1 2 Compress a massive object to a small sketch 2 Compress a - - PowerPoint PPT Presentation

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• Compress a massive object to a small sketch

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• Compress a massive object to a small sketch
• Rich theories: high-dimensional vectors, matrices, graphs

n d

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• Compress a massive object to a small sketch
• Rich theories: high-dimensional vectors, matrices, graphs
• Similarity search, compressed sensing, numerical linear algebra

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• Compress a massive object to a small sketch
• Rich theories: high-dimensional vectors, matrices, graphs
• Similarity search, compressed sensing, numerical linear algebra
• Dimension reduction (Johnson, Lindenstrauss 1984): random

projection on a low-dimensional subspace preserves distances

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• Compress a massive object to a small sketch
• Rich theories: high-dimensional vectors, matrices, graphs
• Similarity search, compressed sensing, numerical linear algebra
• Dimension reduction (Johnson, Lindenstrauss 1984): random

projection on a low-dimensional subspace preserves distances

When is sketching possible?

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• Motivation: similarity search

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• Motivation: similarity search
• Model dis-similarity as a metric

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• Motivation: similarity search
• Model dis-similarity as a metric

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• Motivation: similarity search
• Model dis-similarity as a metric
• Sketching may speed-up computation

and allow indexing

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• Motivation: similarity search
• Model dis-similarity as a metric
• Sketching may speed-up computation

and allow indexing

• Interesting metrics:

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• Motivation: similarity search
• Model dis-similarity as a metric
• Sketching may speed-up computation

and allow indexing

• Interesting metrics:
• Euclidean ℓ2: d(x, y) = (∑i|xi – yi|2)1/2

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• Motivation: similarity search
• Model dis-similarity as a metric
• Sketching may speed-up computation

and allow indexing

• Interesting metrics:
• Euclidean ℓ2: d(x, y) = (∑i|xi – yi|2)1/2
• Manhattan, Hamming ℓ1: d(x, y) = ∑i|xi – yi|

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• Motivation: similarity search
• Model dis-similarity as a metric
• Sketching may speed-up computation

and allow indexing

• Interesting metrics:
• Euclidean ℓ2: d(x, y) = (∑i|xi – yi|2)1/2
• Manhattan, Hamming ℓ1: d(x, y) = ∑i|xi – yi|
• ℓp distances d(x, y) = (∑i|xi – yi|p)1/p for p ≥ 1

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• Motivation: similarity search
• Model dis-similarity as a metric
• Sketching may speed-up computation

and allow indexing

• Interesting metrics:
• Euclidean ℓ2: d(x, y) = (∑i|xi – yi|2)1/2
• Manhattan, Hamming ℓ1: d(x, y) = ∑i|xi – yi|
• ℓp distances d(x, y) = (∑i|xi – yi|p)1/p for p ≥ 1
• Edit Distance, Earth Mover’s Distance etc.

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• Alice and Bob each hold a point from a

metric space (say x and y)

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Alice Bob Charlie

x y

• Alice and Bob each hold a point from a

metric space (say x and y)

• Both send s-bit sketches to Charlie

sketch(x) sketch(y)

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Alice Bob Charlie

x y

• Alice and Bob each hold a point from a

metric space (say x and y)

• Both send s-bit sketches to Charlie
• For r > 0 and D > 1 distinguish
• d(x, y) ≤ r
• d(x, y) ≥ Dr

sketch(x) sketch(y)

d(x, y) ≤ r or d(x, y) ≥ Dr?

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Alice Bob Charlie

x y

• Alice and Bob each hold a point from a

metric space (say x and y)

• Both send s-bit sketches to Charlie
• For r > 0 and D > 1 distinguish
• d(x, y) ≤ r
• d(x, y) ≥ Dr
• Shared randomness, allow 1%

probability of error sketch(x) sketch(y)

d(x, y) ≤ r or d(x, y) ≥ Dr?

0 1 1 0 … 1

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Alice Bob Charlie

x y

• Alice and Bob each hold a point from a

metric space (say x and y)

• Both send s-bit sketches to Charlie
• For r > 0 and D > 1 distinguish
• d(x, y) ≤ r
• d(x, y) ≥ Dr
• Shared randomness, allow 1%

probability of error

• Trade-off between s and D

sketch(x) sketch(y)

d(x, y) ≤ r or d(x, y) ≥ Dr?

