Lip ipsch chitz itz an and ou d oute ter bi r bi-Lip ipschi - PowerPoint PPT Presentation

Lip ipsch chitz itz an and ou d oute ter bi r bi-Lip ipschi chitz tz ex exte tendabi ndabilit lity Yury Makarychev, TTIC Sepideh Mahabadi, Columbia ⇒ TTIC Konstantin Makarychev, Northwestern Ilya Razenshteyn, Microsoft Research University of Notre Dame, March 21, 2018

Plan • The Lipschitz extendability problem • Known results and open problems • Vertex sparsifiers • Connection between vertex sparsifiers and the Lipschitz extendability problem • Outer bi-Lipschitz extendability: definition, results, and open problems

Hahn – Banach Theorem H. Hahn S. Banah Let 𝑊 be a normed space and 𝑀 ⊂ 𝑊 be its linear subspace. Every bounded linear map 𝑔: 𝑀 → ℝ can be extended to ሚ 𝑔 ∶ 𝑊 → ℝ so that ሚ 𝑔 = 𝑔 Is there an analogue for • Lipschitz maps? • maps into ℝ 𝑒 or other normed spaces?

Preliminaries 𝑔: 𝑌 → 𝑍 is Lipschitz if for every 𝑣, 𝑤 ∈ 𝑌 𝑒 𝑍 𝑔 𝑣 , 𝑔 𝑤 ≤ 𝐷𝑒 𝑌 (𝑣, 𝑤) The Lipschitz constant 𝑔 𝑀𝑗𝑞 of 𝑔 is the minimum 𝐷 s.t. that the inequality holds. 𝑔 is bi-Lipschitz if for some 𝐷 1 , 𝐷 2 > 0 , every 𝑣, 𝑤 ∈ 𝑌 𝐷 1 𝑒 𝑌 𝑣, 𝑤 ≤ 𝑒 𝑍 𝑔 𝑣 , 𝑔 𝑤 ≤ 𝐷 2 𝑒 𝑌 (𝑣, 𝑤) The bi-Lipschitz constant or distortion 𝐸(𝑔) of 𝑔 is the minimum of 𝐷 2 /𝐷 1 s.t. that the inequality holds.

Preliminaries 𝑒 is ℝ 𝑒 equipped with the ⋅ 𝑞 norm: ℓ 𝑞 1/𝑞 ෍ 𝑦 𝑗 𝑞 𝑦 𝑞 = 𝑦 ∞ = max |𝑦 𝑗 |

McShane – Whitney Theorem “non -linear Hahn – Banach ” E. McShane H. Whitney Let (𝑌, 𝑒) be a metric space and 𝐵 ⊂ 𝑌 . Every Lipschitz map 𝑔: 𝐵 → ℝ can be extended to ሚ 𝑔 ∶ 𝑌 → ℝ so that ሚ 𝑔 𝑀𝑗𝑞 = 𝑔 𝑀𝑗𝑞 ℝ ሚ 𝑔(𝑦) 𝑌

Kirszbraun Theorem 𝑜 can 𝑛 . Every Lipschitz map 𝑔: 𝐵 → ℓ 2 Let 𝐵 ⊂ ℓ 2 𝑜 so that 𝑛 → ℓ 2 be extended to ሚ 𝑔 ∶ ℓ 2 ሚ 𝑔 𝑀𝑗𝑞 = 𝑔 𝑀𝑗𝑞 𝑛 𝑜 ℓ 2 ℓ 2 𝑔 ⟶

Lipschitz Extension Constant Let 𝑌 be a metric space and 𝑍 be a normed space. 𝑓 𝑙 (𝑌, 𝑍) is the min 𝐷 s.t. for every 𝐵 ⊂ 𝑌 of size ≤ 𝑙 and 𝑔: 𝐵 → 𝑍 there exists an extension ሚ 𝑔: 𝑌 → 𝑍 with ሚ 𝑔 𝑀𝑗𝑞 ≤ 𝐷 𝑔 𝑀𝑗𝑞 McShane – Whitney: 𝑓 𝑙 𝑌, ℝ = 1 Kirszbraun: 𝑓 𝑙 ℓ 2 , ℓ 2 = 1

