Introduction Dichotomies Counter Example A Proof

Metric Dichotomies Manor Mendel The Open University of Israel

May ’07

Manor Mendel

Metric Dichotomies

Introduction Dichotomies Counter Example A Proof

Metric Embedding

(X , ρ), (H, ν): Metric spaces. f : (X , ρ) → (H, ν) embedding. (Multiplicative) Distortion: dist(f ) = kf kLip ·kf −1 kLip = max x6=y

ρ(x, y ) ν(f (x), f (y )) ·max . x6=y ν(f (x), f (y )) ρ(x, y )

Distortion: A notion of faithfulness of metric embeddings. cH (X ) = inf f :X →H dist(f ): “the best embedding” of X in H. In this talk, X is always finite.

Manor Mendel

Metric Dichotomies

Introduction Dichotomies Counter Example A Proof

History & Motivation BiLipschitz embedding of finite metrics was studied at least since the ’60s (Enflo). During ’80s: studied by Banach space experts [Bourgain,Milman, Wolfsom, Johnson, Lindenstrauss, Matouˇsek, ...] Theorem (Bourgain) cL2 (X ) = O(log |X |), and this is tight in general. Since mid ’90s: by computer scientists (starting with [Linial, London, Rabinovich]). CS motivation: Embedding is used as a step of simplifying the (metric) data.

Manor Mendel

Metric Dichotomies

Introduction Dichotomies Counter Example A Proof

Distortion Dichotomies Property (Dichotomy (X: a class of finite metric spaces)) For any host space H, either ∀X ∈ X, cH (X ) = 1, or ∃∞ X ∈ X for which cH (X ) is “large”. CS motivation (Arora,Lov´asz, Newman, Rabani, Rabinovich, Vempala et. al.) 1

Embedding is used as a step of simplifying the data.

2

Distortion affects the quality of the algorithm. A priori: Expect a trade off between the simplicity of the host space, and the distortion.

3

Would translate to algorithmic trade off between efficiency and quality. 4

Distortion dichotomies give a one sided limit to this approach. Manor Mendel

Metric Dichotomies

Introduction Dichotomies Counter Example A Proof

Distortion Dichotomies Property (Dichotomy (X: a class of finite metric spaces)) For any host space H, either ∀X ∈ X, cH (X ) = 1, or ∃∞ X ∈ X for which cH (X ) is “large”. CS motivation (Arora,Lov´asz, Newman, Rabani, Rabinovich, Vempala et. al.) 1

Embedding is used as a step of simplifying the data.

2

Distortion affects the quality of the algorithm. A priori: Expect a trade off between the simplicity of the host space, and the distortion.

3

Would translate to algorithmic trade off between efficiency and quality. 4

Distortion dichotomies give a one sided limit to this approach. Manor Mendel

Metric Dichotomies

Introduction Dichotomies Counter Example A Proof

Classes of Metrics

In this talk X will be one of: Metric class Finite subsets of R Finite subsets of L1 Finite metric spaces Finite tree metrics

Nickname Line metrics L1 metrics L∞ metrics Tree metrics

Manor Mendel

Regular subclass Path metrics Hamming cubes L∞ grids Binary trees

Metric Dichotomies

Introduction Dichotomies Counter Example A Proof

“Weak” Dichotomies [Matouˇsek]

Property (Weak dichotomy for X) For any host space H, either supX ∈X cH (X ) = 1, or supX ∈X cH (X ) = ∞. Theorem ([Matouˇsek]) Lp metrics (p ∈ [1, ∞]), line metrics, and equilateral metrics have the weak dichotomy property.

