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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

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SLIDE 1

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 β‡’ TTIC Konstantin Makarychev, Northwestern Ilya Razenshteyn, Microsoft Research

University of Notre Dame, March 21, 2018

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SLIDE 2

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

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SLIDE 3

Hahn–Banach Theorem

Let π‘Š be a normed space and 𝑀 βŠ‚ π‘Š be its linear

  • subspace. Every bounded linear map

𝑔: 𝑀 β†’ ℝ can be extended to ሚ 𝑔 ∢ π‘Š β†’ ℝ so that ሚ 𝑔 = 𝑔 Is there an analogue for

  • Lipschitz maps?
  • maps into ℝ𝑒 or other normed spaces?
  • H. Hahn
  • S. Banah
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SLIDE 4

Preliminaries

𝑔: π‘Œ β†’ 𝑍 is Lipschitz if for every 𝑣, 𝑀 ∈ π‘Œ 𝑒𝑍 𝑔 𝑣 , 𝑔 𝑀 ≀ π·π‘’π‘Œ(𝑣, 𝑀) The Lipschitz constant 𝑔 π‘€π‘—π‘ž of 𝑔 is the minimum 𝐷 s.t. that the inequality holds. 𝑔 is bi-Lipschitz if for some 𝐷1, 𝐷2 > 0, every 𝑣, 𝑀 ∈ π‘Œ 𝐷1π‘’π‘Œ 𝑣, 𝑀 ≀ 𝑒𝑍 𝑔 𝑣 , 𝑔 𝑀 ≀ 𝐷2π‘’π‘Œ(𝑣, 𝑀) The bi-Lipschitz constant or distortion 𝐸(𝑔) of 𝑔 is the minimum of 𝐷2/𝐷1 s.t. that the inequality holds.

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SLIDE 5

Preliminaries

β„“π‘ž

𝑒 is ℝ𝑒 equipped with the β‹… π‘ž norm:

𝑦 π‘ž = ෍ 𝑦𝑗 π‘ž

1/π‘ž

𝑦 ∞ = max |𝑦𝑗|

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SLIDE 6

McShane–Whitney Theorem

β€œnon-linear Hahn–Banach”

Let (π‘Œ, 𝑒) be a metric space and 𝐡 βŠ‚ π‘Œ. Every Lipschitz map 𝑔: 𝐡 β†’ ℝ can be extended to ሚ 𝑔 ∢ π‘Œ β†’ ℝ so that ሚ 𝑔 π‘€π‘—π‘ž = 𝑔 π‘€π‘—π‘ž

  • E. McShane
  • H. Whitney

ℝ π‘Œ ሚ 𝑔(𝑦)

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SLIDE 7

Kirszbraun Theorem

Let 𝐡 βŠ‚ β„“2

𝑛. Every Lipschitz map 𝑔: 𝐡 β†’ β„“2 π‘œ can

be extended to ሚ 𝑔 ∢ β„“2

𝑛 β†’ β„“2 π‘œ so that

ሚ 𝑔 π‘€π‘—π‘ž = 𝑔 π‘€π‘—π‘ž ⟢

β„“2

𝑛

β„“2

π‘œ

𝑔

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SLIDE 8

Lipschitz Extension Constant

Let π‘Œ be a metric space and 𝑍 be a normed space. 𝑓𝑙(π‘Œ, 𝑍) is the min 𝐷 s.t. for every 𝐡 βŠ‚ π‘Œ of size ≀ 𝑙 and 𝑔: 𝐡 β†’ 𝑍 there exists an extension ሚ 𝑔: π‘Œ β†’ 𝑍 with ሚ 𝑔 π‘€π‘—π‘ž ≀ 𝐷 𝑔 π‘€π‘—π‘ž McShane–Whitney: 𝑓𝑙 π‘Œ, ℝ = 1 Kirszbraun: 𝑓𝑙 β„“2, β„“2 = 1

