A Union of Euclidean Spaces is Euclidean Konstantin Makarychev, - PowerPoint PPT Presentation

A Union of Euclidean Spaces is Euclidean Konstantin Makarychev, Northwestern Yury Makarychev, TTIC AMS Meeting, New York, May 7, 2017

Problem by Assaf Naor Suppose that metric space (𝑌, 𝑒) is a union of two metric spaces 𝐵 and 𝐶 that isometrically embed into ℓ 2 . Does 𝑌 necessarily embed into ℓ 2 with a constant distortion? 𝐵 ℓ 2 𝐶

Motivation The problem is closely connected to research in theoretical computer science on “local -global properties” of metric spaces [Arora, Lovász, Newman, Rabani, Rabinovich, Vempala `06; Charikar, M, Makarychev `07] Why are computer scientists interested? Results imply strong lower bounds for Sherali-Adams linear programming relaxations for many combinatorial optimization problems, including Sparsest Cut, Vertex Cover, Max Cut, Unique Games. [Charikar, M, Makarychev `09]

Our Results Q: Suppose that metric space (𝑌, 𝑒) is a union of two metric spaces 𝐵 and 𝐶 that embed isometrically into ℓ 2 . Does 𝑌 necessarily embed into ℓ 2 with a constant distortion? A: Yes, 𝑌 embeds into ℓ 2 with distortion at most 8.93. 𝑏 with distortion 𝛽 , 𝐶 ↪ ℓ 2 𝑐 with distortion 𝛾 𝐵 ↪ ℓ 2 ⇓ 𝑏+𝑐+1 with distortion at most 11𝛽𝛾 𝑌 = 𝐵 ∪ 𝐶 ↪ ℓ 2

Approach This talk: consider the isometric case. 𝜒 1 : 𝐵 ↪ ℓ 2 𝜒 2 : 𝐶 ↪ ℓ 2 We will define 3 maps: • ത 𝜒 1 : 𝐵 ∪ 𝐶 ↪ ℓ 2 , a 7-Lipschitz extension of 𝜒 1 to 𝑌 • ത 𝜒 2 : 𝐵 ∪ 𝐶 ↪ ℓ 2 , a 7-Lipschitz extension of 𝜒 2 to 𝑌 • Δ 𝑦 = 𝑒 𝑦, 𝐵 − 𝑒(𝑦, 𝐶) 𝜔 = ത 𝜒 1 ⊕ ത 𝜒 2 ⊕ Δ

Approach 𝜔 = ത 𝜒 1 ⊕ ത 𝜒 2 ⊕ Δ Assume that we have • ത 𝜒 1 : 𝐵 ∪ 𝐶 ↪ ℓ 2 , a 7-Lipschitz extension of 𝜒 1 to 𝑌 • ത 𝜒 2 : 𝐵 ∪ 𝐶 ↪ ℓ 2 , a 7-Lipschitz extension of 𝜒 2 to 𝑌 • Δ 𝑦 = 𝑒 𝑦, 𝐵 − 𝑒(𝑦, 𝐶) First, 7 2 + 7 2 + 2 2 𝜔 𝑀𝑗𝑞 = 𝜒 1 ⊕ ത ത 𝜒 2 ⊕ Δ 𝑀𝑗𝑞 ≤ since Δ 𝑀𝑗𝑞 ≤ 2 .

Approach 𝜔 = ത 𝜒 1 ⊕ ത 𝜒 2 ⊕ Δ • ത 𝜒 1 ensures that distances between points in 𝐵 don’t decrease: 𝜒 1 ȁ 𝐵 = 𝜒 1 is an isometric embedding of 𝐵 into ℓ 2 . ത • ത 𝜒 2 ensures that distances between points in 𝐶 don’t decrease. • Δ ensures that distances between points 𝑏 ∈ 𝐵 and 𝑐 ∈ 𝐶 don’t decrease by more than a constant factor.

Approach 𝜔 = ത 𝜒 1 ⊕ ത 𝜒 2 ⊕ Δ 𝑏 𝐵 𝑏′ 𝑐 𝐶 If 𝑒 𝑏, 𝑏′ ≪ 𝑒(𝑏, 𝑐) then 𝜒 2 𝑏 ′ 𝜒 2 (𝑏) − ത ത 𝜒 2 (𝑐) ≈ ത − ത 𝜒 2 𝑐 = 𝑒 𝑏 ′ , 𝑐 ≈ 𝑒(𝑏, 𝑐) If 𝑒 𝑏, 𝑏′ ≈ 𝑒(𝑏, 𝑐) then ≥ 𝑒(𝑏, 𝑏 ′ ) ≈ 𝑒(𝑏, 𝑐) Δ(𝑏) − Δ(𝑐) 𝑏′ is the closest point to 𝑏 in 𝐶