Manor Mendel

Metric Dichotomies

Introduction Dichotomies Counter Example A Proof

Line Metrics Theorem (Matouˇsek) For every metric space H, either supL⊂R cH (L) = 1, or ∃β > 0, such that cH (Pn ) ≥ Ω(nβ ), where Pn is n-point path metric. Proof. Quantitative differentiation argument. Outline in the end of the talk, if time permits. Tightness: ∀β ∈ (0, 1], ∃Hβ such that ∀L line metric, cHβ = O(|L|β ), and cHβ (Pn ) ≥ (n − 1)β . Manor Mendel

Metric Dichotomies

Introduction Dichotomies Counter Example A Proof

Line Metrics Theorem (Matouˇsek) For every metric space H, either supL⊂R cH (L) = 1, or ∃β > 0, such that cH (Pn ) ≥ Ω(nβ ), where Pn is n-point path metric. Proof. Quantitative differentiation argument. Outline in the end of the talk, if time permits. Tightness: ∀β ∈ (0, 1], ∃Hβ such that ∀L line metric, cHβ = O(|L|β ), and cHβ (Pn ) ≥ (n − 1)β . Manor Mendel

Metric Dichotomies

Introduction Dichotomies Counter Example A Proof

L1 Metrics Theorem (Bourgain, Milman, Wolfson) For every metric space H, either supX ⊂L1 cH (X ) = 1, or ∃β > 0, such that cH ({0, 1}n ) ≥ Ω(nβ ), where {0, 1}n is n-dim’ Hamming cube. Proof. The proof uses a non-linear analogue of the “type property” of Banach spaces. Tightness: When H = L2 , √ cL2 ({0, 1}n ) ≥ n, and p ∀X L1 metric, cL2 (X ) = O( |X | log log |X |) [Arora, Lee, Naor]. Manor Mendel

Metric Dichotomies

Introduction Dichotomies Counter Example A Proof

L1 Metrics Theorem (Bourgain, Milman, Wolfson) For every metric space H, either supX ⊂L1 cH (X ) = 1, or ∃β > 0, such that cH ({0, 1}n ) ≥ Ω(nβ ), where {0, 1}n is n-dim’ Hamming cube. Proof. The proof uses a non-linear analogue of the “type property” of Banach spaces. Tightness: When H = L2 , √ cL2 ({0, 1}n ) ≥ n, and p ∀X L1 metric, cL2 (X ) = O( |X | log log |X |) [Arora, Lee, Naor]. Manor Mendel

Metric Dichotomies

Introduction Dichotomies Counter Example A Proof

L∞ metrics Theorem (M., Naor) For every metric space H, either supX cH (X ) = 1, or ∃β > 0, such that cH ([n]n∞ ) ≥ Ω(nβ ), where [m]n∞ is the [m]n grid with the L∞ distance. Proof. The proof uses a non-linear analogue of the “cotype property” of Banach spaces. Tightness: when H = L2 , cL2 (X ) = O(log |X |), and this is tight [Bourgain]. Open Question Find the strongest quantitative dichotomy for L∞ metrics. Manor Mendel

Metric Dichotomies

Introduction Dichotomies Counter Example A Proof

L∞ metrics Theorem (M., Naor) For every metric space H, either supX cH (X ) = 1, or ∃β > 0, such that cH ([n]n∞ ) ≥ Ω(nβ ), where [m]n∞ is the [m]n grid with the L∞ distance. Proof. The proof uses a non-linear analogue of the “cotype property” of Banach spaces. Tightness: when H = L2 , cL2 (X ) = O(log |X |), and this is tight [Bourgain]. Open Question Find the strongest quantitative dichotomy for L∞ metrics. Manor Mendel

Metric Dichotomies

Introduction Dichotomies Counter Example A Proof

L∞ metrics Theorem (M., Naor) For every metric space H, either supX cH (X ) = 1, or ∃β > 0, such that cH ([n]n∞ ) ≥ Ω(nβ ), where [m]n∞ is the [m]n grid with the L∞ distance. Proof. The proof uses a non-linear analogue of the “cotype property” of Banach spaces. Tightness: when H = L2 , cL2 (X ) = O(log |X |), and this is tight [Bourgain]. Open Question Find the strongest quantitative dichotomy for L∞ metrics. Manor Mendel

Metric Dichotomies

Introduction Dichotomies Counter Example A Proof

Tree Metrics The class of tree metrics do not have a comparable dichotomy. Theorem (M., Naor) For any A ≥ 1, there exists a metric space H such that supT cH (T ) = A, where T ranges over the finite tree metrics. root

The metric H = HA : The infinite binary tree B∞ , with “horizontal” distances being contracted by a factor of A. Easy to check that: 1

dA is a metric.