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SLIDE 9

Lipschitz Extension Constant

In general, 𝑓𝑙 π‘Œ, 𝑍 > 1 E.g., 𝑓3 β„“1

3, β„“2 2 β‰₯

3

𝐡 = 𝑏, 𝑐, 𝑑 𝑏 = 1,0,0 𝑐 = 0,1,0 𝑑 = 0,0,1 𝑒 = (0,0,0) 𝑔(𝑏) 𝑔(𝑐) 𝑔(𝑑) β„“2

2

2 2 2

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SLIDE 10

π‘Œ β†’ 𝑍 𝑓𝑙(π‘Œ, 𝑍) any β†’ ℝ or β„“βˆž 1 McShane, Whitney ’34 β„“2 β†’ β„“2 1

Kirszbraun ’34

β„“π‘ž β†’ β„“2 π‘ž ≀ 2 ≀ π·π‘ž log 𝑙

1 π‘žβˆ’1 2

Marcus, Pisier ’84 1 < π‘ž < 2 β‰₯ π‘‘π‘ž log 𝑙 log log 𝑙

1 π‘žβˆ’1 2 Johnson, Lindenstrauss

’84 any β†’ β„“2 ≀ 𝐷 log 𝑙 JL ’84 β„“π‘ž β†’ β„“π‘Ÿ 1 < π‘Ÿ ≀ 2 ≀ π‘ž ≀ 24

π‘žβˆ’1 π‘Ÿβˆ’1

Naor, Peres, Schramm, Sheffield ’04

  • ther

π‘ž, π‘Ÿ ∈ (1, ∞) 𝑓𝑙 β†’ ∞ as 𝑙 β†’ ∞

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SLIDE 11

Extension Results

Johnson–Lindenstruass–Schechtman ’86 𝑓𝑙 π‘Œ, 𝑍 ≀ 𝐷 log 𝑙 Lee–Naor ’03 𝑓𝑙 π‘Œ, 𝑍 ≀ 𝐷 log 𝑙 log log 𝑙 Best lower bounds are: 𝑓𝑙 β‰Ώ 𝑑 log 𝑙 Open Problem: what is the dependence of 𝑓𝑙 on 𝑙 ?

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[JL ’84] Technique for proving lower bounds on 𝑓𝑙 π‘Œ, 𝑍

Prove a lower bound for linear extensions Reinterpret it as a lower bound for Lipschitz extensions

  • Linear extension (β€œprojection”)

constant is up to dim 𝑍

  • [JL β€˜84] 𝑓𝑙 π‘Œ, 𝑍 β‰₯ 𝑑

log 𝑙 log log 𝑙 .

π‘Œ

𝑍

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SLIDE 13

Open Problems

  • Can the upper bound of ∼ log 𝑙 / log log 𝑙 be

improved?

  • Are there any π‘Œ and 𝑍 with 𝑓𝑙(π‘Œ, 𝑍) ≫

log 𝑙 ?

  • Ball β€˜92: Is it true that 𝑓𝑙 β„“2, β„“1 ≀ 𝐷 < ∞ ?
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SLIDE 14

Graph Sparsification

Given: a huge graph 𝐻 Goal: find a β€œsimpler” graph 𝐼 β€œsimilar” to 𝐻

  • compact representation
  • algorithms work faster on the new graph
  • can obtain better approximation results
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Bottleneck & Routing Problems

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SLIDE 16

Bottleneck Problem

  • graph 𝐻 = (π‘Š, 𝐹) with edge capacities 𝑑𝑓
  • set of terminals π‘ˆ βŠ‚ π‘Š

For 𝑇 βŠ‚ π‘ˆ, bk𝐻 𝑇 is the capacity of the minimum capacity cut in 𝐻 that separates 𝑇 and π‘ˆ βˆ– 𝑇 in 𝐻.

