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

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

May 10, 2023 260 likes β€’563 views

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 (, ) is a union of two metric spaces and

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A Union of Euclidean Spaces is Euclidean

Konstantin Makarychev, Northwestern Yury Makarychev, TTIC

AMS Meeting, New York, May 7, 2017

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Problem by Assaf Naor

Suppose that metric space (π‘Œ, 𝑒) is a union of two metric spaces 𝐡 and 𝐢 that isometrically embed into β„“2. Does π‘Œ necessarily embed into β„“2 with a constant distortion? β„“2 𝐡 𝐢

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Motivation

The problem is closely connected to research in theoretical computer science on β€œlocal-global properties” of metric spaces [Arora, LovΓ‘sz, 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]

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Our Results

Q: Suppose that metric space (π‘Œ, 𝑒) is a union of two metric spaces 𝐡 and 𝐢 that embed isometrically into β„“2. Does π‘Œ necessarily embed into β„“2 with a constant distortion? A: Yes, π‘Œ embeds into β„“2 with distortion at most 8.93. 𝐡 β†ͺ β„“2

𝑏 with distortion 𝛽, 𝐢 β†ͺ β„“2 𝑐 with distortion 𝛾

⇓

π‘Œ = 𝐡 βˆͺ 𝐢 β†ͺ β„“2

𝑏+𝑐+1 with distortion at most 11𝛽𝛾

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Approach

This talk: consider the isometric case. πœ’1: 𝐡 β†ͺ β„“2 πœ’2: 𝐢 β†ͺ β„“2 We will define 3 maps:

  • ΰ΄€

πœ’1: 𝐡 βˆͺ 𝐢 β†ͺ β„“2 , a 7-Lipschitz extension of πœ’1 to π‘Œ

  • ΰ΄€

πœ’2: 𝐡 βˆͺ 𝐢 β†ͺ β„“2 , a 7-Lipschitz extension of πœ’2 to π‘Œ

  • Ξ” 𝑦 = 𝑒 𝑦, 𝐡 βˆ’ 𝑒(𝑦, 𝐢)

πœ” = ΰ΄€ πœ’1 βŠ• ΰ΄€ πœ’2 βŠ• Ξ”

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Approach πœ” = ΰ΄€ πœ’1 βŠ• ΰ΄€ πœ’2 βŠ• Ξ”

Assume that we have

  • ΰ΄€

πœ’1: 𝐡 βˆͺ 𝐢 β†ͺ β„“2 , a 7-Lipschitz extension of πœ’1 to π‘Œ

  • ΰ΄€

πœ’2: 𝐡 βˆͺ 𝐢 β†ͺ β„“2 , a 7-Lipschitz extension of πœ’2 to π‘Œ

  • Ξ” 𝑦 = 𝑒 𝑦, 𝐡 βˆ’ 𝑒(𝑦, 𝐢)

First, πœ” π‘€π‘—π‘ž = ΰ΄€ πœ’1 βŠ• ΰ΄€ πœ’2 βŠ• Ξ” π‘€π‘—π‘ž ≀ 72 + 72 + 22 since Ξ” π‘€π‘—π‘ž ≀ 2.

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Approach πœ” = ΰ΄€ πœ’1 βŠ• ΰ΄€ πœ’2 βŠ• Ξ”

  • ΰ΄€

πœ’1 ensures that distances between points in 𝐡 don’t decrease: ΰ΄€ πœ’1ȁ𝐡 = πœ’1 is an isometric embedding of 𝐡 into β„“2.

  • ΰ΄€

πœ’2 ensures that distances between points in 𝐢 don’t decrease.

  • Ξ” ensures that distances between points 𝑏 ∈ 𝐡 and

𝑐 ∈ 𝐢 don’t decrease by more than a constant factor.

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Approach πœ” = ΰ΄€ πœ’1 βŠ• ΰ΄€ πœ’2 βŠ• Ξ”

If 𝑒 𝑏, 𝑏′ β‰ͺ 𝑒(𝑏, 𝑐) then ΰ΄€ πœ’2(𝑏) βˆ’ ΰ΄€ πœ’2(𝑐) β‰ˆ ΰ΄€ πœ’2 𝑏′ βˆ’ ΰ΄€ πœ’2 𝑐 = 𝑒 𝑏′, 𝑐 β‰ˆ 𝑒(𝑏, 𝑐) If 𝑒 𝑏, 𝑏′ β‰ˆ 𝑒(𝑏, 𝑐) then Ξ”(𝑏) βˆ’ Ξ”(𝑐) β‰₯ 𝑒(𝑏, 𝑏′) β‰ˆ 𝑒(𝑏, 𝑐) 𝑏 𝑏′ 𝑐 𝐡 𝐢

𝑏′ is the closest point to 𝑏 in 𝐢

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Approach πœ” = ΰ΄€ πœ’1 βŠ• ΰ΄€ πœ’2 βŠ• Ξ”