Lipschitz Extension Constant In general, 𝑓 𝑙 𝑌, 𝑍 > 1 2 ≥ 3 , ℓ 2 E.g., 𝑓 3 ℓ 1 3 2 ℓ 2 𝑔(𝑏) 𝐵 = 𝑏, 𝑐, 𝑑 𝑏 = 1,0,0 2 2 𝑐 = 0,1,0 𝑑 = 0,0,1 𝑒 = (0,0,0) 2 𝑔(𝑐) 𝑔(𝑑)

𝑌 → 𝑍 𝑓 𝑙 (𝑌, 𝑍) McShane, Whitney ’34 any → ℝ or ℓ ∞ 1 Kirszbraun ’34 ℓ 2 → ℓ 2 1 𝑞−1 1 ℓ 𝑞 → ℓ 2 Marcus, Pisier ’84 2 ≤ 𝐷 𝑞 log 𝑙 𝑞 ≤ 2 𝑞−1 1 2 Johnson, Lindenstrauss log 𝑙 1 < 𝑞 < 2 ≥ 𝑑 𝑞 ’84 log log 𝑙 any → ℓ 2 JL ’84 ≤ 𝐷 log 𝑙 ℓ 𝑞 → ℓ 𝑟 𝑞−1 ≤ 24 𝑟−1 1 < 𝑟 ≤ 2 ≤ 𝑞 Naor, Peres, Schramm, Sheffield ’04 other 𝑓 𝑙 → ∞ 𝑞, 𝑟 ∈ (1, ∞) as 𝑙 → ∞

Extension Results Johnson – Lindenstruass – Schechtman ’86 𝑓 𝑙 𝑌, 𝑍 ≤ 𝐷 log 𝑙 Lee –Naor ’03 𝑓 𝑙 𝑌, 𝑍 ≤ 𝐷 log 𝑙 log log 𝑙 Best lower bounds are: 𝑓 𝑙 ≿ 𝑑 log 𝑙 Open Problem: what is the dependence of 𝑓 𝑙 on 𝑙 ?

[JL ’84] Technique for proving lower bounds on 𝑓 𝑙 𝑌, 𝑍 Prove a lower bound for linear extensions Reinterpret it as a lower bound for Lipschitz extensions • Linear extension (“projection”) constant is up to dim 𝑍 𝑍 𝑌 log 𝑙 • [JL ‘84] 𝑓 𝑙 𝑌, 𝑍 ≥ 𝑑 log log 𝑙 .

Open Problems • Can the upper bound of ∼ log 𝑙 / log log 𝑙 be improved? • Are there any 𝑌 and 𝑍 with 𝑓 𝑙 (𝑌, 𝑍) ≫ log 𝑙 ? • Ball ‘92: Is it true that 𝑓 𝑙 ℓ 2 , ℓ 1 ≤ 𝐷 < ∞ ?

Graph Sparsification Given: a huge graph 𝐻 Goal: find a “simpler” graph 𝐼 “similar” to 𝐻 • compact representation • algorithms work faster on the new graph • can obtain better approximation results

Bottleneck & Routing Problems

Bottleneck Problem • graph 𝐻 = (𝑊, 𝐹) with edge capacities 𝑑 𝑓 • set of terminals 𝑈 ⊂ 𝑊 𝑢 2 𝑢 4 𝑢 1 𝑢 3 For 𝑇 ⊂ 𝑈 , bk 𝐻 𝑇 is the capacity of the minimum capacity cut in 𝐻 that separates 𝑇 and 𝑈 ∖ 𝑇 in 𝐻 .

