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!