0 1 1 0 … 1

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Alice Bob Charlie

x y

Which metrics can we sketch efficiently?

(Kanpur 2006)

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• Near Neighbor Search (NNS):
• Given n-point dataset P
• A query q within r from some data point
• Return any data point within Dr from q

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• Near Neighbor Search (NNS):
• Given n-point dataset P
• A query q within r from some data point
• Return any data point within Dr from q
• Sketches of size s imply NNS with

space nO(s) and a 1-probe query

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• Near Neighbor Search (NNS):
• Given n-point dataset P
• A query q within r from some data point
• Return any data point within Dr from q
• Sketches of size s imply NNS with

space nO(s) and a 1-probe query

• Proof idea: amplify probability of

error to 1/n by increasing the size to O(s log n); sketch of q determines the answer

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• Near Neighbor Search (NNS):
• Given n-point dataset P
• A query q within r from some data point
• Return any data point within Dr from q
• Sketches of size s imply NNS with

space nO(s) and a 1-probe query

• Proof idea: amplify probability of

error to 1/n by increasing the size to O(s log n); sketch of q determines the answer

• For many metrics: the only approach

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Which metrics can we sketch efficiently?

(Kanpur 2006)

ℓ

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ℓ

• (Indyk 2000): can sketch ℓp for 0 < p ≤ 2 via random projections using

p-stable distributions

• For D = 1 + ε one gets s = O(1 / ε2)
• Tight by (Woodruff 2004)

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ℓ

• (Indyk 2000): can sketch ℓp for 0 < p ≤ 2 via random projections using

p-stable distributions

• For D = 1 + ε one gets s = O(1 / ε2)
• Tight by (Woodruff 2004)
• For p > 2 sketching ℓp is somewhat hard (Alon, Matias, Szegedy 1995),

(Bar-Yossef, Jayram, Kumar, Sivakumar 2002), (Indyk, Woodruff 2005)

• To achieve D = O(1) one needs sketch size to be s = Θ~(d1-2/p)

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• Distinguish |x – y| ≤ 1 vs.

|x – y| ≥ 1 + ε

x y

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• Distinguish |x – y| ≤ 1 vs.

|x – y| ≥ 1 + ε

• Randomly shifted pieces of

length 1 + ε/2

x y 1

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• Distinguish |x – y| ≤ 1 vs.

|x – y| ≥ 1 + ε

• Randomly shifted pieces of

length 1 + ε/2

• Repeat O(1 / ε2) times

x y 1

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• Distinguish |x – y| ≤ 1 vs.

|x – y| ≥ 1 + ε

• Randomly shifted pieces of

length 1 + ε/2

• Repeat O(1 / ε2) times
• Overall:
• D = 1 + ε
• s = O(1 / ε2)

x y 1

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ℓ

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ℓ

• (Indyk 2000): can reduce sketching of ℓp with 0 < p ≤ 2 to sketching

reals via random projections

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ℓ

• (Indyk 2000): can reduce sketching of ℓp with 0 < p ≤ 2 to sketching

reals via random projections

• If (G1, G2, …, Gd) are i.i.d. N(0, 1)’s, then ∑i xiGi – ∑i yiGi is distributed as

‖x - y‖2• N(0, 1)

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ℓ

• (Indyk 2000): can reduce sketching of ℓp with 0 < p ≤ 2 to sketching

reals via random projections

• If (G1, G2, …, Gd) are i.i.d. N(0, 1)’s, then ∑i xiGi – ∑i yiGi is distributed as

‖x - y‖2• N(0, 1)

• For 0 < p < 2 use p-stable distributions instead

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ℓ

• (Indyk 2000): can reduce sketching of ℓp with 0 < p ≤ 2 to sketching

reals via random projections

• If (G1, G2, …, Gd) are i.i.d. N(0, 1)’s, then ∑i xiGi – ∑i yiGi is distributed as

‖x - y‖2• N(0, 1)

• For 0 < p < 2 use p-stable distributions instead
• Again, get D = 1 + ε with s = O(1 / ε2)

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ℓ

• (Indyk 2000): can reduce sketching of ℓp with 0 < p ≤ 2 to sketching

reals via random projections

• If (G1, G2, …, Gd) are i.i.d. N(0, 1)’s, then ∑i xiGi – ∑i yiGi is distributed as

‖x - y‖2• N(0, 1)

• For 0 < p < 2 use p-stable distributions instead
• Again, get D = 1 + ε with s = O(1 / ε2)
• (1 + ε)-NNS: space nO(1/ε^2), query time poly((log n) / ε)

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Which metrics can we sketch with constant sketch size and approximation?