Bottleneck Problem ′ is [Moitra ‘09] Graph 𝐼 = (𝑈, 𝐹′) with capacities 𝑑 𝑓 a vertex cut sparsifier for 𝐻 with distortion 𝐸 ≥ 1 if bk 𝐻 𝑇 ≤ bk 𝐼 𝑇 ≤ 𝐸 ⋅ bk 𝐻 𝑇 ∀𝑇 ⊂ 𝑈 𝑢 2 𝑢 4 𝑢 1 𝑢 3 Given 𝐼 , can easily compute bottlenecks between terminals in the network!

Network Routing Problem Routing problem: send a certain amount of data 𝑒 𝑗𝑘 from each terminal 𝑢 𝑗 to 𝑢 𝑘 so that the total amount sent over each edge 𝑓 is at most its capacity 𝑑 𝑓 . [LM ‘10] A vertex flow sparsifier is an analogue of a vertex cut sparsifier for the network routing problem .

Known Results Moitra ‘09 and Leigthon and Moitra ’10 𝑙 = 𝑈 • 𝐷 log 𝑙 / log log 𝑙 existential upper bound • 𝐷 log 2 𝑙 / log log 𝑙 algorithmic upper bound • 𝐷 > 1 lower bound for cut sparsifiers • Ω(log log 𝑙) lower bound for flow sparsifiers Open Questions: • 𝐷 log 𝑙 / log log 𝑙 algorithmic upper bound? • Better lower bounds? • Better upper bounds?

Papers on Vertex Sparsification Charikar , Leighton, Li and Moitra ’10 Englert, Gupta, Krauthgamer, Räcke, Talgam and Talwar ’10 Makarychev and Makarychev ’10

Main Results • Define “Metric Sparsifiers” • Give 𝐷 log 𝑙 / log log 𝑙 algorithmic upper bound [independently, CLLM ‘10, EGKRTT ‘10] • Establish a direct connection between Vertex Sparsifiers and Lipschitz Extendability 𝑑𝑣𝑢 = 𝑓 𝑙 (ℓ 1 , ℓ 1 ) 𝑅 𝑙 𝑔𝑚𝑝𝑥 = 𝑓 𝑙 ℓ ∞ , ℓ ∞ ⊕ 1 … ⊕ 1 ℓ ∞ 𝑅 𝑙

Lower bounds via Lipschitz Extendability Using lower bounds for “projection constants” [Grünbaum ’60] , we get 𝑔𝑚𝑝𝑥 ≥ 𝑓 𝑙 (ℓ ∞ , ℓ 1 ) ≥ 𝐷 log 𝑙/log log 𝑙 𝑅 𝑙 Figiel, Johnson, and Schechtman ’88 implies 𝑑𝑣𝑢 = 𝑓 𝑙 (ℓ 1 , ℓ 1 ) ≥ 𝐷 log 𝑙 𝑅 𝑙 log log 𝑙

𝑑𝑣𝑢 ≤ 𝑅 ≡ 𝑓 𝑙 (ℓ 1 , ℓ 1 ) Proof Idea: 𝑅 𝑙 Consider a game: 𝐻 and {𝑑 𝑓 } are fixed ′ Alice: defines 𝐼 by providing 𝑑 𝑓 Bob: presents 𝑇 1 , 𝑈 ∖ 𝑇 1 and (𝑇 2 , 𝑈 ∖ 𝑇 2 ) bk 𝐻 S 1 ≤ bk 𝐼 S 1 and bk 𝐼 S 2 ≤ 𝑅 bk 𝐻 S 2 yes no Alice wins Bob wins

𝑑𝑣𝑢 ≤ 𝑅 ≡ 𝑓 𝑙 (ℓ 1 , ℓ 1 ) Proof Idea: 𝑅 𝑙 Consider a game: 𝐻 and {𝑑 𝑓 } are fixed Bob: distribution of 𝑇 1 , 𝑈 ∖ 𝑇 1 and (𝑇 2 , 𝑈 ∖ 𝑇 2 ) ′ Alice: defines 𝐼 by providing 𝑑 𝑓 𝔽 bk 𝐻 S 1 ≤ 𝔽 bk 𝐼 S 1 𝔽 bk 𝐼 S 2 ≤ 𝑅 𝔽 bk 𝐻 S 2 yes no Alice wins Bob wins

