higher-order fourier analysis: applications to algebraic property...
TRANSCRIPT
![Page 1: Higher-Order Fourier Analysis: Applications to Algebraic Property …research.nii.ac.jp/~yyoshida/resources/Talks/2016-CCC... · 2017. 7. 9. · Higher order Fourier analysis We look](https://reader035.vdocument.in/reader035/viewer/2022071022/5fd6614edb6a48475f1ccebd/html5/thumbnails/1.jpg)
Higher-Order Fourier Analysis:
Applications to Algebraic Property Testing
Yuichi Yoshida
National Institute of Informatics, andPreferred Infrastructure, Inc
May 28, 2016
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 1 / 33
![Page 2: Higher-Order Fourier Analysis: Applications to Algebraic Property …research.nii.ac.jp/~yyoshida/resources/Talks/2016-CCC... · 2017. 7. 9. · Higher order Fourier analysis We look](https://reader035.vdocument.in/reader035/viewer/2022071022/5fd6614edb6a48475f1ccebd/html5/thumbnails/2.jpg)
Decision Problems
• Function f : Fn2 → {0, 1}.
• Function property P .(such as linearity: f (x) + f (y) ≡ f (x + y) mod 2 for all x , y .)
Q. How long does it take to decide f satisfies P?
A. Trivially, it takes 2n time.
Q. Can we do something in sublinear or even in constant time?
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 2 / 33
![Page 3: Higher-Order Fourier Analysis: Applications to Algebraic Property …research.nii.ac.jp/~yyoshida/resources/Talks/2016-CCC... · 2017. 7. 9. · Higher order Fourier analysis We look](https://reader035.vdocument.in/reader035/viewer/2022071022/5fd6614edb6a48475f1ccebd/html5/thumbnails/3.jpg)
Decision Problems
• Function f : Fn2 → {0, 1}.
• Function property P .(such as linearity: f (x) + f (y) ≡ f (x + y) mod 2 for all x , y .)
Q. How long does it take to decide f satisfies P?A. Trivially, it takes 2n time.
Q. Can we do something in sublinear or even in constant time?
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 2 / 33
![Page 4: Higher-Order Fourier Analysis: Applications to Algebraic Property …research.nii.ac.jp/~yyoshida/resources/Talks/2016-CCC... · 2017. 7. 9. · Higher order Fourier analysis We look](https://reader035.vdocument.in/reader035/viewer/2022071022/5fd6614edb6a48475f1ccebd/html5/thumbnails/4.jpg)
Decision Problems
• Function f : Fn2 → {0, 1}.
• Function property P .(such as linearity: f (x) + f (y) ≡ f (x + y) mod 2 for all x , y .)
Q. How long does it take to decide f satisfies P?A. Trivially, it takes 2n time.
Q. Can we do something in sublinear or even in constant time?
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 2 / 33
![Page 5: Higher-Order Fourier Analysis: Applications to Algebraic Property …research.nii.ac.jp/~yyoshida/resources/Talks/2016-CCC... · 2017. 7. 9. · Higher order Fourier analysis We look](https://reader035.vdocument.in/reader035/viewer/2022071022/5fd6614edb6a48475f1ccebd/html5/thumbnails/5.jpg)
Property testing
Definition
f : {0, 1}n → {0, 1} is ε-far from P if,
dP(f ) := ming∈P
#{x ∈ {0, 1}n | f (x) 6= g(x)}2n
≥ ε.
Accept w.p. 2/3
Reject w.p. 2/3
P
ε-far
A tester for a property P :Given
• f : {0, 1}n → {0, 1}as a query access.
• proximity parameter ε > 0.
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 3 / 33
![Page 6: Higher-Order Fourier Analysis: Applications to Algebraic Property …research.nii.ac.jp/~yyoshida/resources/Talks/2016-CCC... · 2017. 7. 9. · Higher order Fourier analysis We look](https://reader035.vdocument.in/reader035/viewer/2022071022/5fd6614edb6a48475f1ccebd/html5/thumbnails/6.jpg)
Property testing
Definition
f : {0, 1}n → {0, 1} is ε-far from P if,
dP(f ) := ming∈P
#{x ∈ {0, 1}n | f (x) 6= g(x)}2n
≥ ε.