ℓ

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ℓ

• A map f: X → Y is an embedding with distortion C, if for a, b from X:

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ℓ

• A map f: X → Y is an embedding with distortion C, if for a, b from X:

dX(a, b) / C ≤ dY(f(a), f(b)) ≤ dX(a, b)

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X Y

ℓ

• A map f: X → Y is an embedding with distortion C, if for a, b from X:

dX(a, b) / C ≤ dY(f(a), f(b)) ≤ dX(a, b)

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X Y

ℓ

• A map f: X → Y is an embedding with distortion C, if for a, b from X:

dX(a, b) / C ≤ dY(f(a), f(b)) ≤ dX(a, b)

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a b

X Y

ℓ

• A map f: X → Y is an embedding with distortion C, if for a, b from X:

dX(a, b) / C ≤ dY(f(a), f(b)) ≤ dX(a, b)

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a b f(a) f(b) f f

X Y

ℓ

• A map f: X → Y is an embedding with distortion C, if for a, b from X:

dX(a, b) / C ≤ dY(f(a), f(b)) ≤ dX(a, b)

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a b f(a) f(b) f f

X Y

ℓ

• A map f: X → Y is an embedding with distortion C, if for a, b from X:

dX(a, b) / C ≤ dY(f(a), f(b)) ≤ dX(a, b)

• Reductions for geometric problems

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a b f(a) f(b) f f

ℓ

• A map f: X → Y is an embedding with distortion C, if for a, b from X:

dX(a, b) / C ≤ dY(f(a), f(b)) ≤ dX(a, b)

• Reductions for geometric problems

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ℓ

• A map f: X → Y is an embedding with distortion C, if for a, b from X:

dX(a, b) / C ≤ dY(f(a), f(b)) ≤ dX(a, b)

• Reductions for geometric problems

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Sketches of size s and approximation D for Y Sketches of size s and approximation CD for X

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• A metric X admits sketches with s, D = O(1), if:
• X = ℓp for p ≤ 2
• X embeds into ℓp for p ≤ 2 with distortion O(1)

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• A metric X admits sketches with s, D = O(1), if:
• X = ℓp for p ≤ 2
• X embeds into ℓp for p ≤ 2 with distortion O(1)
• Are there any other metrics with efficient sketches (D and s are O(1))?

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• A metric X admits sketches with s, D = O(1), if:
• X = ℓp for p ≤ 2
• X embeds into ℓp for p ≤ 2 with distortion O(1)
• Are there any other metrics with efficient sketches (D and s are O(1))?
• We don’t know!
• Some new techniques are waiting to be discovered?
• No new techniques?!

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If a normed space X admits sketches of size s and approximation D, then for every ε > 0 the space X embeds (linearly) into ℓ1 – ε with distortion O(sD / ε)

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If a normed space X admits sketches of size s and approximation D, then for every ε > 0 the space X embeds (linearly) into ℓ1 – ε with distortion O(sD / ε)

Embedding into ℓp, p ≤ 2 Efficient sketches

(Kushilevitz, Ostrovsky, Rabani 1998) (Indyk 2000) For norms

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If a normed space X admits sketches of size s and approximation D, then for every ε > 0 the space X embeds (linearly) into ℓ1 – ε with distortion O(sD / ε)

• A vector space X with ‖.‖: X → R≥0 is a normed space, if
• ‖x‖ = 0 iff x = 0
• ‖αx‖ = |α|‖x‖
• ‖x + y‖ ≤ ‖x‖ + ‖y‖
• Every norm gives rise to a metric: define d(x, y) = ‖x - y‖

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• [Li, Nguyen, Woodruff 2014]: streaming any function is equivalent to

linear sketches

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• [Li, Nguyen, Woodruff 2014]: streaming any function is equivalent to

linear sketches

• [Braverman, Chestnut, Krauthgamer, Yang 2015]: streaming

symmetric norms

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No embeddings with distortion O(1) into ℓ1 – ε No sketches* of size and approximation O(1)