𝑒1 𝑒2 𝑒3 𝑒4

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SLIDE 17

Bottleneck Problem

[Moitra β€˜09] Graph 𝐼 = (π‘ˆ, 𝐹′) with capacities 𝑑𝑓

β€² is

a vertex cut sparsifier for 𝐻 with distortion 𝐸 β‰₯ 1 if bk𝐻 𝑇 ≀ bk𝐼 𝑇 ≀ 𝐸 β‹… bk𝐻 𝑇 βˆ€π‘‡ βŠ‚ π‘ˆ Given 𝐼, can easily compute bottlenecks between terminals in the network!

𝑒1 𝑒2 𝑒3 𝑒4

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SLIDE 18

Network Routing Problem

Routing problem: send a certain amount of data π‘’π‘—π‘˜ from each terminal 𝑒𝑗 to π‘’π‘˜ so that the total amount sent over each edge 𝑓 is at most its capacity 𝑑𝑓. [LM β€˜10] A vertex flow sparsifier is an analogue of a vertex cut sparsifier for the network routing problem.

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SLIDE 19

Known Results

Moitra β€˜09 and Leigthon and Moitra ’10

𝑙 = π‘ˆ

  • 𝐷 log 𝑙 / log log 𝑙

existential upper bound

  • 𝐷 log2 𝑙 / log log 𝑙

algorithmic upper bound

  • 𝐷 > 1

lower bound for cut sparsifiers

  • Ξ©(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?
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SLIDE 20

Papers on Vertex Sparsification

Charikar, Leighton, Li and Moitra ’10 Englert, Gupta, Krauthgamer, RΓ€cke, Talgam and Talwar ’10 Makarychev and Makarychev ’10

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SLIDE 21

Main Results

  • Define β€œMetric Sparsifiers”
  • Give 𝐷 log 𝑙 / log log 𝑙 algorithmic upper bound

[independently, CLLM β€˜10, EGKRTT β€˜10]

  • Establish a direct connection between Vertex

Sparsifiers and Lipschitz Extendability

𝑅𝑙

𝑑𝑣𝑒 = 𝑓𝑙(β„“1, β„“1)

𝑅𝑙

π‘”π‘šπ‘π‘₯ = 𝑓𝑙 β„“βˆž, β„“βˆž βŠ•1 … βŠ•1 β„“βˆž

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SLIDE 22

Lower bounds via Lipschitz Extendability

Using lower bounds for β€œprojection constants” [GrΓΌnbaum ’60], we get 𝑅𝑙

π‘”π‘šπ‘π‘₯ β‰₯ 𝑓𝑙(β„“βˆž, β„“1) β‰₯ 𝐷 log 𝑙/log log 𝑙

Figiel, Johnson, and Schechtman ’88 implies 𝑅𝑙

𝑑𝑣𝑒 = 𝑓𝑙(β„“1, β„“1) β‰₯ 𝐷 log 𝑙

log log 𝑙

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Proof Idea: 𝑅𝑙

𝑑𝑣𝑒 ≀ 𝑅 ≑ 𝑓𝑙(β„“1, β„“1)

Consider a game: 𝐻 and {𝑑𝑓} are fixed Alice: defines 𝐼 by providing 𝑑𝑓

β€²

Bob: presents 𝑇1, π‘ˆ βˆ– 𝑇1 and (𝑇2, π‘ˆ βˆ– 𝑇2) bk𝐻 S1 ≀ bk𝐼 S1 and bk𝐼 S2 ≀ 𝑅 bk𝐻 S2 Alice wins Bob wins

yes no

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SLIDE 24

Proof Idea: 𝑅𝑙

𝑑𝑣𝑒 ≀ 𝑅 ≑ 𝑓𝑙(β„“1, β„“1)

Consider a game: 𝐻 and {𝑑𝑓} are fixed Bob: distribution of 𝑇1, π‘ˆ βˆ– 𝑇1 and (𝑇2, π‘ˆ βˆ– 𝑇2) Alice: defines 𝐼 by providing 𝑑𝑓

