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**Sparse Approximations**

Nick Harvey University of British Columbia TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAA

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**Approximating Dense Objects by Sparse Objects**

Floor joists Wood Joists Engineered Joists

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**Approximating Dense Objects by Sparse Objects**

Bridges Masonry Arch Truss Arch

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**Approximating Dense Objects by Sparse Objects**

Bones Human Femur Robin Bone

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Mathematically Can an object with many pieces be approximately represented by fewer pieces? Independent random sampling usually does well Theme of this talk: When can we beat random sampling? 6 -1 4 5 7 6 -1 5 -3 2 8 1 Dense Matrix Sparse Matrix Dense Graph Sparse Graph

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**Talk Outline Vignette #1: Discrepancy theory**

Vignette #2: Singular values and eigenvalues Vignette #3: Graphs Theorem on “Spectrally Thin Trees”

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Discrepancy Given vectors v1,…,vn2Rd with kvikp bounded. Want y2{-1,1}n with ki yivikq small. Eg1: If kvik1·1 then E ki yi vik1 · Eg2: If kvik1·1 then 9y s.t. ki yi vik1 · Spencer ‘85: Partial Coloring + Entropy Method Gluskin ‘89: Sidak’s Lemma Giannopoulos ‘97: Partial Coloring + Sidak Bansal ‘10: Brownian Motion + Semidefinite Program Bansal-Spencer ‘11: Brownian Motion + Potential function Lovett-Meka ‘12: Brownian Motion Non-algorithmic Algorithmic

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Discrepancy Given vectors v1,…,vn2Rd with kvikp bounded. Want y2{-1,1}n with ki yivikq small. Eg1: If kvik1·1 then E ki yi vik1 · Eg2: If kvik1·1 then 9y s.t. ki yi vik1 · Eg3: If kvik1·¯, kvik1·±, and ki vik1·1, then 9y with ki yi vik1 · Harvey ’13: Using Lovasz Local Lemma. Question: Can log(±/¯2) factor be improved?

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**Talk Outline Vignette #1: Discrepancy theory**

Vignette #2: Singular values and eigenvalues Vignette #3: Graphs Theorem on “Spectrally Thin Trees”

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**Partitioning sums of rank-1 matrices**

Let v1,…,vn2Rd satisfy i viviT=I and kvik2·±. Want y2{-1,1}n with ki yiviviTk2 small. Random sampling: E ki yiviviTk2 · Rudelson ’96: Proofs using majorizing measures, then nc-Khintchine Marcus-Spielman-Srivastava ’13: 9y2{-1,1}n with ki yiviviTk2 · 2

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**Partitioning sums of matrices**

Given dxd symmetric matrices M1,…,Mn2Rd with i Mi=I and kMik2·±. Want y2{-1,1}n with ki yiMik2 small. Random sampling: E ki yiMik2 · Also follows from nc-Khintchine. Ahlswede-Winter ’02: Using matrix moment generating function. Tropp ‘12: Using matrix cumulant generating function.

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**Partitioning sums of matrices**

Given dxd symmetric matrices M1,…,Mn2Rd with i Mi=I and kMik2·±. Want y2{-1,1}n with ki yiMik2 small. Random sampling: E ki yiMik2 · Question: 9y2{-1,1}n with ki yiMik2 · ? Conjecture: Suppose i Mi=I and kMikSch-1·±. 9y2{-1,1}n with ki yiMik2 · ? MSS ’13: Rank-one case is true Harvey ’13: Diagonal case is true (ignoring log(¢) factor) False!

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**Partitioning sums of matrices**

Given dxd symmetric matrices M1,…,Mn2Rd with i Mi=I and kMik2·±. Want y2{-1,1}n with ki yiMik2 small. Random sampling: E ki yiMik2 · Question: Suppose only that kMik2·1. 9y2{-1,1}n with ki yiMik2 · ? Spencer/Gluskin: Diagonal case is true

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**Column-subset selection**

Given vectors v1,…,vn2Rd with kvik2=1. Let st.rank=n/ki viviTk2. Let 9y2{0,1}n s.t. i yi=k and (1-²)2 · ¸k( i yiviviT ). Spielman-Srivastava ’09: Potential function argument Youssef ’12: Let y2{0,1}n s.t. i yi=k, (1-²)2 · ¸k( i yiviviT ) and ¸1( i yiviviT ) · (1+²)2.