• Convert non-embeddability into lower bounds for sketches in a black

box way

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No embeddings with distortion O(1) into ℓ1 – ε No sketches* of size and approximation O(1)

• Convert non-embeddability into lower bounds for sketches in a black

box way

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No embeddings with distortion O(1) into ℓ1 – ε No sketches* of size and approximation O(1)

*in fact, any communication

protocols

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• ℓp spaces: p > 2 is hard, 1 ≤ p ≤ 2 is easy, p < 1 is not a norm

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• ℓp spaces: p > 2 is hard, 1 ≤ p ≤ 2 is easy, p < 1 is not a norm
• Can classify mixed norms ℓp(ℓq): in particular, ℓ1(ℓ2) is easy, while

ℓ2(ℓ1) is hard! (Jayram, Woodruff 2009), (Kalton 1985)

ℓq ℓp

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• ℓp spaces: p > 2 is hard, 1 ≤ p ≤ 2 is easy, p < 1 is not a norm
• Can classify mixed norms ℓp(ℓq): in particular, ℓ1(ℓ2) is easy, while

ℓ2(ℓ1) is hard! (Jayram, Woodruff 2009), (Kalton 1985)

• A non-example: edit distance is not a norm, sketchability is largely
• pen (Ostrovsky, Rabani 2005), (Andoni, Jayram, Pătraşcu 2010)

ℓq ℓp

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• For x: R[Δ]×[Δ] → R with ∑i,j xi,j = 0, define the Earth Mover’s Distance

‖x‖EMD as the cost of the best transportation of the positive part of x to the negative part (Monge-Kantorovich norm)

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• For x: R[Δ]×[Δ] → R with ∑i,j xi,j = 0, define the Earth Mover’s Distance

‖x‖EMD as the cost of the best transportation of the positive part of x to the negative part (Monge-Kantorovich norm)

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Original motivation of this work!

• For x: R[Δ]×[Δ] → R with ∑i,j xi,j = 0, define the Earth Mover’s Distance

‖x‖EMD as the cost of the best transportation of the positive part of x to the negative part (Monge-Kantorovich norm)

• Best upper bounds:
• D = O(1 / ε) and s = Δε (Andoni, Do Ba, Indyk, Woodruff 2009)
• D = O(log Δ) and s = O(1) (Charikar 2002), (Indyk, Thaper 2003), (Naor,

Schechtman 2005)

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Original motivation of this work!

• For x: R[Δ]×[Δ] → R with ∑i,j xi,j = 0, define the Earth Mover’s Distance

‖x‖EMD as the cost of the best transportation of the positive part of x to the negative part (Monge-Kantorovich norm)

• Best upper bounds:
• D = O(1 / ε) and s = Δε (Andoni, Do Ba, Indyk, Woodruff 2009)
• D = O(log Δ) and s = O(1) (Charikar 2002), (Indyk, Thaper 2003), (Naor,

Schechtman 2005)

No embedding into ℓ1 – ε with distortion O(1) (Naor, Schechtman 2005) No sketches with D = O(1) and s = O(1)

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• For an n × n matrix A define the Trace Norm (the Nuclear Norm) ‖A‖

to be the sum of the singular values

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• For an n × n matrix A define the Trace Norm (the Nuclear Norm) ‖A‖

to be the sum of the singular values

• Previously: lower bounds only for certain restricted classes of

sketches (Li, Nguyen, Woodruff 2014)

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• For an n × n matrix A define the Trace Norm (the Nuclear Norm) ‖A‖

to be the sum of the singular values

• Previously: lower bounds only for certain restricted classes of

sketches (Li, Nguyen, Woodruff 2014)

Any embedding into ℓ1 requires distortion Ω(n1/2) (Pisier 1978) Any sketch must satisfy sD = Ω(n1/2 / log n)

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• For an n × n matrix A define the Trace Norm (the Nuclear Norm) ‖A‖

to be the sum of the singular values

• Previously: lower bounds only for certain restricted classes of

sketches (Li, Nguyen, Woodruff 2014)

Any embedding into ℓ1 requires distortion Ω(n1/2) (Pisier 1978) Any sketch must satisfy sD = Ω(n1/2 / log n)