β€²

𝔽bk𝐻 S1 ≀ 𝔽bk𝐼 S1 𝔽bk𝐼 S2 ≀ 𝑅 𝔽bk𝐻 S2 Alice wins Bob wins

yes no

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SLIDE 25

Distribution of cuts

Distribution 𝒠 of cuts (𝑇, π‘ˆ βˆ– 𝑇) on π‘ˆ defines a map 𝑔: π‘ˆ β†’ 𝑀1(Ξ©, 𝜈): 𝑔 𝑣 = α‰Š0, if 𝑦 ∈ 𝑇 1, if 𝑦 βˆ‰ 𝑇

𝑒1 𝑒2 𝑒3 𝑒4

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SLIDE 26

Distribution of cuts

Distribution 𝒠 of cuts (𝑇, π‘ˆ βˆ– 𝑇) on π‘ˆ defines a map 𝑔: π‘ˆ β†’ 𝑀1(Ξ©, 𝜈): 𝑔 𝑣 = α‰Š0, if 𝑦 ∈ 𝑇 1, if 𝑦 βˆ‰ 𝑇

(0,0) (1,1) (1,0) (0,1)

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SLIDE 27

Distribution of cuts

𝑔 𝑣 = α‰Š0, if 𝑦 ∈ 𝑇 1, if 𝑦 βˆ‰ 𝑇 Pr 𝑣, 𝑀 are separated by 𝑇 = 𝑔 𝑣 βˆ’ 𝑔 𝑀

1

𝔽 bk𝐼 𝑇 = ෍

𝑣,π‘€βˆˆπ‘ˆ

𝑑′ 𝑣, 𝑀 β‹… 𝑔 𝑣 βˆ’ 𝑔 𝑀

1

𝔽 bk𝐻 𝑇 = min

ሚ 𝑔

෍

𝑣,π‘€βˆˆπ‘Š

𝑑′ 𝑣, 𝑀 β‹… ሚ 𝑔 𝑣 βˆ’ ሚ 𝑔 𝑀

1

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SLIDE 28

Bob: gives maps 𝑔

1: π‘ˆ β†’ 𝑀1 and 𝑔 2: π‘ˆ β†’ 𝑀1

Need: bk𝐻 ΰ·© 𝑔

1 ≀ bk𝐼 𝑔 1

bk𝐼 𝑔

2 ≀ 𝑅 bk𝐻 ΰ·©

𝑔

2

𝑀1 𝑀1 𝑔

1

𝑔

2

Relate 𝑔

1𝑔 2 βˆ’1and ΰ·©

𝑔

1 ΰ·©

𝑔

2 βˆ’1

𝑔

1𝑔 2 βˆ’1

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SLIDE 29

Bob: gives maps 𝑔

1: π‘ˆ β†’ 𝑀1 and 𝑔 2: π‘ˆ β†’ 𝑀1

Need: bk𝐻 ΰ·© 𝑔

1 ≀ bk𝐼 𝑔 1

bk𝐼 𝑔

2 ≀ 𝑅 bk𝐻 ΰ·©

𝑔

2

𝑀1 𝑀1 ΰ·© 𝑔

1

ΰ·© 𝑔

2

ΰ·© 𝑔

1 ΰ·©

𝑔

2 βˆ’1

Relate 𝑔

1𝑔 2 βˆ’1and ΰ·©

𝑔

1 ΰ·©

𝑔

2 βˆ’1

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SLIDE 30

Ball’s Open Problem & Sparsification

[MM β€˜10] 𝑅𝑙

𝑑𝑣𝑒 = 𝑓𝑙 β„“1, β„“1 ≀ 𝑓𝑙 β„“2, β„“1 β‹… 𝐷 log 𝑙 log log 𝑙

here, 𝐷 log 𝑙 log log 𝑙 is the distortion of the FrΓ©chet embedding of β„“1 into β„“2 by Arora, Lee, Naor ’07. If 𝑓𝑙 β„“2, β„“1 ≀ 𝐷Ball then 𝑅𝑙

𝑑𝑣𝑒 ≀ 𝐷′ log 𝑙 log log 𝑙

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SLIDE 31

Outer bi-Lipschitz extendibility

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SLIDE 32

1

Bi-Lipschitz Kirszbraun Theorem?