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**Column-subset selection up to the stable rank**

Given vectors v1,…,vn2Rd with kvik2=1. Let st.rank=n/ki viviTk2. Let For y2{0,1}n s.t. i yi=k, can we control ¸k( i yiviviT ) and ¸1( i yiviviT ) ? ¸k can be very small, say O(1/d). Rudelson’s theorem: can get ¸1 · O(log d) and ¸k>0. Harvey-Olver ’13: ¸1 · O(log d / log log d) and ¸k>0. MSS ‘13: If i viviT =I, can get ¸1 · O(1) and ¸k>0.

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**Talk Outline Vignette #1: Discrepancy theory**

Vignette #2: Singular values and eigenvalues Vignette #3: Graphs Theorem on “Spectrally Thin Trees”

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**weighted degree of node c**

Graph Laplacian 5 10 Graph with weights u: a c d 2 1 b a b c d Laplacian Matrix: a 7 -2 -5 3 -1 16 -10 10 negative of u(ac) b Lu = D-A = c weighted degree of node c d Effective Resistance from s to t: voltage difference when each edge e is a (1/ue)-ohm resistor and a 1-amp current source placed between s and t = (es-et)T Luy (es-et) Effective Conductance: cst = 1 / (effective resistance from s to t)

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**Spectral approximation of graphs**

Edge weights u Edge weights w 5 -1 4 6 7 6 -1 -5 5 -3 2 8 -8 1 10 Lu = Lw = ®-spectral sparsifier: Lu ¹ Lw ¹ ®¢Lu

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**Ramanujan Graphs Suppose Lu is complete graph on n vertices (ue=1 8e).**

Lubotzky-Phillips-Sarnak ’86: For infinitely many d and n, 9w2{0,1}E such that e we=dn/ (actually Lw is d-regular) and MSS ‘13: Holds for all d¸3, and all n=c¢2k. Friedman ‘04: If Lw is a random d-regular graph, then 8²>0 with high probability.

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Arbitrary graphs Spielman-Srivastava ’08: For any graph Lu with n=|V|, 9w2RE such that |support(w)| = O(n log(n)/²2) and Proof: Follows from Rudelson’s theorem MSS ’13: For any graph Lu with n=|V|, 9w2RE such that we 2 £(²2) ¢ N ¢ (effective conductance of e) |support(w)| = O(n/²2) and

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**Spectrally-thin trees**

Question: Let G be an unweighted graph with n vertices. Let C = mine (effective conductance of edge e). Want a subtree T of G with Equivalent to Goddyn’s Conjecture ‘85: There is a subtree T with Relates to conjectures of Tutte (‘54) on nowhere-zero flows, and to approximations of the traveling salesman problem.

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**Spectrally-thin trees**

Question: Let G be an unweighted graph with n vertices. Let C = mine (effective conductance of edge e). Want a subtree T of G with Rudelson’s theorem: Easily gives ® = O(log n). Harvey-Olver ‘13: ® = O(log n / log log n). Moreover, there is an efficient algorithm to find such a tree. MSS ’13: ® = O(1), but not algorithmic.

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**Talk Outline Vignette #1: Discrepancy theory**

Vignette #2: Singular values and eigenvalues Vignette #3: Graphs Theorem on “Spectrally Thin Trees”

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**Spectrally Thin Trees Proof overview:**

Given an (unweighted) graph G with eff. conductances ¸ C. Can find an unweighted tree T with Proof overview: Show independent sampling gives spectral thinness, but not a tree. ► Sample every edge e independently with prob. xe=1/ce Show dependent sampling gives a tree, and spectral thinness still works.

