Interlacing Families II: Mixed Characteristic Polynomials and the - PowerPoint PPT Presentation

In One Dimension Main Theorem . Suppose 𝑤 1 , … , 𝑤 𝑛 ∈ 𝑺 1 are numbers w ith |𝑤 𝑗 | 2 ≤ 𝜗 2 and energy one 2 = 1. 𝑤 𝑗 𝑗 Then there is a partition 𝑈 1 ∪ 𝑈 2 such that each part has energy close to half 2 = 1 𝑤 𝑗 2 ± 5𝜗 𝑗∈𝑈 𝑘

In Higher Dimensions Main Theorem . Suppose 𝑤 1 , … , 𝑤 𝑛 ∈ 𝑺 𝑜 are vectors ||𝑤 𝑗 || ≤ 𝜗 and energy one in each direction: 𝑣, 𝑤 𝑗 2 = 1 ∀||𝑣|| = 1 𝑗 Then there is a partition 𝑈 1 ∪ 𝑈 2 such that each part has energy close to half in each direction: 𝑣, 𝑤 𝑗 2 = 1 ∀||𝑣|| = 1 2 ± 5𝜗 𝑗∈𝑈 𝑘

In Higher Dimensions Main Theorem . Suppose 𝑤 1 , … , 𝑤 𝑛 ∈ 𝑺 𝑜 are vectors ||𝑤 𝑗 || ≤ 𝜗 and energy one in each direction: 𝑣, 𝑤 𝑗 2 = 1 ∀||𝑣|| = 1 𝑗 Then there is a partition 𝑈 1 ∪ 𝑈 2 such that each part has energy close to half in each direction: 𝑣, 𝑤 𝑗 2 = 1 ∀||𝑣|| = 1 2 ± 5𝜗 𝑗∈𝑈 𝑘 Optimal in high dim

Matrix Notation Given vectors 𝑤 1 , … , 𝑤 𝑛 write quadratic form as 𝑤 𝑗 , 𝑣 2 = 𝑣 𝑈 𝑈 𝑅 𝑣 = 𝑤 𝑗 𝑤 𝑗 𝑣 𝑗 𝑗

Matrix Notation Given vectors 𝑤 1 , … , 𝑤 𝑛 write quadratic form as 𝑤 𝑗 , 𝑣 2 = 𝑣 𝑈 𝑈 𝑅 𝑣 = 𝑤 𝑗 𝑤 𝑗 𝑣 𝑗 𝑗 Isotropy: 𝑈 = 𝐽 𝑜 𝑤 𝑗 𝑤 𝑗 𝑗

Matrix Notation Given vectors 𝑤 1 , … , 𝑤 𝑛 write quadratic form as 𝑤 𝑗 , 𝑣 2 = 𝑣 𝑈 𝑈 𝑅 𝑣 = 𝑤 𝑗 𝑤 𝑗 𝑣 𝑗 𝑗 Isotropy: 𝑈 = 𝐽 𝑜 𝑤 𝑗 𝑤 𝑗 𝑗 Comparision: 𝐵 ≼ 𝐶 ⟺ 𝑣 𝑈 𝐵𝑣 ≤ 𝑣 𝑈 𝐶 𝑣 ∀𝑣

Matrix Notation Main Theorem . Suppose 𝑤 1 , … , 𝑤 𝑛 ∈ 𝑺 𝑜 are vectors ||𝑤 𝑗 || ≤ 𝜗 and 𝑈 = 𝐽 𝑜 𝑤 𝑗 𝑤 𝑗 𝑗 Then there is a partition 𝑈 1 ∪ 𝑈 2 such that 1 1 𝑈 ≼ 2 − 5𝜗 𝐽 ≼ 𝑤 𝑗 𝑤 𝑗 2 + 5𝜗 𝐽 𝑗∈𝑈 𝑘

Matrix Notation Main Theorem . Suppose 𝑤 1 , … , 𝑤 𝑛 ∈ 𝑺 𝑜 are vectors ||𝑤 𝑗 || 2 ≤ 𝜗 and 𝑈 = 𝐽 𝑜 𝑤 𝑗 𝑤 𝑗 𝑗 Then there is a partition 𝑈 1 ∪ 𝑈 2 such that 1 1 𝑈 ≼ 2 − 5 𝜗 𝐽 ≼ 𝑤 𝑗 𝑤 𝑗 2 + 5 𝜗 𝐽 𝑗∈𝑈 𝑘