If 𝑒 𝑏, 𝑏′ β‰ͺ 𝑒(𝑏, 𝑐) then ΰ΄€ πœ’2(𝑏) βˆ’ ΰ΄€ πœ’2(𝑐) β‰ˆ ΰ΄€ πœ’2 𝑏′ βˆ’ ΰ΄€ πœ’2 𝑐 = 𝑒 𝑏′, 𝑐 β‰ˆ 𝑒(𝑏, 𝑐) If 𝑒 𝑏, 𝑏′ β‰ˆ 𝑒(𝑏, 𝑐) then Ξ”(𝑏) βˆ’ Ξ”(𝑐) β‰₯ 𝑒(𝑏, 𝑏′) β‰ˆ 𝑒(𝑏, 𝑐) 𝑏 𝑏′ 𝑐 𝐡 𝐢

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Constructing maps ΰ΄€ πœ’1 and ΰ΄€ πœ’2

Goal: Given a map πœ’ ≑ πœ’2: 𝐢 β†’ β„“2 find a Lipschitz extension ΰ΄€ πœ’: 𝐡 βˆͺ 𝐢 β†’ β„“2 of πœ’. β„“2 𝐡 𝐢 πœ’ ΰ΄€ πœ’

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Constructing maps ΰ΄€ πœ’1 and ΰ΄€ πœ’2

Assume that 𝐢 βŠ‚ β„“2 and πœ’ = 𝑗𝑒; 𝐡 βˆͺ 𝐢 < ∞. β„“2 𝐡 𝐢 ΰ΄€ πœ’

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Constructing map ΰ΄€ πœ’

Idea 1: map every 𝑏 to the closest 𝑏′ ∈ 𝐢 w.r.t. 𝑒. Issue: the map may not be Lipschitz. 𝐡 𝐢 𝑏 𝑏′ ΰ΄€ πœ’

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Cover for 𝐡

Let 𝑆𝑏 = 𝑒 𝑏, 𝐢 for 𝑏 ∈ 𝐡. 𝐷 βŠ‚ 𝐡 is a cover for 𝐡 if

  • for every 𝑏 ∈ 𝐡, there is 𝑑 ∈ 𝐷 s.t.

𝑒 𝑏, 𝑑 ≀ 𝑆𝑏 and 𝑆𝑑 ≀ 𝑆𝑏

  • for every 𝑑, 𝑒 ∈ 𝐷: 𝑒 𝑑, 𝑒 β‰₯ min 𝑆𝑑, 𝑆𝑒 .

𝑆𝑦 𝑏 𝑑 𝑆𝑦 𝑒 𝑑

𝑏 ∈ 𝐡 is close to some 𝑑 ∈ 𝐷 points in 𝐷 are β€œseparated”

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Cover for 𝐡

Prove by induction that there is always a cover 𝑫. Let 𝑑 ∈ 𝐡 be the point in 𝐡 with the least value of 𝑆𝑑. By induction, there is a cover 𝐷′ for 𝐡 βˆ– Ball 𝑑, 𝑆𝑑 . Let 𝐷 = 𝐷′ βˆͺ {𝑑}. 𝐡

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Cover for 𝐡

Prove by induction that there is always a cover 𝑫. Let 𝑑 ∈ 𝐡 be the point in 𝐡 with the least value of 𝑆𝑑. By induction, there is a cover 𝐷′ for 𝐡 βˆ– Ball 𝑑, 𝑆𝑑 . Let 𝐷 = 𝐷′ βˆͺ {𝑑}. 𝑑

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Cover for 𝐡

Prove by induction that there is always a cover 𝑫. Let 𝑑 ∈ 𝐡 be the point in 𝐡 with the least value of 𝑆𝑑. By induction, there is a cover 𝐷′ for 𝐡 βˆ– Ball 𝑑, 𝑆𝑑 . Let 𝐷 = 𝐷′ βˆͺ {𝑑}. 𝑑

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Cover for 𝐡

Prove by induction that there is always a cover 𝑫. Let 𝑑 ∈ 𝐡 be the point in 𝐡 with the least value of 𝑆𝑑. By induction, there is a cover 𝐷′ for 𝐡 βˆ– Ball 𝑑, 𝑆𝑑 . Let 𝐷 = 𝐷′ βˆͺ {𝑑}. 𝑑

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Constructing map ΰ΄€ πœ’

Idea 2: map every 𝑑 ∈ 𝐷 to the closest 𝑑′ ∈ 𝐢. The map is 4-Lipschitz. 𝐡 𝐢 𝑔 𝑑 𝑑′

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Constructing map ΰ΄€ πœ’

Idea 2: map every 𝑑 ∈ 𝐷 to the closest 𝑑′ ∈ 𝐢. The map is 4-Lipschitz. Assume 𝑆𝑑 ≀ 𝑆𝑒. 𝐡 𝐢 𝑒 𝑒′ 𝑑 𝑑′

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Constructing map ΰ΄€ πœ’