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• Subsequent work (Li, Woodruff 2016): for D = 1 + ε, s ≥ n1 - f(ε)
• One-way communication complexity

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If a normed space X admits sketches of size s and approximation D, then for every ε > 0 the space X embeds (linearly) into ℓ1 – ε with distortion O(sD / ε)

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If a normed space X admits sketches of size s and approximation D, then for every ε > 0 the space X embeds (linearly) into ℓ1 – ε with distortion O(sD / ε) Sketches Weak embedding into ℓ2 Linear embedding into ℓ1 – ε

Information theory Nonlinear functional analysis

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If a normed space X admits sketches of size s and approximation D, then for every ε > 0 the space X embeds (linearly) into ℓ1 – ε with distortion O(sD / ε) Sketches Weak embedding into ℓ2 Linear embedding into ℓ1 – ε

Information theory Nonlinear functional analysis

A map f: X → Y is (s1, s2, τ1, τ2)-threshold, if

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If a normed space X admits sketches of size s and approximation D, then for every ε > 0 the space X embeds (linearly) into ℓ1 – ε with distortion O(sD / ε) Sketches Weak embedding into ℓ2 Linear embedding into ℓ1 – ε

Information theory Nonlinear functional analysis

A map f: X → Y is (s1, s2, τ1, τ2)-threshold, if

• dX(x1, x2) ≤ s1 implies dY(f(x1), f(x2)) ≤ τ1

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If a normed space X admits sketches of size s and approximation D, then for every ε > 0 the space X embeds (linearly) into ℓ1 – ε with distortion O(sD / ε) Sketches Weak embedding into ℓ2 Linear embedding into ℓ1 – ε

Information theory Nonlinear functional analysis

A map f: X → Y is (s1, s2, τ1, τ2)-threshold, if

• dX(x1, x2) ≤ s1 implies dY(f(x1), f(x2)) ≤ τ1
• dX(x1, x2) ≥ s2 implies dY(f(x1), f(x2)) ≥ τ2

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If a normed space X admits sketches of size s and approximation D, then for every ε > 0 the space X embeds (linearly) into ℓ1 – ε with distortion O(sD / ε) Sketches Weak embedding into ℓ2 Linear embedding into ℓ1 – ε

Information theory Nonlinear functional analysis

A map f: X → Y is (s1, s2, τ1, τ2)-threshold, if

• dX(x1, x2) ≤ s1 implies dY(f(x1), f(x2)) ≤ τ1
• dX(x1, x2) ≥ s2 implies dY(f(x1), f(x2)) ≥ τ2

(1, O(sD), 1, 10)-threshold map from X to ℓ2

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→

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→

X has a sketch of size s and approximation D There is a (1, O(sD), 1, 10)- threshold map from X to ℓ2

21

→

X has a sketch of size s and approximation D There is a (1, O(sD), 1, 10)- threshold map from X to ℓ2 No (1, O(sD), 1, 10)- threshold map from X to ℓ2

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→

X has a sketch of size s and approximation D There is a (1, O(sD), 1, 10)- threshold map from X to ℓ2 No (1, O(sD), 1, 10)- threshold map from X to ℓ2 Poincaré-type inequalities on X

Convex duality

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→

X has a sketch of size s and approximation D There is a (1, O(sD), 1, 10)- threshold map from X to ℓ2 No (1, O(sD), 1, 10)- threshold map from X to ℓ2 Poincaré-type inequalities on X

Convex duality

ℓk

∞(X) has no sketches

• f size Ω(k) and

approximation Θ(sD)

(Andoni, Jayram, Pătraşcu 2010) (direct sum theorem for information complexity)

21

→

X has a sketch of size s and approximation D There is a (1, O(sD), 1, 10)- threshold map from X to ℓ2 No (1, O(sD), 1, 10)- threshold map from X to ℓ2 Poincaré-type inequalities on X

Convex duality

ℓk

∞(X) has no sketches

• f size Ω(k) and

approximation Θ(sD)

(Andoni, Jayram, Pătraşcu 2010) (direct sum theorem for information complexity)

