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

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

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

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

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

Our Theorem Given S,, and query access to F ⊂ [0,1] n, there’s an O(1/) -query (nonadaptive) algorithm s.t.: Says YES whp if perim(F) ≤ S; Says NO whp if F is -far from all G with perim(G) ≤ (κ n +δ ) S. 1 δ 2.5

Prior work whodim.queriesapprox factor κ [KR98] [BBBY12] [KNOW14] [Nee14] 1O(1/)1/ 1 O(1/ 4 ) 1 n O(1/) < 1.28 ∀ n any 1+δ if n=1 nO(1/) any 1+δ 1 O(1/ 3.5 ) 1

Prior work whodim.queriesapprox factor κ [KR98] [BBBY12] [KNOW14] [Nee14] 1O(1/)1/ 1 O(1/ 4 ) 1 n O(1/) < 1.28 ∀ n any 1+δ if n=1 nO(1/) any 1+δ 1 O(1/ 3.5 ) 1 Remark: We obtained same results in Gaussian space. So did Neeman.

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 “-far” stuff. We don’t assume convexity, curvature bounds, connectedness — nothing.

Property Testing framework is necessary

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

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

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

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

Algorithm and Completeness Recall: NS F (λ) = Pr [ 1 F (x) ≠ 1 F (y) ] x ~ ℝ n / ℤ n y ~ B λ (x) ’ ≤ c n · λ · perim(F) 0. Given S,, set λ such that =.01 · c n · λ · S. 1. Empirically estimate NS F (λ). 2. Say YES iff ≤ (1 +δ ) · c n · λ · S. Query complexity, Completeness: ✔

Soundness? Recall: NS F (λ) = Pr [ 1 F (x) ≠ 1 F (y) ] x ~ ℝ n / ℤ n y ~ B λ (x) ’ ≤ c n · λ · perim(F) 0. Given S,, set λ such that =.01 · c n · λ · S. 1. Empirically estimate NS F (λ). 2. Say YES iff ≤ (1 +δ ) · c n · λ · S. Query complexity, Completeness: ✔

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

Recall: NS F (λ) = Pr [ 1 F (x) ≠ 1 F (y) ] x ~ ℝ n / ℤ n y ~ B λ (x) ’ ≤ c n · λ · perim(F) Soundness? Our Theorem: For every F ⊂ ℝ n / ℤ n and every λ, F is O(NS F (λ))-close to a set G with perim(G) ≤ C n λ –1 · NS F (λ). (Here C n /c n =: κ n ∈ [1, 4/π] for all n.)

Our Theorem: For every F ⊂ ℝ n / ℤ n and every λ, F is O(NS F (λ))-close to a set G with perim(G) ≤ C n λ –1 · NS F (λ). Given F, how do you “find” G?

Finding G from F F

1. Define g : ℝ n / ℤ n → [0,1] by y~B λ (x) g(x) = Pr [ y ∈ F ]. F

Finding G from F 1. Define g : ℝ n / ℤ n → [0,1] by y~B λ (x) g(x) = Pr [ y ∈ F ]. 2. Choose θ ∈ [0,1] from the triangular distribution: 01 2 pdf: φ θ 3. G := {x : g(x) > θ}.

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

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

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

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

Thanks!