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

ℓ • A map f: X → Y is an embedding with distortion C , if for a, b from X: d X (a, b ) / C ≤ d Y (f(a), f(b) ) ≤ d X (a, b) • Reductions for geometric problems Sketches of size s and Sketches of size s and approximation D for Y approximation CD for X 12

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• A metric X admits sketches with s, D = O(1), if: • X = ℓ p for p ≤ 2 • X embeds into ℓ p for p ≤ 2 with distortion O(1) 13

• A metric X admits sketches with s, D = O(1), if: • X = ℓ p for p ≤ 2 • X embeds into ℓ p for p ≤ 2 with distortion O(1) • Are there any other metrics with efficient sketches (D and s are O(1))? 13

• A metric X admits sketches with s, D = O(1), if: • X = ℓ p for p ≤ 2 • X embeds into ℓ p 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?! 13

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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 ℓ 1 – ε with distortion O(sD / ε ) 14

If a normed space X admits sketches of size s and approximation D, then for every ε > 0 the space X embeds (linearly) into ℓ 1 – ε with distortion O(sD / ε ) Embedding into ℓ p , p ≤ 2 (Kushilevitz, Ostrovsky, For norms Rabani 1998) (Indyk 2000) Efficient sketches 14

If a normed space X admits sketches of size s and approximation D, then for every ε > 0 the space X embeds (linearly) into ℓ 1 – 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‖ 14

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• [Li, Nguyen, Woodruff 2014]: streaming any function is equivalent to linear sketches 15

• [Li, Nguyen, Woodruff 2014]: streaming any function is equivalent to linear sketches • [Braverman, Chestnut, Krauthgamer, Yang 2015]: streaming symmetric norms 15

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No embeddings with distortion O(1) into ℓ 1 – ε No sketches * of size and approximation O(1) 16

• Convert non-embeddability into lower bounds for sketches in a black box way No embeddings with distortion O(1) into ℓ 1 – ε No sketches * of size and approximation O(1) 16

• Convert non-embeddability into lower bounds for sketches in a black box way No embeddings with distortion O(1) into ℓ 1 – ε * in fact, any communication No sketches * of size and protocols approximation O(1) 16

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• ℓ p spaces: p > 2 is hard, 1 ≤ p ≤ 2 is easy, p < 1 is not a norm 17

• ℓ p spaces: p > 2 is hard, 1 ≤ p ≤ 2 is easy, p < 1 is not a norm • Can classify mixed norms ℓ p (ℓ q ): in particular, ℓ 1 (ℓ 2 ) is easy, while ℓ 2 (ℓ 1 ) is hard! (Jayram, Woodruff 2009), (Kalton 1985) ℓ p ℓ q 17

• ℓ p spaces: p > 2 is hard, 1 ≤ p ≤ 2 is easy, p < 1 is not a norm • Can classify mixed norms ℓ p (ℓ q ): in particular, ℓ 1 (ℓ 2 ) is easy, while ℓ 2 (ℓ 1 ) is hard! (Jayram, Woodruff 2009), (Kalton 1985) • A non-example: edit distance is not a norm, sketchability is largely open (Ostrovsky, Rabani 2005), (Andoni, Jayram, Pătraşcu 2010) ℓ p ℓ q 17

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• For x: R [ Δ ]×[ Δ ] → R with ∑ i,j x i,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 ) 18

• For x: R [ Δ ]×[ Δ ] → R with ∑ i,j x i,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 ) Original motivation of this work! 18

• For x: R [ Δ ]×[ Δ ] → R with ∑ i,j x i,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 = Δ ε (Andoni, Do Ba, Indyk, Woodruff 2009) • D = O(log Δ ) and s = O(1) (Charikar 2002), (Indyk, Thaper 2003), (Naor, Schechtman 2005) Original motivation of this work! 18

• For x: R [ Δ ]×[ Δ ] → R with ∑ i,j x i,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 = Δ ε (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 ℓ 1 – ε No sketches with D = O(1) with distortion O(1) and s = O(1) (Naor, Schechtman 2005) 18

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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 19

• 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) 19

• 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 sketch must satisfy Any embedding into ℓ 1 requires distortion Ω (n 1/2 ) (Pisier 1978) sD = Ω (n 1/2 / log n) 19

• 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 sketch must satisfy Any embedding into ℓ 1 requires distortion Ω (n 1/2 ) (Pisier 1978) sD = Ω (n 1/2 / log n) • Subsequent work (Li, Woodruff 2016): for D = 1 + ε , s ≥ n 1 - f( ε ) • One-way communication complexity 19

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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 ℓ 1 – ε with distortion O(sD / ε ) 20

