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Testing Surface Area

Testing Surface Area. Ryan O’Donnell Carnegie Mellon & Bo ğaziçi University. joint work with Pravesh Kothari (UT Austin), Amir Nayyeri (Oregon), Chenggang Wu (Tsinghua). In 2 dimensions…. BTW: This is one shape, that happens to be disconnected. “surface area” is called “ perimeter ”.

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Testing Surface Area

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  1. Testing Surface Area Ryan O’Donnell Carnegie Mellon & Boğaziçi University joint work with Pravesh Kothari(UT Austin),Amir Nayyeri (Oregon), Chenggang Wu (Tsinghua)

  2. In 2 dimensions… BTW: This is one shape,that happens to bedisconnected “surface area” is called “perimeter”

  3. In 1 dimension… BTW: This is one shape,that happens to bedisconnected “surface area” equals “# of endpoints” = “2 × # of intervals”

  4. Our Theorem Given S, ϵ, and query access to F ⊂ [0,1]n, there’s an O(1/ϵ)-query (nonadaptive) algorithm s.t.: vol(F∆G) > ϵ • Says YES whp if perim(F) ≤ S; • Says NO whp if F is ϵ-far from all Gwith perim(G) ≤1.28S.

  5. Our Theorem Given S, ϵ, and query access to F ⊂ [0,1]n, there’s an O(1/ϵ)-query (nonadaptive) algorithm s.t.: • Says YES whp if perim(F) ≤ S; • Says NO whp if F is ϵ-far from all Gwith perim(G) ≤1.28S. No Curse Of Dimensionality! No assumptions about F!

  6. Our Theorem Given S, ϵ, and query access to F ⊂ [0,1]n, there’s an O(1/ϵ) -query (nonadaptive) algorithm s.t.: 1 ϵδ2.5 • Says YES whp if perim(F) ≤ S; • Says NO whp if F is ϵ-far from all Gwith perim(G) ≤(κn+δ)S.

  7. Prior work who dim. queries approx factor κ [KR98] 1 O(1/ϵ) 1/ϵ O(1/ϵ4) [BBBY12] 1 1 O(1/ϵ3.5) 1 1 [KNOW14] < 1.28 ∀nany 1+δ if n=1 O(1/ϵ) n any 1+δ [Nee14] n O(1/ϵ)

  8. Prior work who dim. queries approx factor κ [KR98] 1 O(1/ϵ) 1/ϵ O(1/ϵ4) [BBBY12] 1 1 O(1/ϵ3.5) 1 1 [KNOW14] < 1.28 ∀nany 1+δ if n=1 Remark: We obtained same results inGaussian space. So did Neeman. O(1/ϵ) n any 1+δ [Nee14] n O(1/ϵ)

  9. Property Testing framework is necessary Theorem [BNN06]: If F ⊂ [0,1]n promised to be convex, can estimate perim(F) to factor 1+δ whp using poly(n/δ) queries. No “ϵ-far” stuff. We don’t assume convexity, curvature bounds,connectedness — nothing.

  10. Property Testing framework is necessary

  11. Property Testing framework is necessary

  12. Soundness theorem challenge:Cut string, smooth side, fill in holes.

  13. Algorithm: Buffon’s Needle Crofton Formula. Let F ⊂ [0,1]n Pick x ~ ℝn/ℤn uniformly. Pick y ~ Bλ(x). Line segment xy called “the needle”. Then… ℝn/ℤn. E[ #(xy∩∂F) ] = cn ·λ· perim(F) y x F

  14. Algorithm: Buffon’s Needle Crofton Formula. Let F ⊂ [0,1]n Pick x ~ ℝn/ℤn uniformly. Pick y ~ Bλ(x). Line segment xy called “the needle”. Then… explicit dimension-dependent constant, Θ(n–1/2) ℝn/ℤn. E[ #(xy∩∂F) ] = cn ·λ· perim(F) y x F