Unnormalized Version Suppose I get some vectors 𝑥 1 , … , 𝑥 𝑛 which are not isotropic: 𝑈 = 𝑋 ≽ 0 𝑥 𝑗 𝑥 𝑗 𝑗 Consider 𝑤 𝑗 ≔ 𝑋 − 1 2 𝑥 𝑗 and apply theorem to 𝑤 𝑗 . Normalized vectors have ||𝑤 𝑗 || 2 = ||𝑋 − 1 2 𝑥 𝑗 || 2 = 𝜗 Thm. gives 1 1 𝑈 ≼ 2 − 5 𝜗 𝐽 ≼ 𝑤 𝑗 𝑤 𝑗 2 + 5 𝜗 𝐽 𝑗∈𝑈 𝑘

Unnormalized Version Suppose I get some vectors 𝑥 1 , … , 𝑥 𝑛 which are not isotropic: 𝑈 = 𝑋 ≽ 0 𝑥 𝑗 𝑥 𝑗 𝑗 Consider 𝑤 𝑗 ≔ 𝑋 − 1 2 𝑥 𝑗 and apply theorem to 𝑤 𝑗 . Normalized vectors have ||𝑤 𝑗 || 2 = ||𝑋 − 1 2 𝑥 𝑗 || 2 = 𝜗 1 1 𝑋 −1 𝑈 𝑋 −1 2 ≼ 2 − 5 𝜗 𝐽 ≼ 2 𝑥 𝑗 𝑥 𝑗 2 + 5 𝜗 𝐽 𝑗∈𝑈 𝑘

Unnormalized Version Suppose I get some vectors 𝑥 1 , … , 𝑥 𝑛 which are not isotropic: 𝑈 = 𝑋 ≽ 0 𝑥 𝑗 𝑥 𝑗 𝑗 Consider 𝑤 𝑗 ≔ 𝑋 − 1 2 𝑥 𝑗 and apply theorem to 𝑤 𝑗 . Normalized vectors have ||𝑤 𝑗 || 2 = ||𝑋 − 1 2 𝑥 𝑗 || 2 = 𝜗 1 1 2 − 5 𝜗 𝐽 ≼ 𝑋 −1 𝑋 −1 𝑈 2 ≼ 𝑥 𝑗 𝑥 𝑗 2 + 5 𝜗 𝐽 2 𝑗∈𝑈 𝑘

Unnormalized Version Suppose I get some vectors 𝑥 1 , … , 𝑥 𝑛 which are Fact : 𝐵 ≼ 𝐶 ⟺ 𝐷𝐵𝐷 ≼ 𝐷𝐶𝐷 not isotropic: for invertible 𝐷 𝑈 = 𝑁 ≽ 0 𝑥 𝑗 𝑥 𝑗 𝑗 Consider 𝑤 𝑗 ≔ 𝑋 − 1 2 𝑥 𝑗 and apply theorem to 𝑤 𝑗 . Normalized vectors have ||𝑤 𝑗 || 2 = ||𝑋 − 1 2 𝑥 𝑗 || 2 = 𝜗 1 1 2 − 5 𝜗 𝐽 ≼ 𝑋 −1 𝑋 −1 𝑈 2 ≼ 𝑥 𝑗 𝑥 𝑗 2 + 5 𝜗 𝐽 2 𝑗∈𝑈 𝑘

Unnormalized Version Suppose I get some vectors 𝑥 1 , … , 𝑥 𝑛 which are not isotropic: 𝑈 = 𝑋 ≽ 0 𝑥 𝑗 𝑥 𝑗 𝑗 Consider 𝑤 𝑗 ≔ 𝑋 − 1 2 𝑥 𝑗 and apply theorem to 𝑤 𝑗 . Normalized vectors have ||𝑤 𝑗 || 2 = ||𝑋 − 1 2 𝑥 𝑗 || 2 = 𝜗 1 1 𝑈 2 − 5 𝜗 𝑋 ≼ 𝑥 𝑗 𝑥 𝑗 ≼ 2 + 5 𝜗 𝑋 𝑗∈𝑈 𝑘

Unnormalized Theorem Given arbitrary vectors 𝑥 1 , … , 𝑥 𝑛 ∈ ℝ 𝑜 there is a partition 𝑛 = 𝑈 1 ∪ 𝑈 2 with 1 𝑈 ≼ 1 𝑈 𝑈 2 − 𝜗 𝑥 𝑗 𝑥 𝑗 ≼ 𝑥 𝑗 𝑥 𝑗 2 − 𝜗 𝑥 𝑗 𝑥 𝑗 𝑗 𝑗∈𝑈 𝑘 𝑗 || 𝑋 − 1 2 𝑥 𝑗 || 2 Where 𝜗 ≔ max 𝑗

