A Union of Euclidean Spaces is Euclidean. Konstantin Makarychev, Northwestern Yury Makarychev, TTIC

Transcription

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

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

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

4 Our Results Q: Suppose that metric space (X, d) is a union of two metric spaces and B that embed isometrically into l 2. Does X necessarily embed into l 2 with a constant distortion? : Yes, X embeds into l 2 with distortion at most l 2 a with distortion α, B l 2 b with distortion β X = B l 2 a+b+1 with distortion at most 11αβ

5 pproach This talk: consider the isometric case. φ 1 : l 2 φ 2 : B l 2 We will define 3 maps: തφ 1 : B l 2, a 7-Lipschitz extension of φ 1 to X തφ 2 : B l 2, a 7-Lipschitz extension of φ 2 to X Δ x = d x, d(x, B) ψ = തφ 1 തφ 2 Δ

6 pproach ψ = തφ 1 തφ 2 Δ ssume that we have തφ 1 : B l 2, a 7-Lipschitz extension of φ 1 to X തφ 2 : B l 2, a 7-Lipschitz extension of φ 2 to X Δ x = d x, d(x, B) First, ψ Lip = തφ 1 തφ 2 Δ Lip since Δ Lip 2.

7 pproach ψ = തφ 1 തφ 2 Δ തφ 1 ensures that distances between points in don t decrease: തφ 1 ȁ = φ 1 is an isometric embedding of into l 2. തφ 2 ensures that distances between points in B don t decrease. Δ ensures that distances between points a and b B don t decrease by more than a constant factor.

8 pproach ψ = തφ 1 തφ 2 Δ a a If d a, a d(a, b) then തφ 2 (a) തφ 2 (b) തφ 2 a തφ 2 b = d a, b d(a, b) If d a, a d(a, b) then Δ(a) Δ(b) b d(a, a ) d(a, b) B a is the closest point to a in B

9 pproach ψ = തφ 1 തφ 2 Δ a a If d a, a d(a, b) then തφ 2 (a) തφ 2 (b) തφ 2 a തφ 2 b = d a, b d(a, b) If d a, a d(a, b) then Δ(a) Δ(b) b d(a, a ) d(a, b) B

10 Constructing maps തφ 1 and തφ 2 Goal: Given a map φ φ 2 : B l 2 find a Lipschitz extension തφ: B l 2 of φ. തφ B φ l 2

11 Constructing maps തφ 1 and തφ 2 ssume that B l 2 and φ = id; B <. തφ l 2 B

12 Constructing map തφ Idea 1: map every a to the closest a B w.r.t. d. Issue: the map may not be Lipschitz. a തφ a B

13 Cover for Let R a = d a, B for a. C is a cover for if for every a, there is c C s.t. d a, c R a and R c R a for every c, d C: d c, d min R c, R d. R x R x a c d c a is close to some c C points in C are separated

14 Cover for Prove by induction that there is always a cover C. Let c be the point in with the least value of R c. By induction, there is a cover C for Ball c, R c. Let C = C {c}.

15 Cover for Prove by induction that there is always a cover C. Let c be the point in with the least value of R c. By induction, there is a cover C for Ball c, R c. Let C = C {c}. c

16 Cover for Prove by induction that there is always a cover C. Let c be the point in with the least value of R c. By induction, there is a cover C for Ball c, R c. Let C = C {c}. c

17 Cover for Prove by induction that there is always a cover C. Let c be the point in with the least value of R c. By induction, there is a cover C for Ball c, R c. Let C = C {c}. c

18 Constructing map തφ Idea 2: map every c C to the closest c B. The map is 4-Lipschitz. c f c B

19 Constructing map തφ Idea 2: map every c C to the closest c B. The map is 4-Lipschitz. ssume R c R d. c c d d B

20 Constructing map തφ Idea 2: map every c C to the closest c B. The map is 4-Lipschitz. ssume R c R d. c c d d B

21 Constructing map തφ Idea 2: map every c C to the closest c B. The map is 4-Lipschitz. ssume R c R d. c c d d B d c, d 2 d c, d + 2 d c, c 4d(c, d)

22 Kirszbraun Theorem Let C D l 2 and f be a Lipschitz map from C to l 2. There exists an extension g: D l 2 of f such g Lip = f Lip C D l 2

23 Constructing map തφ Idea 2: map every c C to the closest c B. Extend f from C to using the Kirszbraun theorem. c f c B

24 Constructing map തφ Idea 2: map every c C to the closest c B. Extend f from C to using the Kirszbraun theorem. f u, if u തφ u = ቊ u, if u B തφ u is 7-Lipschitz: തφȁ is 4-Lipschitz തφȁ B is 1-Lipschitz തφ a തφ b = f(a) b

25 Constructing map തφ a f(a) b c f(c) B

26 Constructing map തφ a R a c R c R a f(a) 4R a f(c) b B f a b 6R a + d a, b 7d(a, b)

27 Constructing map തφ a R a c R c R a f(a) 4R a f(c) b B f a b 6R a + d a, b 7d(a, b) Q.E.D.

28 Lower Bound There exists a metric space X = B s.t. and B isometrically embed into l 2 every embedding of X into l 2 has distortion at least 3 ε n, where n = = ȁbȁ and ε n 0 as n

29 Open Problems 1. Find the least value of D s.t. if, B l 2 isometrically, then B l 2 with distortion at most D. We know that D 3, Study the problem for other l p. We conjecture that the answer is negative for every p {2, }. 3. What happens if X = 1 k and each i l 2 isometrically? We only know that c log k D 2 Ck. 4. ssume that every subset of X of size ȁxȁ isometrically embeds into l 2. What is the least distortion with which X l 2? More results and open problems in the paper!