Accept w.p. 2/3
Reject w.p. 2/3
P
ε-far
A tester for a property P :Given
• f : {0, 1}n → {0, 1}as a query access.
• proximity parameter ε > 0.
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 3 / 33
![Page 7: Higher-Order Fourier Analysis: Applications to Algebraic Property …research.nii.ac.jp/~yyoshida/resources/Talks/2016-CCC... · 2017. 7. 9. · Higher order Fourier analysis We look](https://reader035.vdocument.in/reader035/viewer/2022071022/5fd6614edb6a48475f1ccebd/html5/thumbnails/7.jpg)
Testing f ≡ 1
Input: a function f : {0, 1}n → {0, 1} and ε > 0.Goal: f (x) = 1 for every x ∈ {0, 1}n?
1: for i = 1 to Θ(1/ε) do2: Sample x ∈ {0, 1}n uniformly at random.3: if f (x) = 0 then reject.4: Accept.
Theorem
• If f ≡ 1, always accepts. (one-sided error)
• If f is ε-far, accepts with probability (1− ε)Θ(1/ε) ≤ 1/3.
• Query complexity is O(1/ε)⇒ constant!
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 4 / 33
![Page 8: Higher-Order Fourier Analysis: Applications to Algebraic Property …research.nii.ac.jp/~yyoshida/resources/Talks/2016-CCC... · 2017. 7. 9. · Higher order Fourier analysis We look](https://reader035.vdocument.in/reader035/viewer/2022071022/5fd6614edb6a48475f1ccebd/html5/thumbnails/8.jpg)
Testing f ≡ 1
Input: a function f : {0, 1}n → {0, 1} and ε > 0.Goal: f (x) = 1 for every x ∈ {0, 1}n?
1: for i = 1 to Θ(1/ε) do2: Sample x ∈ {0, 1}n uniformly at random.3: if f (x) = 0 then reject.4: Accept.
Theorem
• If f ≡ 1, always accepts. (one-sided error)
• If f is ε-far, accepts with probability (1− ε)Θ(1/ε) ≤ 1/3.
• Query complexity is O(1/ε)⇒ constant!
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 4 / 33
![Page 9: Higher-Order Fourier Analysis: Applications to Algebraic Property …research.nii.ac.jp/~yyoshida/resources/Talks/2016-CCC... · 2017. 7. 9. · Higher order Fourier analysis We look](https://reader035.vdocument.in/reader035/viewer/2022071022/5fd6614edb6a48475f1ccebd/html5/thumbnails/9.jpg)
Testing f ≡ 1
Input: a function f : {0, 1}n → {0, 1} and ε > 0.Goal: f (x) = 1 for every x ∈ {0, 1}n?
1: for i = 1 to Θ(1/ε) do2: Sample x ∈ {0, 1}n uniformly at random.3: if f (x) = 0 then reject.4: Accept.
Theorem
• If f ≡ 1, always accepts. (one-sided error)
• If f is ε-far, accepts with probability (1− ε)Θ(1/ε) ≤ 1/3.
• Query complexity is O(1/ε)⇒ constant!
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 4 / 33
![Page 10: Higher-Order Fourier Analysis: Applications to Algebraic Property …research.nii.ac.jp/~yyoshida/resources/Talks/2016-CCC... · 2017. 7. 9. · Higher order Fourier analysis We look](https://reader035.vdocument.in/reader035/viewer/2022071022/5fd6614edb6a48475f1ccebd/html5/thumbnails/10.jpg)
Testing f ≡ 1
Input: a function f : {0, 1}n → {0, 1} and ε > 0.Goal: f (x) = 1 for every x ∈ {0, 1}n?
1: for i = 1 to Θ(1/ε) do2: Sample x ∈ {0, 1}n uniformly at random.3: if f (x) = 0 then reject.4: Accept.
Theorem
• If f ≡ 1, always accepts. (one-sided error)
• If f is ε-far, accepts with probability (1− ε)Θ(1/ε) ≤ 1/3.
• Query complexity is O(1/ε)⇒ constant!