Let 𝐡 βŠ‚ β„“2

𝑛. Can we extend a bi-Lipschitz map 𝑔: 𝐡 β†’ β„“2 π‘œ

to a bi-Lipschitz map ሚ 𝑔: β„“2

𝑛 β†’ β„“2 π‘œ ?

No!

  • ℝ2 β†’ ℝ. Assume ℝ βŠ‚ ℝ2. Extend f = 𝑗𝑒ℝ ?

There is even no injective extension of 𝑔 to ℝ2.

  • ℝ β†’ ℝ. 𝑔 maps 0, 1, 2 to 0, 2, 1, respectively. There is

even no continuous one-to-one extension. 2 1 2

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SLIDE 33

Bi-Lipschitz Kirszbraun Theorem?

Can overcome these obstacles by using extra dimensions!

  • ℝ2 β†’ ℝ. Assume ℝ βŠ‚ ℝ2. Extend f = 𝑗𝑒ℝ ?

Assume that target ℝ βŠ‚ ℝ2. Then ሚ 𝑔 = 𝑗𝑒ℝ2→ℝ2

  • ℝ β†’ ℝ. 𝑔 maps 0, 1, 2 to 0, 2, 1, respectively.

1 2

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SLIDE 34

Outer bi-Lipschitz extension

Given 𝐡 βŠ‚ β„“2

𝑛 and bi-Lipschitz 𝑔: 𝐡 β†’ β„“2 π‘œ .

Assume β„“2

π‘œ βŠ‚ β„“2 𝑂.

ሚ 𝑔: β„“2

𝑛 β†’ β„“2 𝑂 is an outer bi-Lipschitz extension of 𝑔 if

ሚ 𝑔 𝑏 = 𝑔(𝑏) for every 𝑏 ∈ 𝐡

and ሚ

𝑔 is bi-Lipschitz.

[MMMR ’18] For every bi-Lipschitz 𝑔: 𝐡 β†’ β„“2

π‘œ, there

exists an outer bi-Lipschitz extension with 𝐸 ሚ 𝑔 ≀ 3𝐸 𝑔 .

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SLIDE 35

Outer bi-Lipschitz extension

Near isometric case. What happens when 𝐸(𝑔) = 1 + 𝜁 ? If 𝜁 = 0, i.e., 𝑔 is an isometric map, there is an isometric extension ሚ 𝑔 Is there a bi-Lipschitz extension with 𝐸 ሚ 𝑔 = 1 + 𝑝(1) as 𝜁 β†’ 0 ? For 𝑔: 𝐡 β†’ ℝ, 𝐡 βŠ‚ ℝ, yes! There is an extension with 𝐸( ሚ 𝑔) = 1 + 1 log2 1/𝜁 The bound is tight.

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SLIDE 36

Outer bi-Lipschitz extension

Near isometric case. What happens when 𝐸(𝑔) = 1 + 𝜁 ? If 𝜁 = 0, i.e., 𝑔 is an isometric map, there is an isometric extension ሚ 𝑔 Is there a bi-Lipschitz extension with 𝐸 ሚ 𝑔 = 1 + 𝑝(1) as 𝜁 β†’ 0 ? For a one-point extension of 𝑔: 𝐡 β†’ β„“2

π‘œ

𝐸( ሚ 𝑔) = 1 + 𝐷 𝜁 The bound is tight.

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SLIDE 37

Summary

Characterized the optimal distortion of cut and flow vertex sparsifiers in terms of Lipschitz extension constants.

  • Find 𝑓𝑙 β„“1, β„“1 and 𝑓𝑙(β„“βˆž, β„“βˆž βŠ•1 … βŠ•1 β„“βˆž)
  • Is 𝑓𝑙 β„“2, β„“1 < ∞ ?

Defined outer bi-Lipschitz extension and proved an analogue of Kirzsbraun theorem for it. Partial results for nearly isometric maps.

  • Understand the nearly isometric case.

Applications to a prioritized dimension reduction.