Unnormalized Theorem Given arbitrary vectors 𝑥 1 , … , 𝑥 𝑛 ∈ ℝ 𝑜 there is a partition 𝑛 = 𝑈 1 ∪ 𝑈 2 with 1 𝑈 ≼ 1 𝑈 𝑈 2 − 𝜗 𝑥 𝑗 𝑥 𝑗 ≼ 𝑥 𝑗 𝑥 𝑗 2 − 𝜗 𝑥 𝑗 𝑥 𝑗 𝑗 𝑗∈𝑈 𝑘 𝑗 || 𝑋 − 1 2 𝑥 𝑗 || 2 Where 𝜗 ≔ max 𝑗 Any quadratic form in which no vector has too much influence can be split into two representative pieces.

Applications

1. Graph Theory Given an undirected graph 𝐻 = (𝑊, 𝐹) , consider its Laplacian matrix: 𝑈 𝑀 𝐻 = 𝜀 𝑗 − 𝜀 𝑘 )(𝜀 𝑗 − 𝜀 𝑘 𝑗𝑘∈𝐹

1. Graph Theory Given an undirected graph 𝐻 = (𝑊, 𝐹) , consider its Laplacian matrix: 𝑈 𝑀 𝐻 = 𝜀 𝑗 − 𝜀 𝑘 )(𝜀 𝑗 − 𝜀 𝑘 𝑗𝑘∈𝐹 i j i j

1. Graph Theory Given an undirected graph 𝐻 = (𝑊, 𝐹) , consider its Laplacian matrix: 𝑈 = 𝐸 − 𝐵 𝑀 𝐻 = 𝜀 𝑗 − 𝜀 𝑘 )(𝜀 𝑗 − 𝜀 𝑘 𝑗𝑘∈𝐹

1. Graph Theory Given an undirected graph 𝐻 = (𝑊, 𝐹) , consider its Laplacian matrix: 𝑈 = 𝐸 − 𝐵 𝑀 𝐻 = 𝜀 𝑗 − 𝜀 𝑘 )(𝜀 𝑗 − 𝜀 𝑘 𝑗𝑘∈𝐹 Quadratic form: 2 𝑔𝑝𝑠 𝑦 ∈ 𝑺 𝑜 𝑦 𝑈 𝑀𝑦 = 𝑦 𝑗 − 𝑦 𝑘 𝑗𝑘∈𝐹

The Laplacian Quadratic Form An example: -1 +1 +2 0 +1 +3

The Laplacian Quadratic Form An example: -1 +1 1 2 3 2 2 2 +2 0 2 2 1 2 3 2 +1 +3 x T Lx =  i,j 2 E ( x (i)- x (j)) 2

The Laplacian Quadratic Form An example: -1 +1 1 2 3 2 2 2 +2 0 2 2 1 2 3 2 +1 +3 x T Lx =  i,j 2 E ( x (i)- x (j)) 2 = 28

The Laplacian Quadratic Form Another example: 0 0 0 +1 0 +1

The Laplacian Quadratic Form Another example: 0 0 1 2 0 2 1 2 0 +1 1 2 0 2 0 2 0 +1 x T L G x = 3

Cuts and the Quadratic Form For characteristic vector

Cuts and the Quadratic Form For characteristic vector The Laplacian Quadratic form encodes the entire cut structure of the graph.

Application to Graphs G

Application to Graphs G 𝑈 𝑀 𝐻 = 𝜀 𝑗 − 𝜀 𝑘 )(𝜀 𝑗 − 𝜀 𝑘 𝑗𝑘∈𝐹

Application to Graphs G Theorem

Application to Graphs G Theorem H 1 𝑀 𝐼 1 𝑀 𝐼 2 H 2

Application to Graphs G Theorem H 1 H 2 𝑦 𝑈 𝑀 𝐼 1 𝑦 ≈ 1 2 𝑦 𝑈 𝑀 𝐻 𝑦

Recursive Application Gives: 1. Graph Sparsification Theorem [Batson- Spielman- S’09]: Every graph G has a weighted O(1)-cut approximation H with O(n) edges. G H 𝑃 𝑜 2 edges 𝑃 𝑜 edges Unweighted Weighted

Approximating One Graph by Another Cut Approximation [Benczur- Karger’96] G H For every cut, weight of edges in G ≈ weight of edges in H

Approximating One Graph by Another Cut Approximation [Benczur- Karger’96] G H For every cut, weight of edges in G ≈ weight of edges in H