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 4 / 33
![Page 11: Higher-Order Fourier Analysis: Applications to Algebraic Property …research.nii.ac.jp/~yyoshida/resources/Talks/2016-CCC... · 2017. 7. 9. · Higher order Fourier analysis We look](https://reader035.vdocument.in/reader035/viewer/2022071022/5fd6614edb6a48475f1ccebd/html5/thumbnails/11.jpg)
Testing f ≡ 1
Input: a function f : {0, 1}n → {0, 1} and ε > 0.Goal: f (x) = 1 for every x ∈ {0, 1}n?
1: for i = 1 to Θ(1/ε) do2: Sample x ∈ {0, 1}n uniformly at random.3: if f (x) = 0 then reject.4: Accept.
Theorem
• If f ≡ 1, always accepts. (one-sided error)
• If f is ε-far, accepts with probability (1− ε)Θ(1/ε) ≤ 1/3.
• Query complexity is O(1/ε)⇒ constant!
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 4 / 33
![Page 12: Higher-Order Fourier Analysis: Applications to Algebraic Property …research.nii.ac.jp/~yyoshida/resources/Talks/2016-CCC... · 2017. 7. 9. · Higher order Fourier analysis We look](https://reader035.vdocument.in/reader035/viewer/2022071022/5fd6614edb6a48475f1ccebd/html5/thumbnails/12.jpg)
Backgrounds
Property testing was introduced by [RS96] for program checking.
Since then, various kinds of objects have been studied.Ex.: Functions, graphs, distributions, geometric objects, images.
Q. Why do we study property testing?A. Interested in
• ultra-efficient algorithms.
• connections to inapproximability, locally testable codes, andlearning.
• the relation between local view and global property.
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 5 / 33
![Page 13: Higher-Order Fourier Analysis: Applications to Algebraic Property …research.nii.ac.jp/~yyoshida/resources/Talks/2016-CCC... · 2017. 7. 9. · Higher order Fourier analysis We look](https://reader035.vdocument.in/reader035/viewer/2022071022/5fd6614edb6a48475f1ccebd/html5/thumbnails/13.jpg)
Backgrounds
Property testing was introduced by [RS96] for program checking.
Since then, various kinds of objects have been studied.Ex.: Functions, graphs, distributions, geometric objects, images.
Q. Why do we study property testing?A. Interested in
• ultra-efficient algorithms.
• connections to inapproximability, locally testable codes, andlearning.
• the relation between local view and global property.
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 5 / 33
![Page 14: Higher-Order Fourier Analysis: Applications to Algebraic Property …research.nii.ac.jp/~yyoshida/resources/Talks/2016-CCC... · 2017. 7. 9. · Higher order Fourier analysis We look](https://reader035.vdocument.in/reader035/viewer/2022071022/5fd6614edb6a48475f1ccebd/html5/thumbnails/14.jpg)
Backgrounds
Property testing was introduced by [RS96] for program checking.
Since then, various kinds of objects have been studied.Ex.: Functions, graphs, distributions, geometric objects, images.
Q. Why do we study property testing?
A. Interested in
• ultra-efficient algorithms.
• connections to inapproximability, locally testable codes, andlearning.
• the relation between local view and global property.
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 5 / 33
![Page 15: Higher-Order Fourier Analysis: Applications to Algebraic Property …research.nii.ac.jp/~yyoshida/resources/Talks/2016-CCC... · 2017. 7. 9. · Higher order Fourier analysis We look](https://reader035.vdocument.in/reader035/viewer/2022071022/5fd6614edb6a48475f1ccebd/html5/thumbnails/15.jpg)
Backgrounds
Property testing was introduced by [RS96] for program checking.
Since then, various kinds of objects have been studied.Ex.: Functions, graphs, distributions, geometric objects, images.
Q. Why do we study property testing?A. Interested in
• ultra-efficient algorithms.
• connections to inapproximability, locally testable codes, andlearning.
• the relation between local view and global property.