Idea 2: map every 𝑑 ∈ 𝐷 to the closest 𝑑′ ∈ 𝐢. The map is 4-Lipschitz. Assume 𝑆𝑑 ≀ 𝑆𝑒. 𝐡 𝐢 𝑒 𝑒′ 𝑑 𝑑′

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Constructing map ΰ΄€ πœ’

Idea 2: map every 𝑑 ∈ 𝐷 to the closest 𝑑′ ∈ 𝐢. The map is 4-Lipschitz. Assume 𝑆𝑑 ≀ 𝑆𝑒. 𝐡 𝐢 𝑑 𝑑′ 𝑒 𝑒′ 𝑒 𝑑′, 𝑒′ ≀ 2 𝑒 𝑑, 𝑒 + 2 𝑒 𝑑, 𝑑′ ≀ 4𝑒(𝑑, 𝑒)

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Kirszbraun Theorem

Let 𝐷 βŠ‚ 𝐸 βŠ‚ β„“2 and 𝑔 be a Lipschitz map from 𝐷 to β„“2. There exists an extension 𝑕: 𝐸 β†’ β„“2 of 𝑔 such 𝑕 π‘€π‘—π‘ž = 𝑔 π‘€π‘—π‘ž 𝐸

β„“2

𝐷

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Constructing map ΰ΄€ πœ’

Idea 2: map every 𝑑 ∈ 𝐷 to the closest 𝑑′ ∈ 𝐢. Extend 𝑔 from 𝐷 to 𝐡 using the Kirszbraun theorem. 𝐡 𝐢 𝑔 𝑑 𝑑′

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Constructing map ΰ΄€ πœ’

Idea 2: map every 𝑑 ∈ 𝐷 to the closest 𝑑′ ∈ 𝐢. Extend 𝑔 from 𝐷 to 𝐡 using the Kirszbraun theorem. ΰ΄€ πœ’ 𝑣 = α‰Šπ‘” 𝑣 , if 𝑣 ∈ 𝐡 𝑣, if 𝑣 ∈ 𝐢 ΰ΄€ πœ’ 𝑣 is 7-Lipschitz:

  • ΰ΄€

πœ’Θπ΅ is 4-Lipschitz

  • ΰ΄€

πœ’ΘπΆ is 1-Lipschitz

  • ΰ΄€

πœ’ 𝑏 βˆ’ ΰ΄€ πœ’ 𝑐 = 𝑔(𝑏) βˆ’ 𝑐 ≀ β‹―

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Constructing map ΰ΄€ πœ’

𝐡 𝐢 𝑏

𝑔(𝑏) 𝑔(𝑑)

𝑑 𝑐

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Constructing map ΰ΄€ πœ’

𝐡 𝐢 𝑏

𝑔(𝑏) 𝑔(𝑑)

𝑐

𝑆𝑑 ≀ 𝑆𝑏 ≀ 𝑆𝑏 ≀ 4𝑆𝑏

𝑑 𝑔 𝑏 βˆ’ 𝑐 ≀ 6𝑆𝑏 + 𝑒 𝑏, 𝑐 ≀ 7𝑒(𝑏, 𝑐)

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Constructing map ΰ΄€ πœ’

𝐡 𝐢 𝑏

𝑔(𝑏) 𝑔(𝑑)

𝑐

𝑆𝑑 ≀ 𝑆𝑏 ≀ 𝑆𝑏 ≀ 4𝑆𝑏

𝑑 𝑔 𝑏 βˆ’ 𝑐 ≀ 6𝑆𝑏 + 𝑒 𝑏, 𝑐 ≀ 7𝑒(𝑏, 𝑐)

Q.E.D.

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Lower Bound

There exists a metric space π‘Œ = 𝐡 βˆͺ 𝐢 s.t.

  • 𝐡 and 𝐢 isometrically embed into β„“2
  • every embedding of π‘Œ into β„“2 has distortion at least

3 βˆ’ πœπ‘œ, where π‘œ = 𝐡 = ȁ𝐢ȁ and πœπ‘œ β†’ 0 as π‘œ β†’ ∞

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Open Problems

  • 1. Find the least value of 𝐸 s.t. if 𝐡, 𝐢 β†ͺ β„“2

isometrically, then 𝐡 βˆͺ 𝐢 β†ͺ β„“2 with distortion at most 𝐸. We know that 𝐸 ∈ 3, 8.93 .

  • 2. Study the problem for other β„“π‘ž. We conjecture

that the answer is negative for every π‘ž βˆ‰ {2, ∞}.

  • 3. What happens if π‘Œ = 𝐡1 βˆͺ β‹― βˆͺ 𝐡𝑙 and each

𝐡𝑗 β†ͺ β„“2 isometrically? We only know that 𝑑 log 𝑙 ≀ 𝐸 ≀ 2𝐷𝑙.

  • 4. Assume that every subset of π‘Œ of size Θπ‘ŒΘ

isometrically embeds into β„“2. What is the least distortion with which π‘Œ β†ͺ β„“2? More results and open problems in the paper!