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Matrix Concentration Theorem: [Tropp ‘12] Let Y1,…,Ym be independent, PSD matrices of size nxn. Let Y=i Yi and Z=E [ Y ]. Suppose Yi ¹ R¢Z a.s. Then

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**Independent sampling Let Suppose Mi ¹ Z.**

Define sampling probabilities xe = 1/ce. It is known that e xe = n–1. Claim: Independent sampling gives T µ E with E [|T|]=n–1 and Theorem [Tropp ‘12]: Let M1,…,Mm be nxn PSD matrices. Let D(x) be a product distribution on {0,1}m with marginals x. Let Suppose Mi ¹ Z. Then Define Me = ce¢Le. Then Z = LG and Me ¹ Z holds. Setting ®=6 log n / log log n, we get whp. But T is not a tree! Yi = Xi Mi Y = \sum_i Yi Z = E[Y] = E[\sum_i Yi] = \sum_i E[Xi Mi] = \sum_i xi Mi. Assume that Yi <= R*Z as, which is equivalent to Mi <= R*Z a.s. It turns out that R = 1. Laplacian of the single edge e Properties of conductances used

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**Spectrally Thin Trees Proof overview:**

Given an (unweighted) graph G with eff. conductances ¸ C. Can find an unweighted tree T with Proof overview: Show independent sampling gives spectral thinness, but not a tree. ► Sample every edge e independently with prob. xe=1/ce Show dependent sampling gives a tree, and spectral thinness still works. ► Run pipage rounding to get tree T with Pr[ e2T ] = xe = 1/ce

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Pipage rounding [Ageev-Svirideno ‘04, Srinivasan ‘01, Calinescu et al. ‘07, Chekuri et al. ‘09] Let P be any matroid polytope. E.g., convex hull of characteristic vectors of spanning trees. 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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**Pipage rounding and concavity**

Say f : Rm ! R is concave under swaps if z ! f( x + z(ea-eb) ) is concave 8x2P, 8a, b2[m]. 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). [Jensen] Let E µ {0,1}m be an event. Let g : [0,1]m ! R be a pessimistic estimator for E, i.e., Claim: Suppose g is concave under swaps. Then Pr[ ÂT 2 E ] · g(X0).

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**Chernoff Bound Chernoff Bound: Fix any w, x 2 [0,1]m and let ¹ = wTx.**

Define Then, Claim: gt,µ is concave under swaps [Elementary calculus] Let X0 be initial point and ÂT be final point visited by pipage rounding. Let ¹ = wTX0. Then Bound achieved by independent sampling also achieved by pipage rounding

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**Matrix Pessimistic Estimators**

Theorem [Tropp ‘12]: Let M1,…,Mm be nxn PSD matrices. Let D(x) be a product distribution on {0,1}m with marginals x. Let Suppose Mi ¹ Z. Let Then and Pessimistic estimator Main Theorem: gt,µ is concave under swaps. Bound achieved by independent sampling also achieved by pipage rounding

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**Spectrally Thin Trees Proof overview:**

Given an (unweighted) graph G with eff. conductances ¸ C. Can find an unweighted tree T with Proof overview: Show independent sampling gives spectral thinness, but not a tree. ► Sample every edge e independently with prob. xe=1/ce Show dependent sampling gives a tree, and spectral thinness still works. ► Run pipage rounding to get tree T with Pr[ e2T ] = xe = 1/ce

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Matrix Analysis Matrix concentration inequalities are usually proven via sophisticated inequalities in matrix analysis Rudelson: non-commutative Khinchine inequality Ahlswede-Winter: Golden-Thompson inequality if A, B symmetric, then tr(eA+B) · tr(eA eB). Tropp: Lieb’s concavity inequality [1973] if A, B Hermitian and C is PD, then z ! tr exp( A + log(C+zB) ) is concave. Key technical result: new variant of Lieb’s theorem if A Hermitian, B1, B2 are PSD, and C1, C2 are PD, then z ! tr exp( A + log(C1+zB1) + log(C2–zB2) ) is concave.

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Questions Can Spencer/Gluskin theorem be extended to matrices? Can MSS’13 be made algorithmic? Can MSS’13 be extended to large-rank matrices? O(1)-spectrally thin trees exist. Can one be found algorithmically? Are O(1)-spectrally thin trees helpful for Goddyn’s conjecture?

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