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 5 / 33
![Page 16: Higher-Order Fourier Analysis: Applications to Algebraic Property …research.nii.ac.jp/~yyoshida/resources/Talks/2016-CCC... · 2017. 7. 9. · Higher order Fourier analysis We look](https://reader035.vdocument.in/reader035/viewer/2022071022/5fd6614edb6a48475f1ccebd/html5/thumbnails/16.jpg)
Local testability of affine-Invariant properties
Definition
P is affine-invariant if a function f : Fn2 → {0, 1} satisfies P , then
f ◦ A satisfies P for any bijective affine transformation A : Fn2 → Fn
2.
Definition
P is (locally) testable if there is a tester for P with q(ε) queries.
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 6 / 33
![Page 17: Higher-Order Fourier Analysis: Applications to Algebraic Property …research.nii.ac.jp/~yyoshida/resources/Talks/2016-CCC... · 2017. 7. 9. · Higher order Fourier analysis We look](https://reader035.vdocument.in/reader035/viewer/2022071022/5fd6614edb6a48475f1ccebd/html5/thumbnails/17.jpg)
Local testability of affine-Invariant properties
Definition
P is affine-invariant if a function f : Fn2 → {0, 1} satisfies P , then
f ◦ A satisfies P for any bijective affine transformation A : Fn2 → Fn
2.
Definition
P is (locally) testable if there is a tester for P with q(ε) queries.
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 6 / 33
![Page 18: Higher-Order Fourier Analysis: Applications to Algebraic Property …research.nii.ac.jp/~yyoshida/resources/Talks/2016-CCC... · 2017. 7. 9. · Higher order Fourier analysis We look](https://reader035.vdocument.in/reader035/viewer/2022071022/5fd6614edb6a48475f1ccebd/html5/thumbnails/18.jpg)
Local testability of affine-Invariant properties
Some specific locally testable affine-invariant properties:
• Degree-d polynomials [AKK+05, BKS+10]
• Fourier sparsity [GOS+11]
• Odd-cycle-freeness: There exist no x1, . . . , x2k+1 ∈ Fn2 such that
f (x1) = · · · = f (x2k+1) = 1 and x1 + · · ·+ x2k+1 ≡ 0 [BGRS12].
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 7 / 33
![Page 19: Higher-Order Fourier Analysis: Applications to Algebraic Property …research.nii.ac.jp/~yyoshida/resources/Talks/2016-CCC... · 2017. 7. 9. · Higher order Fourier analysis We look](https://reader035.vdocument.in/reader035/viewer/2022071022/5fd6614edb6a48475f1ccebd/html5/thumbnails/19.jpg)
The goal
Q. Can we characterize locally testable affine-invariantproperties? [KS08]
A. Yes.
In this talk, we review how we have attacked this question.
• One-sided error testable ≈ Affine-subspace hereditary
• Testable ⇔ Estimable
• Two-sided error testable ⇔ Regular-reducible
Higher order Fourier analysis has played a crucial role!
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 8 / 33
![Page 20: Higher-Order Fourier Analysis: Applications to Algebraic Property …research.nii.ac.jp/~yyoshida/resources/Talks/2016-CCC... · 2017. 7. 9. · Higher order Fourier analysis We look](https://reader035.vdocument.in/reader035/viewer/2022071022/5fd6614edb6a48475f1ccebd/html5/thumbnails/20.jpg)
The goal
Q. Can we characterize locally testable affine-invariantproperties? [KS08]
A. Yes.
In this talk, we review how we have attacked this question.
• One-sided error testable ≈ Affine-subspace hereditary
• Testable ⇔ Estimable
• Two-sided error testable ⇔ Regular-reducible
Higher order Fourier analysis has played a crucial role!
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 8 / 33
![Page 21: Higher-Order Fourier Analysis: Applications to Algebraic Property …research.nii.ac.jp/~yyoshida/resources/Talks/2016-CCC... · 2017. 7. 9. · Higher order Fourier analysis We look](https://reader035.vdocument.in/reader035/viewer/2022071022/5fd6614edb6a48475f1ccebd/html5/thumbnails/21.jpg)
The goal
Q. Can we characterize locally testable affine-invariantproperties? [KS08]
A. Yes.
In this talk, we review how we have attacked this question.
