testing surface area ryan o’donnell carnegie mellon & boğaziçi university joint work with...
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Testing Surface Area
Ryan O’DonnellCarnegie 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, 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 G with perim(G) ≤ 1.28 S.
vol(F∆G) > ϵ
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 G with perim(G) ≤ 1.28 S.
No Curse Of Dimensionality!
No assumptions about F!
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 G
with perim(G) ≤ (κn+δ) S.
1
ϵ δ2.5
Prior work
who dim. queries approx factor κ
[KR98]
[BBBY12]
[KNOW14]
[Nee14]
1 O(1/ϵ) 1/ϵ
1 O(1/ϵ4) 1
n O(1/ϵ) < 1.28 ∀nany 1+δ if n=1
n O(1/ϵ) any 1+δ
1 O(1/ϵ3.5) 1
Prior work
who dim. queries approx factor κ
[KR98]
[BBBY12]
[KNOW14]
[Nee14]
1 O(1/ϵ) 1/ϵ
1 O(1/ϵ4) 1
n O(1/ϵ) < 1.28 ∀nany 1+δ if n=1
n O(1/ϵ) any 1+δ
1 O(1/ϵ3.5) 1
Remark: We obtained same results inGaussian 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+δwhp using poly(n/δ) queries.
No “ϵ-far” stuff.
We don’t assume convexity, curvature bounds,
connectedness — nothing.
Property Testing framework is necessary
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 ~ ℝn / ℤn uniformly.
Pick y ~ Bλ(x).
Line segment xy called “the needle”.Then…
ℝn / ℤn.
xyE[ #( xy∩∂F ) ]
=
cn · λ · perim(F) F
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.
xyE[ #( xy∩∂F ) ]
=
cn · λ · perim(F)
explicit dimension-dependent constant,
Θ(n–1/2)
F
xyE[ #( xy∩∂F ) ]
=
cn · λ · perim(F) F
E[ 1{x∈F, y∉F, or vice versa} ]
≤
Pr[ 1F(x)≠1F(y) ]
=
NSF(λ)
:=
The “Noise Sensitivity” of F:
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: ✔
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: ✔
Recall: NSF(λ) = Pr [ 1F(x) ≠ 1F(y) ]x ~ ℝn/ℤn
y ~ Bλ(x)’ ≤ cn · λ · perim(F)
Soundness?
Q: If NSF(λ) ≤ cn · λ · S, is perim(F) ≾ S?
A: Not necessarily. (F may “wiggle at a scale ≪ λ”.)
Q: I.e., is perim(F) ≾ (cn λ)–1 · NSF(λ) always?
Q: Is F at least close to some G with
perim(G) ≾ (cn λ)–1 · NSF(λ) ? YES!
Recall: NSF(λ) = Pr [ 1F(x) ≠ 1F(y) ]x ~ ℝn/ℤn
y ~ Bλ(x)’ ≤ cn · λ · perim(F)
Soundness?
Our Theorem:
For every F ⊂ ℝn / ℤn and every λ,
F is O(NSF(λ))-close to a set G with
perim(G) ≤ Cn λ–1 · NSF(λ).
(Here Cn/cn =: κn ∈ [1, 4/π] for all n.)
Our Theorem:
For every F ⊂ ℝn / ℤn and every λ,
F is O(NSF(λ))-close to a set G with
perim(G) ≤ Cn λ–1 · NSF(λ).
Given F, how do you “find” G?
Finding G from F
F
Finding G from F
1. Define g : ℝn / ℤn → [0,1]
by y~Bλ(x)
g(x) = Pr [ y ∈ F ].
F
Finding G from F
1. Define g : ℝn / ℤn → [0,1]
by y~Bλ(x)
g(x) = Pr [ y ∈ F ].
2. Choose θ ∈ [0,1] from
the triangular distribution:
0 1
2 pdf: φθ
3. G := {x : g(x) > θ}.
1-Slide Sketch of Analysis
G being O(NSF(λ))-close to F (whp) is easy.
Theorem: E[ perim(G) ?
1-Slide Sketch of Analysis
G being O(NSF(λ))-close to F (whp) is easy.
Theorem: E[ perim(G) ] ?
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) ] = Cn λ–1 · NSF(λ)
[Neeman 14]’s version
• Picks needles of Gaussian length,rather than uniform on a ball.
• Uses a more clever pdf φθ.
Thanks!