‖(x1, …, xk)‖ = maxi ‖xi‖

21

→

X has a sketch of size s and approximation D There is a (1, O(sD), 1, 10)- threshold map from X to ℓ2 No (1, O(sD), 1, 10)- threshold map from X to ℓ2 Poincaré-type inequalities on X

Convex duality

ℓk

∞(X) has no sketches

• f size Ω(k) and

approximation Θ(sD)

(Andoni, Jayram, Pătraşcu 2010) (direct sum theorem for information complexity)

X has no sketches of size s and approximation D ‖(x1, …, xk)‖ = maxi ‖xi‖

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X has sketches of size s and approximation D ℓk

∞(X) has sketches of size O(s)

and approximation Dk

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X has sketches of size s and approximation D ℓk

∞(X) has sketches of size O(s)

and approximation Dk

Alice Bob

(a1, a2, …, ak) (b1, b2, …, bk)

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X has sketches of size s and approximation D ℓk

∞(X) has sketches of size O(s)

and approximation Dk

Alice Bob

(a1, a2, …, ak) (b1, b2, …, bk)

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(σ1, σ2, …, σk) — random ±1’s

X has sketches of size s and approximation D ℓk

∞(X) has sketches of size O(s)

and approximation Dk

Alice Bob

(a1, a2, …, ak) (b1, b2, …, bk) ∑i σiai ∑i σibi

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(σ1, σ2, …, σk) — random ±1’s

X has sketches of size s and approximation D ℓk

∞(X) has sketches of size O(s)

and approximation Dk

Alice Bob

(a1, a2, …, ak) (b1, b2, …, bk) ∑i σiai ∑i σibi sketch(∑i σi ai) sketch(∑i σi bi)

22

(σ1, σ2, …, σk) — random ±1’s

X has sketches of size s and approximation D ℓk

∞(X) has sketches of size O(s)

and approximation Dk

Alice Bob

(a1, a2, …, ak) (b1, b2, …, bk) ∑i σiai ∑i σibi sketch(∑i σi ai) sketch(∑i σi bi) maxi ‖ai - bi‖ ≤ ‖∑i σi(ai – bi)‖ ≤ ∑i ‖ai - bi‖ ≤ k maxi ‖ai - bi‖

22

(σ1, σ2, …, σk) — random ±1’s

X has sketches of size s and approximation D ℓk

∞(X) has sketches of size O(s)

and approximation Dk

Alice Bob

(a1, a2, …, ak) (b1, b2, …, bk) ∑i σiai ∑i σibi sketch(∑i σi ai) sketch(∑i σi bi) maxi ‖ai - bi‖ ≤ ‖∑i σi(ai – bi)‖ ≤ ∑i ‖ai - bi‖ ≤ k maxi ‖ai - bi‖

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(σ1, σ2, …, σk) — random ±1’s with probability 1/2

X has sketches of size s and approximation D ℓk

∞(X) has sketches of size O(s)

and approximation Dk

Alice Bob

(a1, a2, …, ak) (b1, b2, …, bk) ∑i σiai ∑i σibi sketch(∑i σi ai) sketch(∑i σi bi) maxi ‖ai - bi‖ ≤ ‖∑i σi(ai – bi)‖ ≤ ∑i ‖ai - bi‖ ≤ k maxi ‖ai - bi‖ Crucially use the linear structure of X (not enough to be merely a metric!)

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(σ1, σ2, …, σk) — random ±1’s with probability 1/2

→

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→

(1, O(sD), 1, 10)-threshold map from X to ℓ2 Linear embedding into ℓ1 – ε with distortion O(sD / ε)

23

→

(1, O(sD), 1, 10)-threshold map from X to ℓ2 Linear embedding into ℓ1 – ε with distortion O(sD / ε) Uniform embedding into ℓ2

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→

(1, O(sD), 1, 10)-threshold map from X to ℓ2 Linear embedding into ℓ1 – ε with distortion O(sD / ε) Uniform embedding into ℓ2

g: X → ℓ2 s.t. L(‖x1 – x2‖) ≤ ‖g(x1) – g(x2)‖ ≤ U(‖x1 – x2‖) where

• L and U are non-decreasing,
• L(t) > 0 for t > 0
• U(t) → 0 as t → 0

23

→

(1, O(sD), 1, 10)-threshold map from X to ℓ2 Linear embedding into ℓ1 – ε with distortion O(sD / ε) Uniform embedding into ℓ2

g: X → ℓ2 s.t. L(‖x1 – x2‖) ≤ ‖g(x1) – g(x2)‖ ≤ U(‖x1 – x2‖) where

• L and U are non-decreasing,
• L(t) > 0 for t > 0
• U(t) → 0 as t → 0

(Aharoni, Maurey, Mityagin 1985) (Nikishin 1973)