• One-sided error testable ≈ Affine-subspace hereditary
• Testable ⇔ Estimable
• Two-sided error testable ⇔ Regular-reducible
Higher order Fourier analysis has played a crucial role!
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 8 / 33
![Page 22: Higher-Order Fourier Analysis: Applications to Algebraic Property …research.nii.ac.jp/~yyoshida/resources/Talks/2016-CCC... · 2017. 7. 9. · Higher order Fourier analysis We look](https://reader035.vdocument.in/reader035/viewer/2022071022/5fd6614edb6a48475f1ccebd/html5/thumbnails/22.jpg)
Fourier analysis
A function f : Fn2 → R can be uniquely decomposed as
f (x) =∑S⊆[n]
f̂ (S)χS(x),
where χS(x) = (−1)∑
i∈S xi .
f̂ (S) measures the correlation of f with χS . (Fourier coefficients)
Fourier analysis is
• powerful enough to study specific properties.
• not powerful enough to obtain general results.
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 9 / 33
![Page 23: Higher-Order Fourier Analysis: Applications to Algebraic Property …research.nii.ac.jp/~yyoshida/resources/Talks/2016-CCC... · 2017. 7. 9. · Higher order Fourier analysis We look](https://reader035.vdocument.in/reader035/viewer/2022071022/5fd6614edb6a48475f1ccebd/html5/thumbnails/23.jpg)
Fourier analysis
A function f : Fn2 → R can be uniquely decomposed as
f (x) =∑S⊆[n]
f̂ (S)χS(x),
where χS(x) = (−1)∑
i∈S xi .
f̂ (S) measures the correlation of f with χS . (Fourier coefficients)
Fourier analysis is
• powerful enough to study specific properties.
• not powerful enough to obtain general results.
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 9 / 33
![Page 24: Higher-Order Fourier Analysis: Applications to Algebraic Property …research.nii.ac.jp/~yyoshida/resources/Talks/2016-CCC... · 2017. 7. 9. · Higher order Fourier analysis We look](https://reader035.vdocument.in/reader035/viewer/2022071022/5fd6614edb6a48475f1ccebd/html5/thumbnails/24.jpg)
Fourier analysis
A function f : Fn2 → R can be uniquely decomposed as
f (x) =∑S⊆[n]
f̂ (S)χS(x),
where χS(x) = (−1)∑
i∈S xi .
f̂ (S) measures the correlation of f with χS . (Fourier coefficients)
Fourier analysis is
• powerful enough to study specific properties.
• not powerful enough to obtain general results.
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 9 / 33
![Page 25: Higher-Order Fourier Analysis: Applications to Algebraic Property …research.nii.ac.jp/~yyoshida/resources/Talks/2016-CCC... · 2017. 7. 9. · Higher order Fourier analysis We look](https://reader035.vdocument.in/reader035/viewer/2022071022/5fd6614edb6a48475f1ccebd/html5/thumbnails/25.jpg)
Higher order Fourier analysis
We look at correlations with polynomials instead of linear functions.
Main technical tools:
• Decomposition theoremA function can be decomposed into a structured part +pseudorandom part (with respect to low-degree polynomials)
• Equidistribution theorem“generic” polynomials look independently distributed.
Caveat: In this talk, we do not touch most of technical foundationssuch as Gowers norm, rank, and bias.
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 10 / 33
![Page 26: Higher-Order Fourier Analysis: Applications to Algebraic Property …research.nii.ac.jp/~yyoshida/resources/Talks/2016-CCC... · 2017. 7. 9. · Higher order Fourier analysis We look](https://reader035.vdocument.in/reader035/viewer/2022071022/5fd6614edb6a48475f1ccebd/html5/thumbnails/26.jpg)
Higher order Fourier analysis
We look at correlations with polynomials instead of linear functions.
Main technical tools:
• Decomposition theoremA function can be decomposed into a structured part +pseudorandom part (with respect to low-degree polynomials)
• Equidistribution theorem“generic” polynomials look independently distributed.
Caveat: In this talk, we do not touch most of technical foundationssuch as Gowers norm, rank, and bias.