Approach 𝜔 = ത 𝜒 1 ⊕ ത 𝜒 2 ⊕ Δ 𝑏 𝐵 𝑐 𝑏′ 𝐶 If 𝑒 𝑏, 𝑏′ ≪ 𝑒(𝑏, 𝑐) then 𝜒 2 𝑏 ′ 𝜒 2 (𝑏) − ത ത 𝜒 2 (𝑐) ≈ ത − ത 𝜒 2 𝑐 = 𝑒 𝑏 ′ , 𝑐 ≈ 𝑒(𝑏, 𝑐) If 𝑒 𝑏, 𝑏′ ≈ 𝑒(𝑏, 𝑐) then ≥ 𝑒(𝑏, 𝑏 ′ ) ≈ 𝑒(𝑏, 𝑐) Δ(𝑏) − Δ(𝑐)

Constructing maps ത 𝜒 1 and ത 𝜒 2 Goal: Given a map 𝜒 ≡ 𝜒 2 : 𝐶 → ℓ 2 find a Lipschitz extension ത 𝜒: 𝐵 ∪ 𝐶 → ℓ 2 of 𝜒 . 𝜒 ത 𝐵 ℓ 2 𝜒 𝐶

Constructing maps ത 𝜒 1 and ത 𝜒 2 Assume that 𝐶 ⊂ ℓ 2 and 𝜒 = 𝑗𝑒 ; 𝐵 ∪ 𝐶 < ∞ . 𝜒 ത ℓ 2 𝐵 𝐶

Constructing map ത 𝜒 Idea 1: map every 𝑏 to the closest 𝑏 ′ ∈ 𝐶 w.r.t. 𝑒 . Issue: the map may not be Lipschitz. 𝜒 ത 𝑏′ 𝑏 𝐵 𝐶

Cover for 𝐵 Let 𝑆 𝑏 = 𝑒 𝑏, 𝐶 for 𝑏 ∈ 𝐵 . 𝐷 ⊂ 𝐵 is a cover for 𝐵 if • for every 𝑏 ∈ 𝐵 , there is 𝑑 ∈ 𝐷 s.t. 𝑒 𝑏, 𝑑 ≤ 𝑆 𝑏 and 𝑆 𝑑 ≤ 𝑆 𝑏 • for every 𝑑, 𝑒 ∈ 𝐷 : 𝑒 𝑑, 𝑒 ≥ min 𝑆 𝑑 , 𝑆 𝑒 . 𝑆 𝑦 𝑆 𝑦 𝑏 𝑑 𝑑 𝑒 𝑏 ∈ 𝐵 is close to some 𝑑 ∈ 𝐷 points in 𝐷 are “separated”

Cover for 𝐵 Prove by induction that there is always a cover 𝑫 . Let 𝑑 ∈ 𝐵 be the point in 𝐵 with the least value of 𝑆 𝑑 . By induction, there is a cover 𝐷′ for 𝐵 ∖ Ball 𝑑, 𝑆 𝑑 . Let 𝐷 = 𝐷 ′ ∪ {𝑑} . 𝐵

Cover for 𝐵 Prove by induction that there is always a cover 𝑫 . Let 𝑑 ∈ 𝐵 be the point in 𝐵 with the least value of 𝑆 𝑑 . By induction, there is a cover 𝐷′ for 𝐵 ∖ Ball 𝑑, 𝑆 𝑑 . Let 𝐷 = 𝐷 ′ ∪ {𝑑} . 𝑑

Cover for 𝐵 Prove by induction that there is always a cover 𝑫 . Let 𝑑 ∈ 𝐵 be the point in 𝐵 with the least value of 𝑆 𝑑 . By induction, there is a cover 𝐷′ for 𝐵 ∖ Ball 𝑑, 𝑆 𝑑 . Let 𝐷 = 𝐷 ′ ∪ {𝑑} . 𝑑

Cover for 𝐵 Prove by induction that there is always a cover 𝑫 . Let 𝑑 ∈ 𝐵 be the point in 𝐵 with the least value of 𝑆 𝑑 . By induction, there is a cover 𝐷′ for 𝐵 ∖ Ball 𝑑, 𝑆 𝑑 . Let 𝐷 = 𝐷 ′ ∪ {𝑑} . 𝑑

Constructing map ത 𝜒 Idea 2: map every 𝑑 ∈ 𝐷 to the closest 𝑑 ′ ∈ 𝐶 . The map is 4-Lipschitz. 𝑔 𝑑′ 𝑑 𝐵 𝐶

Constructing map ത 𝜒 Idea 2: map every 𝑑 ∈ 𝐷 to the closest 𝑑 ′ ∈ 𝐶 . The map is 4-Lipschitz. Assume 𝑆 𝑑 ≤ 𝑆 𝑒 . 𝑑′ 𝑑 𝑒 𝑒′ 𝐵 𝐶