Distribution of cuts Distribution 𝒠 of cuts (𝑇, 𝑈 ∖ 𝑇) on 𝑈 defines a map 𝑔: 𝑈 → 𝑀 1 (Ω, 𝜈) : 𝑔 𝑣 = ቊ0, if 𝑦 ∈ 𝑇 1, if 𝑦 ∉ 𝑇 𝑢 2 𝑢 4 𝑢 1 𝑢 3

Distribution of cuts Distribution 𝒠 of cuts (𝑇, 𝑈 ∖ 𝑇) on 𝑈 defines a map 𝑔: 𝑈 → 𝑀 1 (Ω, 𝜈) : 𝑔 𝑣 = ቊ0, if 𝑦 ∈ 𝑇 1, if 𝑦 ∉ 𝑇 (0,1) (0,0) (1,1) (1,0)

Distribution of cuts 𝑔 𝑣 = ቊ 0, if 𝑦 ∈ 𝑇 1, if 𝑦 ∉ 𝑇 Pr 𝑣, 𝑤 are separated by 𝑇 = 𝑔 𝑣 − 𝑔 𝑤 1 𝑑 ′ 𝑣, 𝑤 ⋅ 𝑔 𝑣 − 𝑔 𝑤 𝔽 bk 𝐼 𝑇 = ෍ 1 𝑣,𝑤∈𝑈 𝑑 ′ 𝑣, 𝑤 ⋅ 𝑔 𝑣 − ሚ ሚ 𝔽 bk 𝐻 𝑇 = min ෍ 𝑔 𝑤 1 ሚ 𝑔 𝑣,𝑤∈𝑊

Bob: gives maps 𝑔 1 : 𝑈 → 𝑀 1 and 𝑔 2 : 𝑈 → 𝑀 1 Need: bk 𝐻 ෩ 𝑔 1 ≤ bk 𝐼 𝑔 1 2 ≤ 𝑅 bk 𝐻 ෩ bk 𝐼 𝑔 𝑔 2 −1 −1 and ෩ 1 ෩ Relate 𝑔 1 𝑔 𝑔 𝑔 2 2 𝑔 𝑀 1 1 −1 𝑔 1 𝑔 2 𝑔 2 𝑀 1

Bob: gives maps 𝑔 1 : 𝑈 → 𝑀 1 and 𝑔 2 : 𝑈 → 𝑀 1 Need: bk 𝐻 ෩ 𝑔 1 ≤ bk 𝐼 𝑔 1 2 ≤ 𝑅 bk 𝐻 ෩ bk 𝐼 𝑔 𝑔 2 −1 −1 and ෩ 1 ෩ Relate 𝑔 1 𝑔 𝑔 𝑔 2 2 ෩ 𝑔 1 𝑀 1 −1 ෩ 1 ෩ ෩ 𝑔 𝑔 𝑔 2 2 𝑀 1

Ball’s Open Problem & Sparsification [MM ‘10] 𝑑𝑣𝑢 = 𝑓 𝑙 ℓ 1 , ℓ 1 ≤ 𝑓 𝑙 ℓ 2 , ℓ 1 ⋅ 𝐷 log 𝑙 log log 𝑙 𝑅 𝑙 here, 𝐷 log 𝑙 log log 𝑙 is the distortion of the Fréchet embedding of ℓ 1 into ℓ 2 by Arora, Lee, Naor ’07. If 𝑓 𝑙 ℓ 2 , ℓ 1 ≤ 𝐷 Ball then 𝑑𝑣𝑢 ≤ 𝐷′ log 𝑙 log log 𝑙 𝑅 𝑙

Outer bi-Lipschitz extendibility