23

→

24

→

• A map f: X → ℓ2 such that
• ‖x1 - x2‖ ≤ 1 implies ‖f(x1) - f(x2)‖ ≤ 1
• ‖x1 - x2‖ ≥ Θ(sD) implies ‖f(x1) - f(x2)‖ ≥ 10

24

→

• A map f: X → ℓ2 such that
• ‖x1 - x2‖ ≤ 1 implies ‖f(x1) - f(x2)‖ ≤ 1
• ‖x1 - x2‖ ≥ Θ(sD) implies ‖f(x1) - f(x2)‖ ≥ 10
• Building on (Johnson, Randrianarivony 2006)

24

→

• A map f: X → ℓ2 such that
• ‖x1 - x2‖ ≤ 1 implies ‖f(x1) - f(x2)‖ ≤ 1
• ‖x1 - x2‖ ≥ Θ(sD) implies ‖f(x1) - f(x2)‖ ≥ 10
• Building on (Johnson, Randrianarivony 2006)
• 1-net N of X; f Lipschitz on N

24

→

• A map f: X → ℓ2 such that
• ‖x1 - x2‖ ≤ 1 implies ‖f(x1) - f(x2)‖ ≤ 1
• ‖x1 - x2‖ ≥ Θ(sD) implies ‖f(x1) - f(x2)‖ ≥ 10
• Building on (Johnson, Randrianarivony 2006)
• 1-net N of X; f Lipschitz on N
• Extend f from N to a Lipschitz function on the whole X

24

25

• Extend to as general class of metrics as possible (Edit Distance?)

25

• Extend to as general class of metrics as possible (Edit Distance?)
• Strengthen to “sketches with O(1) size and approximation imply

embedding into ℓ1 with distortion O(1)”?

• Equivalent to an old open problem from Functional Analysis (Kwapien 1969)

25

• Extend to as general class of metrics as possible (Edit Distance?)
• Strengthen to “sketches with O(1) size and approximation imply

embedding into ℓ1 with distortion O(1)”?

• Equivalent to an old open problem from Functional Analysis (Kwapien 1969)
• Keep in mind negative-type metrics that do not embed into ℓ1

(Khot, Vishnoi 2005) (Cheeger, Kleiner, Naor 2009)

25

• Extend to as general class of metrics as possible (Edit Distance?)
• Strengthen to “sketches with O(1) size and approximation imply

embedding into ℓ1 with distortion O(1)”?

• Equivalent to an old open problem from Functional Analysis (Kwapien 1969)
• Keep in mind negative-type metrics that do not embed into ℓ1

(Khot, Vishnoi 2005) (Cheeger, Kleiner, Naor 2009)

• Spaces that require s = Ω(d) for D = O(1) besides ℓ∞?

25

• Extend to as general class of metrics as possible (Edit Distance?)
• Strengthen to “sketches with O(1) size and approximation imply

embedding into ℓ1 with distortion O(1)”?

• Equivalent to an old open problem from Functional Analysis (Kwapien 1969)
• Keep in mind negative-type metrics that do not embed into ℓ1

(Khot, Vishnoi 2005) (Cheeger, Kleiner, Naor 2009)

• Spaces that require s = Ω(d) for D = O(1) besides ℓ∞?
• Linear sketches with f(s) measurements and g(D) approximation?

25

• Extend to as general class of metrics as possible (Edit Distance?)
• Strengthen to “sketches with O(1) size and approximation imply

embedding into ℓ1 with distortion O(1)”?

• Equivalent to an old open problem from Functional Analysis (Kwapien 1969)
• Keep in mind negative-type metrics that do not embed into ℓ1

(Khot, Vishnoi 2005) (Cheeger, Kleiner, Naor 2009)

• Spaces that require s = Ω(d) for D = O(1) besides ℓ∞?
• Linear sketches with f(s) measurements and g(D) approximation?

25