If a normed space X admits sketches of size s and approximation D, then for every ε > 0 the space X embeds (linearly) into ℓ 1 – ε with distortion O(sD / ε ) Weak embedding Linear embedding Sketches into ℓ 2 into ℓ 1 – ε Information theory Nonlinear functional analysis 20

If a normed space X admits sketches of size s and approximation D, then for every ε > 0 the space X embeds (linearly) into ℓ 1 – ε with distortion O(sD / ε ) Weak embedding Linear embedding Sketches into ℓ 2 into ℓ 1 – ε Information theory Nonlinear functional analysis A map f: X → Y is (s 1 , s 2 , τ 1 , τ 2 )-threshold , if 20

If a normed space X admits sketches of size s and approximation D, then for every ε > 0 the space X embeds (linearly) into ℓ 1 – ε with distortion O(sD / ε ) Weak embedding Linear embedding Sketches into ℓ 2 into ℓ 1 – ε Information theory Nonlinear functional analysis A map f: X → Y is (s 1 , s 2 , τ 1 , τ 2 )-threshold , if • d X (x 1 , x 2 ) ≤ s 1 implies d Y (f(x 1 ), f(x 2 )) ≤ τ 1 20

If a normed space X admits sketches of size s and approximation D, then for every ε > 0 the space X embeds (linearly) into ℓ 1 – ε with distortion O(sD / ε ) Weak embedding Linear embedding Sketches into ℓ 2 into ℓ 1 – ε Information theory Nonlinear functional analysis A map f: X → Y is (s 1 , s 2 , τ 1 , τ 2 )-threshold , if • d X (x 1 , x 2 ) ≤ s 1 implies d Y (f(x 1 ), f(x 2 )) ≤ τ 1 • d X (x 1 , x 2 ) ≥ s 2 implies d Y (f(x 1 ), f(x 2 )) ≥ τ 2 20

If a normed space X admits sketches of size s and approximation D, then for every ε > 0 the space X embeds (linearly) into ℓ 1 – ε with distortion O(sD / ε ) Weak embedding Linear embedding Sketches into ℓ 2 into ℓ 1 – ε Information theory Nonlinear functional analysis A map f: X → Y is (s 1 , s 2 , τ 1 , τ 2 )-threshold , if (1, O(sD), 1, 10)-threshold • d X (x 1 , x 2 ) ≤ s 1 implies d Y (f(x 1 ), f(x 2 )) ≤ τ 1 map from X to ℓ 2 • d X (x 1 , x 2 ) ≥ s 2 implies d Y (f(x 1 ), f(x 2 )) ≥ τ 2 20

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→ X has a sketch of size s There is a (1, O(sD), 1, 10)- and approximation D threshold map from X to ℓ 2 21

→ X has a sketch of size s There is a (1, O(sD), 1, 10)- and approximation D threshold map from X to ℓ 2 No (1, O(sD), 1, 10)- threshold map from X to ℓ 2 21

→ X has a sketch of size s There is a (1, O(sD), 1, 10)- and approximation D threshold map from X to ℓ 2 No (1, O(sD), 1, 10)- threshold map from X to ℓ 2 Convex duality Poincaré-type inequalities on X 21

→ X has a sketch of size s There is a (1, O(sD), 1, 10)- and approximation D threshold map from X to ℓ 2 No (1, O(sD), 1, 10)- threshold map from X to ℓ 2 (Andoni, Jayram, Pătraşcu 2010) Convex duality ℓ k ∞ (X) has no sketches (direct sum theorem for information complexity) Poincaré-type of size Ω (k) and inequalities on X approximation Θ (sD) 21

→ X has a sketch of size s There is a (1, O(sD), 1, 10)- and approximation D threshold map from X to ℓ 2 No (1, O(sD), 1, 10)- threshold map from X to ℓ 2 (Andoni, Jayram, Pătraşcu 2010) Convex duality ℓ k ∞ (X) has no sketches (direct sum theorem for information complexity) Poincaré-type of size Ω (k) and inequalities on X approximation Θ (sD) ‖ (x 1 , …, x k ) ‖ = max i ‖x i ‖ 21

→ X has a sketch of size s There is a (1, O(sD), 1, 10)- and approximation D threshold map from X to ℓ 2 X has no sketches of size No (1, O(sD), 1, 10)- s and approximation D threshold map from X to ℓ 2 (Andoni, Jayram, Pătraşcu 2010) Convex duality ℓ k ∞ (X) has no sketches (direct sum theorem for information complexity) Poincaré-type of size Ω (k) and inequalities on X approximation Θ (sD) ‖ (x 1 , …, x k ) ‖ = max i ‖x i ‖ 21

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ℓ k X has sketches of size s ∞ (X) has sketches of size O(s) and approximation D and approximation Dk 22