Approximating One Graph by Another Cut Approximation [Benczur- Karger’96] G H For every cut, weight of edges in G ≈ weight of edges in H

Approximating One Graph by Another Cut Approximation [Benczur- Karger’96] G H For every cut, weight of edges in G ≈ weight of edges in H

Approximating One Graph by Another Cut Approximation [Benczur- Karger’96] G H For every cut, weight of edges in G ≈ weight of edges in H

Approximating One Graph by Another G and H have same cuts. Equivalent for min Cut Approximation [Benczur- Karger’96] cut, max cut, sparsest cut… G H For every cut, weight of edges in G ≈ weight of edges in H

Recursive Application Gives: 2. Unweighted Graph Sparsification Every transitive graph G has an unweighted O(1)-cut approximation H with O(n) edges. 𝐿 𝑜 𝐼 Expander graph

Recursive Application Gives: 2. Unweighted Graph Sparsification Every transitive graph G can be partitioned into O(1)-cut approximations with O(n) edges. 𝐿 𝑜 𝐼 1 … 𝐼 𝑜 Expander graphs

Recursive Application Gives: 2. Unweighted Graph Sparsification Every transitive graph G can be partitioned into O(1)-cut approximations with O(n) edges. 𝐿 𝑜 𝐼 1 … 𝐼 𝑜 Expander graphs Generalizes [Frieze-Molloy]

Recursive Application Gives: 2. Unweighted Graph Sparsification Every transitive graph G can be partitioned into O(1)-cut approximations with O(n) edges. H 1 G H 2 Same cut structure

2. Uncertainty Principles Signal 𝑦 ∈ ℂ 𝑜 . Discrete Fourier Transform 𝑏2𝜌𝑗𝑙 𝑦 a = 〈𝑦, exp(− ) 𝑙≤𝑜 〉 𝑜

2. Uncertainty Principles Signal 𝑦 ∈ ℂ 𝑜 . Discrete Fourier Transform 𝑏2𝜌𝑗𝑙 𝑦 a = 〈𝑦, exp(− ) 𝑙≤𝑜 〉 𝑜 Uncertainty Principle : 𝑦 and 𝑦 cannot be simultaneously localized. 𝑡𝑣𝑞𝑞 𝑦 × 𝑡𝑣𝑞𝑞 𝑦 ≥ 𝑜

2. Uncertainty Principles Signal 𝑦 ∈ ℂ 𝑜 . Discrete Fourier Transform 𝑏2𝜌𝑗𝑙 𝑦 a = 〈𝑦, exp(− ) 𝑙≤𝑜 〉 𝑜 Uncertainty Principle : 𝑦 and 𝑦 cannot be simultaneously localized. 𝑡𝑣𝑞𝑞 𝑦 × 𝑡𝑣𝑞𝑞 𝑦 ≥ 𝑜 If 𝑦 is supported on 𝑇 = √𝑜 coordinates, 𝑡𝑣𝑞𝑞 𝑦 ≥ 𝑜

2. Uncertainty Principles Signal 𝑦 ∈ ℂ 𝑜 . Discrete Fourier Transform 𝑏2𝜌𝑗𝑙 𝑦 a = 〈𝑦, exp(− ) 𝑙≤𝑜 〉 𝑜 Stronger Uncertainty Principle: For every subset 𝑇 = 𝑜 , there is a partition 𝑜 = 𝑈 1 ∪ ⋯ 𝑈 𝑜 1 ||𝑦| 𝑇 || 2 ≈ 𝑦| 𝑈 𝑗 || 2 𝑜 || for all 𝑦 and 𝑈 𝑗

2. Uncertainty Principles Signal 𝑦 ∈ ℂ 𝑜 . Discrete Fourier Transform 𝑏2𝜌𝑗𝑙 𝑦 a = 〈𝑦, exp(− ) 𝑙≤𝑜 〉 𝑜 Stronger Uncertainty Principle: For every subset 𝑇 = 𝑜 , there is a partition 𝑜 = 𝑈 1 ∪ ⋯ 𝑈 𝑜 1 ||𝑦| 𝑇 || 2 ≈ 𝑦| 𝑈 𝑗 || 2 𝑜 || for all 𝑦 and 𝑈 𝑗

2. Uncertainty Principles Proof. 𝑏2𝜌𝑗𝑙 Let 𝑔 𝑙 = exp(− ) 𝑙≤𝑜 be the Fourier basis. 𝑜 Fix a subset 𝑇 ⊂ 𝑜 of √𝑜 coords. The restricted norm is: ||𝑦| 𝑇 || 2 = 𝑙 𝑦| 𝑇 , 𝑔 𝑙 2 a quadratic form in 𝑜 dimensions. Apply the theorem.