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 10 / 33
![Page 27: Higher-Order Fourier Analysis: Applications to Algebraic Property …research.nii.ac.jp/~yyoshida/resources/Talks/2016-CCC... · 2017. 7. 9. · Higher order Fourier analysis We look](https://reader035.vdocument.in/reader035/viewer/2022071022/5fd6614edb6a48475f1ccebd/html5/thumbnails/27.jpg)
Higher order Fourier analysis
We look at correlations with polynomials instead of linear functions.
Main technical tools:
• Decomposition theoremA function can be decomposed into a structured part +pseudorandom part (with respect to low-degree polynomials)
• Equidistribution theorem“generic” polynomials look independently distributed.
Caveat: In this talk, we do not touch most of technical foundationssuch as Gowers norm, rank, and bias.
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 10 / 33
![Page 28: Higher-Order Fourier Analysis: Applications to Algebraic Property …research.nii.ac.jp/~yyoshida/resources/Talks/2016-CCC... · 2017. 7. 9. · Higher order Fourier analysis We look](https://reader035.vdocument.in/reader035/viewer/2022071022/5fd6614edb6a48475f1ccebd/html5/thumbnails/28.jpg)
Oblivious tester
Definition
1
0
1
0
0
0
1
1H
f |H
fAn oblivious tester works as follows:
• Take a restriction f |H .• H: random affine
subspace of dimension h(ε).
• Output based only on f |H .
Motivation: avoid “unnatural” properties such as f ∈ P ⇔ n is even.For natural properties, ∃ a tester ⇒ ∃ an oblivious tester [BGS10].
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 11 / 33
![Page 29: Higher-Order Fourier Analysis: Applications to Algebraic Property …research.nii.ac.jp/~yyoshida/resources/Talks/2016-CCC... · 2017. 7. 9. · Higher order Fourier analysis We look](https://reader035.vdocument.in/reader035/viewer/2022071022/5fd6614edb6a48475f1ccebd/html5/thumbnails/29.jpg)
Decomposition theorem
µf ,h: the distribution of f |H .
Observation
A tester cannot distinguish f from g if µf ,h ≈ µg ,h.
Theorem (Decomposition theorem)
Any function can be decomposed as f = f1 + f2 + f3 for d = d(ε, h):
• f1 = Γ(P1, . . . ,PC ) for “generic” degree-d polynomials {Pi}.• f2: small L2 norm.
• f3: uncorrelated with degree-d polynomials.
The pseudorandom parts f2 and f3 do not affect µf ,h much.⇒ we can focus on the structured part f1.
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 12 / 33
![Page 30: Higher-Order Fourier Analysis: Applications to Algebraic Property …research.nii.ac.jp/~yyoshida/resources/Talks/2016-CCC... · 2017. 7. 9. · Higher order Fourier analysis We look](https://reader035.vdocument.in/reader035/viewer/2022071022/5fd6614edb6a48475f1ccebd/html5/thumbnails/30.jpg)
Decomposition theorem
µf ,h: the distribution of f |H .
Observation
A tester cannot distinguish f from g if µf ,h ≈ µg ,h.
Theorem (Decomposition theorem)
Any function can be decomposed as f = f1 + f2 + f3 for d = d(ε, h):
• f1 = Γ(P1, . . . ,PC ) for “generic” degree-d polynomials {Pi}.• f2: small L2 norm.
• f3: uncorrelated with degree-d polynomials.
The pseudorandom parts f2 and f3 do not affect µf ,h much.⇒ we can focus on the structured part f1.
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 12 / 33
![Page 31: Higher-Order Fourier Analysis: Applications to Algebraic Property …research.nii.ac.jp/~yyoshida/resources/Talks/2016-CCC... · 2017. 7. 9. · Higher order Fourier analysis We look](https://reader035.vdocument.in/reader035/viewer/2022071022/5fd6614edb6a48475f1ccebd/html5/thumbnails/31.jpg)
Decomposition theorem
µf ,h: the distribution of f |H .
Observation
A tester cannot distinguish f from g if µf ,h ≈ µg ,h.
Theorem (Decomposition theorem)
Any function can be decomposed as f = f1 + f2 + f3 for d = d(ε, h):
• f1 = