Ultrametric skeletons Manor Mendel Open University of Israel

January 2012

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Goal

Given: A metric space (X , d ) & probability measure µ on X . Goal: Finding a structured subset S ⊂ X that captures useful properties of (X , d , µ).

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Structured metrics: Ultrametrics Definition (X , ρ) is called ultrametric if: ∀x, y , z ∈ X , ρ(x, z) ≤ max{ρ(x, y ), ρ(y , z)}.

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Structured metrics: Ultrametrics Definition (X , ρ) is called ultrametric if: ∀x, y , z ∈ X , ρ(x, z) ≤ max{ρ(x, y ), ρ(y , z)}.

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Structured metrics: Ultrametrics Definition (X , ρ) is called ultrametric if: ∀x, y , z ∈ X , ρ(x, z) ≤ max{ρ(x, y ), ρ(y , z)}.

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Structured metrics: Ultrametrics Definition (X , ρ) is called ultrametric if: ∀x, y , z ∈ X , ρ(x, z) ≤ max{ρ(x, y ), ρ(y , z)}.

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Structured metrics: Ultrametrics Definition (X , ρ) is called ultrametric if: ∀x, y , z ∈ X , ρ(x, z) ≤ max{ρ(x, y ), ρ(y , z)}.

Definition (Alternative) ∆ = 10 ∆=8

∆=8

∆=9

∆=7

x1

xn

X

d(xi , xj ) = ∆(lca(xi , xj )) where u = parent(v) ⇒ ∆(v) ≤ ∆(u).

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Properties of UM

Ultrametrics: 1

are tree metrics.

2

are Euclidean metrics

3

have hierarchical clustering

4

support compact DS for fast queries of distances.

5

are amenable for DP optimization of clustering problems like k-median, k-minsum, Σ`p clustering.

6

represents evolution trees. .. .

7

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Exact UM subset is useless

Finding meaningful UM subsets in general metric spaces is hopeless: x

z

y |x − z| > max{|x − y|, y − z|}

No 3 points on the line can be an UM. So the largest UM subset of [0, 1] is of size 2. An UM subset S ⊂ [0, 1] can “capture" the diameter, but not much more. Solution: allowing S to have an approximate UM structure.

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Approximate UM Definition (Approximate UM) (S, d ) is D-approx’ UM if ∃ρ UM on S such that d (x, y ) ≤ ρ(x, y ) ≤ D · d (x, y ),

∀x, y ∈ S.

∆

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Approximate UM Definition (Approximate UM) (S, d ) is D-approx’ UM if ∃ρ UM on S such that d (x, y ) ≤ ρ(x, y ) ≤ D · d (x, y ),

≥

∆ D

≥

∆ D

≥

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∀x, y ∈ S.

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∆ D

≥

∆ D

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Approximate UM Definition (Approximate UM) (S, d ) is D-approx’ UM if ∃ρ UM on S such that d (x, y ) ≤ ρ(x, y ) ≤ D · d (x, y ),

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∀x, y ∈ S.

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Large UM subset of the discrete line: Cantor subset

D=4

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Large UM subset of the discrete line: Cantor subset 32 12

12

5 4

1

5 3

4

1

2

D=4 1

Number of points left: ≈ 2log8/3 n = n log(8/3) .

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Large UM subset of the discrete line: Cantor subset 32 12

12

5 4

1

5 3

4

1

2

D=4 1

Number of points left: ≈ 2log8/3 n = n log(8/3) . Generalize: Throw out the middle ε-fraction. 1 ε

approx’ UM. log

Number of points: ≈ 2

2 1−ε

n

This is best possible for the line.

≈ n1−cε .

Approximates the size of the space up-to a power. M. Mendel (OUI)

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Beyond size

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Beyond size

S

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Beyond size

We want to capture both approximate size and approximate diameter and more by a UM subset. M. Mendel (OUI)

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Main Theorem

Theorem ([M. Naor ’11]) ∀ε > 0, ∀(X , d ) compact metric, ∀µ prob’ measure on X : ∃S ⊂ X , ∃ν prob’ measure supported on S s.t. (S, d ) is 9/ε-approx’ UM. ν(B(x, r )) ≤ µ(B(x, Cε r ))1−ε , Here Cε = 2

O(1/ε2 )

∀x ∈ X , ∀r > 0

depends only on ε.

The tradeoff between the approximation and the power of µ is asymptotically tight.

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The diameter property Recall: ν supported on S ⊂ X , and ν(B(x, r )) ≤ µ(B(x, Cε r ))1−ε , ∀x ∈ X , ∀r > 0.

Claim ∆(S) ≥ ∆(supp(µ))/(2Cε ).

Proof.

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The diameter property Recall: ν supported on S ⊂ X , and ν(B(x, r )) ≤ µ(B(x, Cε r ))1−ε , ∀x ∈ X , ∀r > 0.

Claim ∆(S) ≥ ∆(supp(µ))/(2Cε ).

Proof.

S

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The diameter property Recall: ν supported on S ⊂ X , and ν(B(x, r )) ≤ µ(B(x, Cε r ))1−ε , ∀x ∈ X , ∀r > 0.

Claim ∆(S) ≥ ∆(supp(µ))/(2Cε ).

Proof.

r

S

ν(B(x, r)) = 1

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The diameter property Recall: ν supported on S ⊂ X , and ν(B(x, r )) ≤ µ(B(x, Cε r ))1−ε , ∀x ∈ X , ∀r > 0.

Claim ∆(S) ≥ ∆(supp(µ))/(2Cε ).

Proof.

µ(B(x, Cε r)) ≥ ν(B(x, r))1/1−ε = 1 Cε r

S

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Large UM subset Recall: ν supported on S ⊂ X , and ν(B(x, r )) ≤ µ(B(x, Cε r ))1−ε ,

∀x ∈ X , ∀r > 0.

(1)

Theorem ([M Naor ’06]) ∀ε > 0 and ∀(X , d ) finite m.s., ∃S ⊆ X s.t. (S, d ) is 9/ε approx’ UM. |S| ≥ |X |1−ε .

Proof. Let n = |X |, and set µ uniform on X , i.e., µ({x}) = n1 , ∀x ∈ X . Then by main thm, ∃S ⊂ X , and ν as above. Applying (1) with r = 0, X X 1= ν({x}) ≤ µ({x})1−ε = |S| · n−(1−ε) . x∈S

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x∈S

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Large UM subset: History I Theorem ([Dvoretzky ’60, Milman ’71]) For every ε > 0 there exists Cε such that any n-dimensional normed space X has a subspace Y ⊂ X of dimension Cε log n on which the induced norm is 1 + ε distorted Euclidean.

Theorem ([Bourgain Figiel Milman ’86]: finite metric space Dvoretzky thm) For every ε > 0 there exists cε such that any n-point m.s. (X , d ) has a subset S ⊂ X , |S| ≥ cε log n, and (S, d ) is 1 + ε approx’ Euclidean. Actually: (S, d ) is 1 + ε approx’ UM. [BFM]’s bound is closely related to classical graph Ramsey theory: Up to 2-approx, every graph can be represented by {1, 2} metric. UM subset must be equilateral which is monochromatic. M. Mendel (OUI)

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Large UM subset: History II

When approx’> 2 is considered, ∆-ineq’ “kicks-in". Thus: Let D > 2. ∀(X , d ) finite, ∃S ⊆ X such that S is D-approx UM and  p  [Blum Karloff Rabani Saks ’92]: |S| ≥ exp cD log n . [Bartal Linial M. Naor ’03]: |S| ≥ |X |αD , D where αD > 0 for D > 2, and αD ≥ 1 − C log D . [MN ’06]: αD ≥ 1 −

C D

(asymptotically tight).

[Naor Tao ’10]: C ≤ 2e.

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k-server problem [Manasse McGeoch Sleator ’88]

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k-server problem [Manasse McGeoch Sleator ’88]

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k-server problem [Manasse McGeoch Sleator ’88]

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k-server problem [Manasse McGeoch Sleator ’88]

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k-server problem [Manasse McGeoch Sleator ’88]

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k-server problem [Manasse McGeoch Sleator ’88]

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k-server problem [Manasse McGeoch Sleator ’88]

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k-server problem [Manasse McGeoch Sleator ’88]

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k-server problem [Manasse McGeoch Sleator ’88]

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k-server problem [Manasse McGeoch Sleator ’88]

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k-server problem [Manasse McGeoch Sleator ’88]

M. Mendel (OUI)

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k-server problem [Manasse McGeoch Sleator ’88]

M. Mendel (OUI)

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k-server problem [Manasse McGeoch Sleator ’88]

M. Mendel (OUI)

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k-server problem [Manasse McGeoch Sleator ’88]

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k-server problem [Manasse McGeoch Sleator ’88]

M. Mendel (OUI)

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k-server problem [Manasse McGeoch Sleator ’88]

Goal: Minimizing the total distance travelled by the trucks. M. Mendel (OUI)

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Competitive ratio

k-server is analysed in an online setting using the competitive ratio. The performance of (randomized) online algorithm A is compared to the optimal offline cost.

Definition (r -Competitiveness of online alg A) ∃C > 0, ∀σ request sequence, E[Cost(A(σ))] ≤ r · Cost(opt(σ)) + C .

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Competitive ratio for the k-server

The deterministic competitive ratio is between k [MMS ’88] and 2k − 1 [Koutsoupias Papadimitriou ’94] on any metric space.

The randomized competitive ratio of k-server attracted much research. On some metric spaces it is Θ(log k). Recent breakthrough: polylog(k, n) competitive randomized algorithm for k-server in n-point spaces [Bansal, Buchbinder, Madry, Naor ’11]. ∃ a lower bound of Ω(log k/ log log k) on any metric space. The l.b. proof uses the large UM subset theorem.

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Large UM subset in l.b. argument

The competitive ratio is changed multiplicatively by at most a factor D on D-approx’ metric. An algorithm for a space X implies an algorithm for subset S ⊆ X . Counter-positive: A l.b. on subset S ⊆ X implies a l.b. on X . A l.b. on all spaces of size n = k + 1 implies a l.b. ∀n > k. The [Karloff Rabani Ravid ’91] paradigm is therefore:

Conceive a class of spaces C for which we can prove a good lower bound (e.g. log k). Prove: ∀X n-pt space ∃S ⊆ X , of size f (n) and S is D-approx’ in C. Deduce a l.b. of log f D(k+1) on the competitive ratio in S. The same lower bound also holds in X .

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k-server lower bounds

Dvoretzky-type theorems and the lower bounds they imply Reference Karloff Rabani Ravid ’91

Dist’ 4

Size log n log log n q

Blum Karloff Rabani Saks ’92 Bartal Bollobas M. ’01 Bartal Linial M. Naor ’03

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4

2

0.1·log n log log n

Type equilateral/lacunary (⊂ UM)

log log n

n0.01

dichotomic triangles (⊂ UM) UM

4

n0.01

UM

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Lower Bound log log k q

log k log log k

log k (log log k)2 log k log log k

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Approximate distance oracle (ADO)

Definition Given a metric space (or a graph) X . Preprocess X “quickly” and obtain a “compact” data structure S. Answer in “constant time” queries of the form Given x, y ∈ X , compute dX (x, y ) approximately.

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Space / approximation trade-off Example (Distance Matrix) Constant query time. Exact answer. Storage: Θ(n2 ) words.

Proposition ([Thorup Zwick ’01]) 1

2

3

If ADO A estimates distances with distortion < 3, then |A| = Ω(n2 ) bits. A widely believed girth conjecture implies: If DS A estimates distances with distortion < 2h + 1, then space |A| = Ω(n1+1/h ) bits. The conjecture is verified for h = 1, 2, 3, 5.

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Thorup-Zwick oracle Theorem ([TZ ’01]) Let h ∈ N. ∃ ADO with the parameters 2h − 1 approximation.

Storage: O(hn1+1/h ) words. Query time O(h). Preprocessing time O(n2 ). 1

Space / approximation trade-off almost optimal.

2

Oracle: For constant approximations, constant query time.

3

Many many follow-ups & variants.

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Thorup-Zwick oracle Theorem ([TZ ’01]) Let h ∈ N. ∃ ADO with the parameters 2h − 1 approximation.

Storage: O(hn1+1/h ) words. Query time O(h). Preprocessing time O(n2 ). 1

Space / approximation trade-off almost optimal.

2

Oracle: For constant approximations, constant query time.

3

Many many follow-ups & variants.

4

Are there “true oracles” with universal constant query time?

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True oracles

Theorem ([M Naor ’06]) Let h ∈ N. ∃ ADO with the parameters

c · h approximation, where c > 2 is a universal constant. Storage: O(n1+1/h ) words.

Query time is a universal constant. Preprocessing time is O(n2 ). [M. Schwob ’09] Construction is based on the large UM subset theorem. 2 < c ≤ 6e [Naor Tao ’10].

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ADO using Dvoretzky [MN ’06] X

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ADO using Dvoretzky [MN ’06] X T1

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ADO using Dvoretzky [MN ’06] X T1

T1

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ADO using Dvoretzky [MN ’06] X T1

Te1

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ADO using Dvoretzky [MN ’06] X

T2

T1

Te1

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ADO using Dvoretzky [MN ’06] X

T2

T1

Te1

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T2

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ADO using Dvoretzky [MN ’06] X

T2

T1

Te1

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Te2

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ADO using Dvoretzky [MN ’06] X

T2

T1

T4 T3

Te1

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Te2

Te3

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Te4

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ADO using Dvoretzky [MN ’06] X

T2

T1

T4 T3

Te1

Te2

x1

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Te4

Te3

x23

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ADO using Dvoretzky [MN ’06] X

T2

T1

T4 T3

Te1

Te2

Te4

Te3

x1

x23

Each tree contains n1−ε black points (ε = 1/h). Therefore, there are O(nε ) trees. The total size of the DS is n · O(nε ) = O(n1+ε ). M. Mendel (OUI)

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Hausdorff dimension Definition (Hausdorff dimension) α H∞ (X )

= inf

( X i

riα

: ∃(xi )i∈N

[ i

B(xi , ri ) ⊇ X

)

α dimH (X ) = inf {α > 0 : H∞ (X ) = 0}

Example 1

For any countable X , dimH (X ) = 0.

2

The interval [0, 1]. Fix α > 1 ∀m > 1,  i i+1 
α Pm−1 α ([0, 1]) ≤ H∞ = m1−α →m→∞ 0 i=0 diam m, m Hence dimH ([0, 1]) ≤ α ⇒ dimH ([0, 1]) ≤ 1. M. Mendel (OUI)

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Dvoretzky-type theorem for Hausdorff dimension

Theorem ([M. Naor ’11]) ∀ε ∈ (0, 1), ∀X compact space, ∃S ⊆ X s.t. dimH (S) ≥ (1 − ε)dimH (X ); S is (9/ε) approx’ UM.

The bound is asymptotically tight even for approx’ Euclidean subsets. Example: Expander-based fractals

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Proof of the Dvoretzky thm for Hausdorff dimension Lemma ([Frostman ’31]) Fix a compact m.s. X . Then β < dimH (X ) iff ∃µ prob’ measure on X such that ∀x ∈ X , ∀r > 0, µ(B(x, r )) ≤ Cr β .

Proof of the Dvoretzky Thm. Fix a compact (X , d ), and fix β < dimH (X ). Let µ be a β- Frostman measure for X , i.e., µ(B(x, r )) ≤ Cr β , ∀x ∈ X , r > 0.

By [Main Thm], ∃S ⊂ X which is such that

9 ε

approx’ UM, & prob’ measure ν on S

ν(B(x, r )) ≤ µ(B(x, Cε r ))1−ε ≤ C 1−ε Cε(1−ε)β · r (1−ε)β . So ν is (1 − ε)β-Frostman measure for S ⇒ dimH (S) ≥ (1 − ε)β. M. Mendel (OUI)

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Urbański problem Lip

[Urbański ’09] ∀X , dimH (X ) > n ⇒ ∃f : X −→ [0, 1]n , f (X ) = [0, 1]n ?

Theorem ([Keleti Máthé ’12]) Yes, assuming (X , d ) is compact.

Proof. 1

if X is also UM then the thm can be proved using Frostman lemma and the n-th dim’ Peano curve.

2

Apply the Dvoretzky thm: ∃S ⊂ X s.t. dimH (S) > n, and S is approx’ UM.

3

4

Since S is (approx) UM, by step (1.), ∃f : S → [0, 1]n Lipschitz and surjective. By general Lipschitz extension thms (e.g. [Lee Naor ’05]) f is extended into f¯ : X → [0, 1]n . M. Mendel (OUI)

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Boundness Gaussian processes

Let (Gx )x∈X be Gaussian process, i.e., Gx is a Gaussian r.v. for every x ∈X . We are interested in estimating E maxx∈X Gx . p The metric d (x, y ) = E[(Gx − Gy )2 ] determines the covariance matrix, and hence all the properties of Gaussian process.

[Dudley ’67] observed that the E maxx∈X Gx can be estimated by the “size" of the metric d .

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Fernique-Talagrand γ2 functional [Fernique ’75] defined γ2 (X , d ) = inf sup µ x∈X

Z

0

∞p

log(1/µ(B(x, r )))dr ,

the inf is taken over all probability measures µ over X Fernique observed that E maxx∈X Gx . γ2 (X , d ). He conjectured that in fact E maxx∈X Gx  γ2 (X , d ).

Theorem ([Talagrand ’87]) ∀(Gx )x∈X finite Gaussian process, E maxx∈X Gx & γ2 (X , d ), p where d (x, y ) = E[(Gx − Gy )2 ]. M. Mendel (OUI)

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Talagrand’s proof Proposition (UM special case [Fernique],[Talagrand]) Let (Gx )x∈U be a Gaussian process, and let d (x, y ) = Assume that (U, d ) is an ultrametric, then

p E[(Gx − Gy )2 ].

