an object that’s sparser, but almost as good as the original object. Here’s an example I learned about when I bought my first house a few years ago: floor joists used to be made of solid wood, but now they’re “engineered”, often with a truss structure, often with other materials. Another familiar example is image compression: we express the image in a particular basis (perhaps a wavelet basis), then keep the most important components and throw away the less important ones.

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**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 that visits every vertex in V at least once, has vivi+12E for every i, and minimizes total weight §1·i·k w(vivi+1).

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**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 that visits 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.

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**Graph Laplacians Laplacian of edge bc Lbc = a c d b a b c d a 1 -1 b c**

1 -1 b Let me begin the talk by motivating what sparsification is and why it’s important. Sparsification is a concept that’s familiar from our daily lives: you want to take some object that’s dense, and replace by an object that’s sparser, but almost as good as the original object. Here’s an example I learned about when I bought my first house a few years ago: floor joists used to be made of solid wood, but now they’re “engineered”, often with a truss structure, often with other materials. Another familiar example is image compression: we express the image in a particular basis (perhaps a wavelet basis), then keep the most important components and throw away the less important ones. Lbc = c d

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**Graph Laplacians Laplacian of graph G LG = §e2E Le = -1 for every edge**

b Laplacian of graph G a b c d a 2 -1 3 1 -1 for every edge b LG = §e2E Le = Let me begin the talk by motivating what sparsification is and why it’s important. Sparsification is a concept that’s familiar from our daily lives: you want to take some object that’s dense, and replace by an object that’s sparser, but almost as good as the original object. Here’s an example I learned about when I bought my first house a few years ago: floor joists used to be made of solid wood, but now they’re “engineered”, often with a truss structure, often with other materials. Another familiar example is image compression: we express the image in a particular basis (perhaps a wavelet basis), then keep the most important components and throw away the less important ones. c degree of node d

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**Spectrally-thin trees**

5 -1 4 6 7 6 -1 -5 5 -3 2 8 -8 1 10 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. Let me begin the talk by motivating what sparsification is and why it’s important. Sparsification is a concept that’s familiar from our daily lives: you want to take some object that’s dense, and replace by an object that’s sparser, but almost as good as the original object. Here’s an example I learned about when I bought my first house a few years ago: floor joists used to be made of solid wood, but now they’re “engineered”, often with a truss structure, often with other materials. Another familiar example is image compression: we express the image in a particular basis (perhaps a wavelet basis), then keep the most important components and throw away the less important ones.

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**Spectrally-thin trees**

5 -1 4 6 7 6 -1 -5 5 -3 2 8 -8 1 10 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. Let me begin the talk by motivating what sparsification is and why it’s important. Sparsification is a concept that’s familiar from our daily lives: you want to take some object that’s dense, and replace by an object that’s sparser, but almost as good as the original object. Here’s an example I learned about when I bought my first house a few years ago: floor joists used to be made of solid wood, but now they’re “engineered”, often with a truss structure, often with other materials. Another familiar example is image compression: we express the image in a particular basis (perhaps a wavelet basis), then keep the most important components and throw away the less important ones.

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**Asymmetric Traveling Salesman Problem**

Recent breakthrough: [Ansari, Oveis-Gharan Dec 2014] Show how to build on the O(1)-spectrally-thin tree result to 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!

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Our Main Result Let 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

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**Our Main Result Let 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).

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**Our Main Result Let 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.

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**Pipage rounding Let P be any matroid polytope. Given fractional x**

[Ageev-Svirideno ‘04, Srinivasan ‘01, Calinescu et al. ‘07, Chekuri et al. ‘09] Let P be any matroid polytope. Given fractional x Find coordinates a and b s.t. line z x + z ( ea – eb ) stays in current face Find two points where line leaves P Randomly choose one of those points s.t. expectation is x Repeat until x = ÂT is integral x is a martingale: expectation of final ÂT is original fractional x. ÂT1 ÂT6 ÂT2 x ÂT3 ÂT5 ÂT4

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**Pessimistic estimators**

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

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Concavity under swaps 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).

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**Matrix Pessimistic Estimators**

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 Pessimistic estimator Main Technical Result: gt,µ is concave under swaps. ) Tropp’s bound for independent sampling also achieved by pipage rounding

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**Our Variant of Lieb’s Theorem:**

PD Our Variant of Lieb’s Theorem:

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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?

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