2. Uncertainty Principles Applications in analytic number theory, harmonic analysis. Proof. 𝑏2𝜌𝑗𝑙 Let 𝑔 𝑙 = exp(− ) 𝑙≤𝑜 be the Fourier basis. 𝑜 Fix a subset 𝑇 ⊂ 𝑜 of √𝑜 coords. The restricted norm is: ||𝑦| 𝑇 || 2 = 𝑙 𝑦| 𝑇 , 𝑔 𝑙 2 a quadratic form in 𝑜 dimensions. Apply the theorem.

3. The Kadison-Singer Problem Dirac 1930’s: Bra-Ket formalism for quantum states. 𝜔 , 𝑞 …

3. The Kadison-Singer Problem Dirac 1930’s: Bra-Ket formalism for quantum states. 𝜔 , 𝑞 … What are Bras and Kets? NOT vectors.

3. The Kadison-Singer Problem Dirac 1930’s: Bra-Ket formalism for quantum states. 𝜔 , 𝑞 … What are Bras and Kets? NOT vectors. Von Neumann 1936: Theory of 𝐷 ∗ algebras.

3. The Kadison-Singer Problem Dirac 1930’s: Bra-Ket formalism for quantum states. 𝜔 , 𝑞 … What are Bras and Kets? NOT vectors. Von Neumann 1936: Theory of 𝐷 ∗ algebras. Kadison-Singer 1959: Does this lead to a satisfactory notion of measurement? Conjecture : about ∞ matrices

3. The Kadison-Singer Problem Kadison-Singer 1959: Does this lead to a satisfactory notion of measurement? Conjecture : about ∞ matrices

3. The Kadison-Singer Problem Kadison-Singer 1959: Does this lead to a satisfactory notion of measurement? Conjecture : about ∞ matrices Anderson 1979 : Reduced to a question about finite matrices. “Paving Conjecture”

3. The Kadison-Singer Problem Kadison-Singer 1959: Does this lead to a satisfactory notion of measurement? Conjecture : about ∞ matrices Anderson 1979 : Reduced to a question about finite matrices. Akemann-Anderson 1991 : Reduced to a question about finite projection matrices.

3. The Kadison-Singer Problem Kadison-Singer 1959: Does this lead to a satisfactory notion of measurement? Conjecture : about ∞ matrices Anderson 1979 : Reduced to a question about finite matrices. Akemann-Anderson 1991 : Reduced to a question about finite projection matrices. Weaver 2002: Discrepancy theoretic formulation of the same question.

3. The Kadison-Singer Problem Kadison-Singer 1959: Does this lead to a satisfactory notion of measurement? Conjecture : about ∞ matrices Anderson 1979 : Reduced to a question about finite matrices. Akemann-Anderson 1991 : Reduced to a question about finite projection matrices. This work: Proof of Weaver’s conjecture.

3. The Kadison-Singer Problem Kadison-Singer 1959: Does this lead to a satisfactory notion of measurement? Conjecture : about ∞ matrices Anderson 1979 : Reduced to a question about finite matrices. Akemann-Anderson 1991 : Reduced to a question about finite projection matrices. This work: Proof of Weaver’s conjecture.

In General Anything that can be encoded as a quadratic form can be split into pieces while preserving certain properties. Many different things can be gainfully encoded this way.

Proof

Main Theorem Suppose 𝑤 1 , … , 𝑤 𝑛 ∈ 𝑺 𝑜 are vectors ||𝑤 𝑗 || 2 ≤ 𝜗 and 𝑈 = 𝐽 𝑜 𝑤 𝑗 𝑤 𝑗 𝑗 Then there is a partition 𝑈 1 ∪ 𝑈 2 such that 1 1 𝑈 ≼ 2 − 5 𝜗 𝐽 ≼ 𝑤 𝑗 𝑤 𝑗 2 + 5 𝜗 𝐽 𝑗∈𝑈 𝑘

Equivalent Theorem Suppose 𝑤 1 , … , 𝑤 𝑛 ∈ 𝑺 𝑜 are vectors ||𝑤 𝑗 || 2 ≤ 𝜗 and 𝑈 = 𝐽 𝑜 𝑤 𝑗 𝑤 𝑗 𝑗 Then there is a partition 𝑈 1 ∪ 𝑈 2 such that 1 𝑈 ≼ 𝑤 𝑗 𝑤 𝑗 2 + 5 𝜗 𝐽 𝑗∈𝑈 𝑘