2

cdA (B∞ ) ≤ A (the identity embedding).

lca(x, y) a

b x

dε (x, y) = b + 2a/A

The proof that limn→∞ cdA (Bn ) = A is lengthy. Manor Mendel

Metric Dichotomies

y

Introduction Dichotomies Counter Example A Proof

Line Metrics

Theorem (Matouˇsek) For every metric space H, either supL⊂R cH (L) = 1, or ∃β > 0, such that cH (Pn ) ≥ Ω(nβ ), where Pn is n-point path metric.

Can be proved by a simple discrete differentiation argument. I’ll show an equivalent argument that can be extended for L1 metrics and L∞ metrics. Consider it as a toy case for the L1 metrics and L∞ metrics.

Manor Mendel

Metric Dichotomies

Introduction Dichotomies Counter Example A Proof

Let Ln (H) be the smallest L such that ∀f : Pn → H ∈ H, dH (f0 , fn−1 ) ≤ L(n − 1)

max

i=0,...n−2

dH (fi , fi+1 ).

Lemma 1 L (H) ≤ 1. n 2

cH (Pn ) ≥ 1/Ln (H).

3

If Ln (H) = 1, then cH (Pn ) = 1.

4

Lmn (H) ≤ Lm (H) · Ln (H).

Proof.

Manor Mendel

Metric Dichotomies

(1)

Introduction Dichotomies Counter Example A Proof

Let Ln (H) be the smallest L such that ∀f : Pn → H ∈ H, dH (f0 , fn−1 ) ≤ L(n − 1)

max

i=0,...n−2

dH (fi , fi+1 ).

Lemma 1 L (H) ≤ 1. n 2

cH (Pn ) ≥ 1/Ln (H).

3

If Ln (H) = 1, then cH (Pn ) = 1.

4

Lmn (H) ≤ Lm (H) · Ln (H).

Proof. 1. Triangle inequality.

Manor Mendel

Metric Dichotomies

(1)

Introduction Dichotomies Counter Example A Proof

Let Ln (H) be the smallest L such that ∀f : Pn → H ∈ H, dH (f0 , fn−1 ) ≤ L(n − 1)

max

i=0,...n−2

dH (fi , fi+1 ).

(1)

Lemma 1 L (H) ≤ 1. n 2

cH (Pn ) ≥ 1/Ln (H).

3

If Ln (H) = 1, then cH (Pn ) = 1.

4

Lmn (H) ≤ Lm (H) · Ln (H).

Proof. 2. Fix f : Pn → H. Plugging into (1) n kf −1 k

≤ dH (f (0), f (n−1)) ≤ Ln (H)n Lip

max

i=0,...n−2

dH (f (i), f (i+1))

≤ Ln (H)nkf kLip , Manor Mendel

Metric Dichotomies

Introduction Dichotomies Counter Example A Proof

Let Ln (H) be the smallest L such that ∀f : Pn → H ∈ H, dH (f0 , fn−1 ) ≤ L(n − 1)

max

i=0,...n−2

dH (fi , fi+1 ).

Lemma 1 L (H) ≤ 1. n 2

cH (Pn ) ≥ 1/Ln (H).

3

If Ln (H) = 1, then cH (Pn ) = 1.

4

Lmn (H) ≤ Lm (H) · Ln (H).

Proof. 3. Ln (H) = 1 ⇒ ∃f : Pn → H for which dH (f0 , fn−1 ) = n maxi=0,...n−2 dH (fi , fi+1 ), implying that for all i ∈ {0, . . . , n − 2}, dH (f0 , fn−1 ) = (n − 1)dH (fi , fi+1 ).