E max Gx & γ2 (U, d ). x∈U

Theorem (Dvoretzky thm for γ2 [Talagrand]) ∀(X , d ) m.s., ∃S ⊂ X such that

(S, d ) is O(1) approximate UM. γ2 (S) & γ2 (X ).

Let S ⊂ X be the subset from the Dvoretzky thm for γ2 : ∗

E max Gx ≥ E max Gx & γ2 (S, d ) & γ2 (X , d ). x∈X

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x∈S

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γ2 and δ2 Let φ(t) =

p

log(1/t) . R∞ γ2 (X ) = inf µ supx∈X 0 φ(µ(B(x, r )))dr .

γ2 looks like a multi-scale covering radius parameter. Similarly, we define δ2 (X ) = sup inf

µ x∈X

Z

∞

φ(µ(B(x, r )))dr .

0

δ2 looks like a multi-scale packing radius parameter. Thus, no wonder that δ2  γ2 . This is in fact an easy observation. It also holds in greater generality [Bednorz ’11]: δφ  γφ , ∀φ continuous, non-increasing, and φ(0) = ∞. It thus sufficient to show a Dvoretzky theorem for δ2 .

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Dvoretzky theorem for δ2 Proposition ∀(X , d ) m.s., ∃S ⊂ X such that

(S, d ) is O(1) approximate UM. δ2 (S) & δ2 (X ).

Proof. Let µ a prob’ measure on X s.t. δ2 (X ) = inf x∈X Apply Main Thm, with ε = 1/2.

R∞ 0

φ(µ(B(x, r )))dr .

Obtain S ⊂ X and prob’s measure ν on S such that:

(S, d ) is 18-approx UM, and ν(B(x, r )) ≤ µ(B(x, Cr ))1/2 . Z ∞ δ2 (S) ≥ inf φ(ν(B(x, r )))dr x∈S 0 Z ∞ 1 ≥ inf φ(µ(B(x, Cr ))1/2 )dr ≥ √ δ2 (X ). x∈S 0 2C M. Mendel (OUI)

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Concluding remarks Theorem (Main theorem) ∀ε > 0, ∀(X , d ) compact metric, ∀µ prob’ measure on X : ∃S ⊂ X , ∃ν prob’ measure supported on S s.t. (S, d ) is 9/ε-approx’ UM. ν(B(x, r )) ≤ µ(B(x, Cε r ))1−ε , Here Cε = 2

O(1/ε2 )

M. Mendel (OUI)

∀x ∈ X , ∀r > 0

depends only ε.

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Concluding remarks Theorem (Main theorem) ∀ε > 0, ∀(X , d ) compact metric, ∀µ prob’ measure on X : ∃S ⊂ X , ∃ν prob’ measure supported on S s.t. (S, d ) is 9/ε-approx’ UM. ν(B(x, r )) ≤ µ(B(x, Cε r ))1−ε , Here Cε = 2

O(1/ε2 )

∀x ∈ X , ∀r > 0

depends only ε.

Proof. arXiv:1106.0879 & arXiv:1112.3416

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Concluding remarks Theorem (Main theorem) ∀ε > 0, ∀(X , d ) compact metric, ∀µ prob’ measure on X : ∃S ⊂ X , ∃ν prob’ measure supported on S s.t. (S, d ) is 9/ε-approx’ UM. ν(B(x, r )) ≤ µ(B(x, Cε r ))1−ε , Here Cε = 2

O(1/ε2 )

∀x ∈ X , ∀r > 0

depends only ε.

Proof. arXiv:1106.0879 & arXiv:1112.3416 The algorithmic applications only used the large subset theorem — which corresponds to r = 0. More sophisticated CS applications? 2 We obtain Cε = 2O(1/ε ) . Can this bound be improved?

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Concluding remarks Theorem (Main theorem) ∀ε > 0, ∀(X , d ) compact metric, ∀µ prob’ measure on X : ∃S ⊂ X , ∃ν prob’ measure supported on S s.t. (S, d ) is 9/ε-approx’ UM. ν(B(x, r )) ≤ µ(B(x, Cε r ))1−ε , Here Cε = 2

O(1/ε2 )

∀x ∈ X , ∀r > 0

depends only ε.

Proof. arXiv:1106.0879 & arXiv:1112.3416 The algorithmic applications only used the large subset theorem — which corresponds to r = 0. More sophisticated CS applications? 2 We obtain Cε = 2O(1/ε ) . Can this bound be improved?

Thank You

M. Mendel (OUI)

UM skeletons

January 2012

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