Constructing map ത 𝜒 Idea 2: map every 𝑑 ∈ 𝐷 to the closest 𝑑 ′ ∈ 𝐶 . The map is 4-Lipschitz. Assume 𝑆 𝑑 ≤ 𝑆 𝑒 . 𝑑′ 𝑑 𝑒 𝑒′ 𝐵 𝐶

Constructing map ത 𝜒 Idea 2: map every 𝑑 ∈ 𝐷 to the closest 𝑑 ′ ∈ 𝐶 . The map is 4-Lipschitz. Assume 𝑆 𝑑 ≤ 𝑆 𝑒 . 𝑑′ 𝑑 𝑒 𝑒′ 𝐵 𝐶 𝑒 𝑑 ′ , 𝑒 ′ ≤ 2 𝑒 𝑑, 𝑒 + 2 𝑒 𝑑, 𝑑 ′ ≤ 4𝑒(𝑑, 𝑒)

Kirszbraun Theorem Let 𝐷 ⊂ 𝐸 ⊂ ℓ 2 and 𝑔 be a Lipschitz map from 𝐷 to ℓ 2 . There exists an extension 𝑕: 𝐸 → ℓ 2 of 𝑔 such 𝑕 𝑀𝑗𝑞 = 𝑔 𝑀𝑗𝑞 𝐷 ℓ 2 𝐸

Constructing map ത 𝜒 Idea 2: map every 𝑑 ∈ 𝐷 to the closest 𝑑 ′ ∈ 𝐶 . Extend 𝑔 from 𝐷 to 𝐵 using the Kirszbraun theorem. 𝑔 𝑑′ 𝑑 𝐵 𝐶

Constructing map ത 𝜒 Idea 2: map every 𝑑 ∈ 𝐷 to the closest 𝑑 ′ ∈ 𝐶 . Extend 𝑔 from 𝐷 to 𝐵 using the Kirszbraun theorem. 𝜒 𝑣 = ቊ𝑔 𝑣 , if 𝑣 ∈ 𝐵 ത 𝑣, if 𝑣 ∈ 𝐶 𝜒 𝑣 is 7-Lipschitz: ത • ത 𝜒ȁ 𝐵 is 4-Lipschitz • ത 𝜒ȁ 𝐶 is 1-Lipschitz • 𝜒 𝑏 − ത ത 𝜒 𝑐 = 𝑔(𝑏) − 𝑐 ≤ ⋯

Constructing map ത 𝜒 𝑐 𝑏 𝑔(𝑏) 𝑑 𝑔(𝑑) 𝐵 𝐶

Constructing map ത 𝜒 𝑐 𝑏 𝑔(𝑏) ≤ 𝑆 𝑏 ≤ 4𝑆 𝑏 𝑆 𝑑 ≤ 𝑆 𝑏 𝑑 𝑔(𝑑) 𝐵 𝐶 𝑔 𝑏 − 𝑐 ≤ 6𝑆 𝑏 + 𝑒 𝑏, 𝑐 ≤ 7𝑒(𝑏, 𝑐)

Constructing map ത 𝜒 𝑐 𝑏 𝑔(𝑏) ≤ 𝑆 𝑏 ≤ 4𝑆 𝑏 𝑆 𝑑 ≤ 𝑆 𝑏 𝑑 𝑔(𝑑) 𝐵 𝐶 𝑔 𝑏 − 𝑐 ≤ 6𝑆 𝑏 + 𝑒 𝑏, 𝑐 ≤ 7𝑒(𝑏, 𝑐) Q.E.D.

Lower Bound There exists a metric space 𝑌 = 𝐵 ∪ 𝐶 s.t. • 𝐵 and 𝐶 isometrically embed into ℓ 2 • every embedding of 𝑌 into ℓ 2 has distortion at least 3 − 𝜁 𝑜 , where 𝑜 = 𝐵 = ȁ𝐶ȁ and 𝜁 𝑜 → 0 as 𝑜 → ∞

Open Problems 1. Find the least value of 𝐸 s.t. if 𝐵, 𝐶 ↪ ℓ 2 isometrically, then 𝐵 ∪ 𝐶 ↪ ℓ 2 with distortion at most 𝐸 . We know that 𝐸 ∈ 3, 8.93 . 2. Study the problem for other ℓ 𝑞 . We conjecture that the answer is negative for every 𝑞 ∉ {2, ∞} . 3. What happens if 𝑌 = 𝐵 1 ∪ ⋯ ∪ 𝐵 𝑙 and each 𝐵 𝑗 ↪ ℓ 2 isometrically? We only know that 𝑑 log 𝑙 ≤ 𝐸 ≤ 2 𝐷𝑙 . 4. Assume that every subset of 𝑌 of size ȁ𝑌ȁ isometrically embeds into ℓ 2 . What is the least distortion with which 𝑌 ↪ ℓ 2 ? More results and open problems in the paper!