Manor Mendel

Metric Dichotomies

(1)

Introduction Dichotomies Counter Example A Proof

Let Ln (H) be the smallest L such that ∀f : Pn → H ∈ H, dH (f0 , fn−1 ) ≤ L(n − 1)

max

i=0,...n−2

dH (fi , fi+1 ).

(1)

Lemma 1 L (H) ≤ 1. n 2

cH (Pn ) ≥ 1/Ln (H).

3

If Ln (H) = 1, then cH (Pn ) = 1.

4

Lmn (H) ≤ Lm (H) · Ln (H).

Proof. 4. Fix f : Pmn → H. Define g : Pn → H, by g (i) = f (im). Applying (1) to g , we obtain dH (f0 , fmn−1 ) ≤ Ln (H)n max dH (fim , f(i+1)m ). i=0...n−1

Manor Mendel

Metric Dichotomies

Introduction Dichotomies Counter Example A Proof

Let Ln (H) be the smallest L such that ∀f : Pn → H ∈ H, dH (f0 , fn−1 ) ≤ L(n − 1)

max

i=0,...n−2

dH (fi , fi+1 ).

(1)

Lemma 1 L (H) ≤ 1. n 2

cH (Pn ) ≥ 1/Ln (H).

3

If Ln (H) = 1, then cH (Pn ) = 1.

4

Lmn (H) ≤ Lm (H) · Ln (H).

Proof. 4. Next, define hi : Pm → H, hi (j) = f (im + j), and apply (1) for each hi , and so dH (fim , f(i+1)m ) ≤ Lm (H)m

max

j=0...m−1

Manor Mendel

dH (fim+j , f(i+1)m+j+1 ).

Metric Dichotomies

Introduction Dichotomies Counter Example A Proof

Lemma 1 L (H) ≤ 1. m 2

cH (Pn ) ≥ 1/Ln (H).

3

If Ln (H) = 1, then cH (Pn ) = 1.

4

Lmn (H) ≤ Lm (H) · Ln (H).

Proof of dichotomy for line metrics. If ∀n ∈ N, Ln (H) = 1, then cH (Pn ) = 1, Note that for any finite line metric L, c(Pn )n (L) = 1, and so cH (L) = 1. If ∃n0 for which Ln0 (H) = η < 1, then let β > 0 be such that n0−β = η, and from the submultiplicativity, Lnk (H) ≤ η k = (n0k )−β , and so cH (Pnk ) ≥ (n0k )β . 0

0

Manor Mendel

Metric Dichotomies

Introduction Dichotomies Counter Example A Proof

Lemma 1 L (H) ≤ 1. m 2

cH (Pn ) ≥ 1/Ln (H).

3

If Ln (H) = 1, then cH (Pn ) = 1.

4

Lmn (H) ≤ Lm (H) · Ln (H).

Proof of dichotomy for line metrics. If ∀n ∈ N, Ln (H) = 1, then cH (Pn ) = 1, Note that for any finite line metric L, c(Pn )n (L) = 1, and so cH (L) = 1. If ∃n0 for which Ln0 (H) = η < 1, then let β > 0 be such that n0−β = η, and from the submultiplicativity, Lnk (H) ≤ η k = (n0k )−β , and so cH (Pnk ) ≥ (n0k )β . 0

0

Manor Mendel

Metric Dichotomies

Introduction Dichotomies Counter Example A Proof

L1 metrics and L∞ metrics 1

The same proof outline works for L1 /L∞ metrics.

2

But we use more sophisticated inequalities.

3

E.g., For L∞ : Let Γn (H) be the smallest Γ such that ∀m ∈ N, and ∀f : Znm → H n X i=1

E dH (f (x), f (x + nej ))2

x∈Znm

≤ Γ2 · n2 · n

E

E dH (f (x), f (x + ε))2 .

ε∈{±1}n x∈Znm

4

We argue the same 4 lemmas for Γn (H) instead of Ln (H).

5

The proofs are considerably more complicated. Manor Mendel

Metric Dichotomies

Introduction Dichotomies Counter Example A Proof

Thank You

Manor Mendel

Metric Dichotomies

Metric Dichotomies

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