  15. The “Noise Sensitivity” of F: NSF(λ) := Pr[ 1F(x)≠1F(y) ] = E[ 1{x∈F, y∉F, or vice versa} ] ≤ E[ #(xy∩∂F) ] = cn ·λ· perim(F) y x F

  16. Algorithm and Completeness Recall: NSF(λ) = Pr [ 1F(x) ≠ 1F(y) ] x ~ ℝn/ℤn y ~ Bλ(x)’ ≤ cn · λ· perim(F) 0. Given S, ϵ, set λ such that ϵ = .01 · cn · λ· S. 1. Empirically estimate NSF(λ). 2. Say YES iff ≤ (1+δ) · cn · λ· S. Query complexity, Completeness: ✔

  17. Soundness? Recall: NSF(λ) = Pr [ 1F(x) ≠ 1F(y) ] x ~ ℝn/ℤn y ~ Bλ(x)’ ≤ cn · λ· perim(F) 0. Given S, ϵ, set λ such that ϵ = .01 · cn · λ· S. 1. Empirically estimate NSF(λ). 2. Say YES iff ≤ (1+δ) · cn · λ· S. Query complexity, Completeness: ✔

  18. Soundness? Recall: NSF(λ) = Pr [ 1F(x) ≠ 1F(y) ] x ~ ℝn/ℤn y ~ Bλ(x)’ ≤ cn · λ· perim(F) Q: If NSF(λ) ≤ cn · λ· S, is perim(F) ≾ S? Q: I.e., is perim(F) ≾ (cnλ)–1 · NSF(λ) always? A: Not necessarily. (F may “wiggle at a scale ≪λ”.) Q: Is F at least close to some G with perim(G) ≾ (cnλ)–1 · NSF(λ) ? YES!

  19. Soundness? Recall: NSF(λ) = Pr [ 1F(x) ≠ 1F(y) ] x ~ ℝn/ℤn y ~ Bλ(x)’ ≤ cn · λ· perim(F) Our Theorem: For every F ⊂ ℝn/ℤn and every λ, F is O(NSF(λ))-close to a set G with perim(G) ≤ Cnλ–1 · NSF(λ). (Here Cn/cn =: κn ∈ [1, 4/π] for all n.)

  20. Our Theorem: For every F ⊂ ℝn/ℤn and every λ, F is O(NSF(λ))-close to a set G with Given F, how do you “find” G? perim(G) ≤ Cnλ–1 · NSF(λ).

  21. Finding G from F F

  22. Finding G from F 1. Define g : ℝn/ℤn → [0,1]by F g(x) = Pr [ y ∈ F ]. y~Bλ(x)

  23. Finding G from F 1. Define g : ℝn/ℤn → [0,1]by 2. Choose θ ∈ [0,1]fromthe triangular distribution: pdf:φθ 2 g(x) = Pr [ y ∈ F ]. y~Bλ(x) 3.G := {x : g(x) > θ}. 0 1

  24. 1-Slide Sketch of Analysis G being O(NSF(λ))-close to F (whp) is easy. Theorem: E[ perim(G) ?

  25. 1-Slide Sketch of Analysis G being O(NSF(λ))-close to F (whp) is easy. Theorem:E[ perim(G) ] ?

  26. 1-Slide Sketch of Analysis G being O(NSF(λ))-close to F (whp) is easy. Theorem: E[ perim(G) ] =E[ φθ(g(x)) · ‖∇g(x)‖ ] x~ℝn/ℤn (“Coarea Formula”) Theorem: E[ perim(G) ]≤ Lip(g) · E[ φθ(g(x)) ] Theorem: E[ perim(G) ]≤ Lip(g) · 4 NSF(λ) Theorem: E[ perim(G) ]≤ O(n–1/2) λ–1 · 4 NSF(λ) Theorem: E[ perim(G) ] = Cnλ–1 · NSF(λ)

  27. [Neeman 14]’s version • Picks needles of Gaussian length,rather than uniform on a ball. • Uses a more clever pdf φθ.

  28. Thanks!

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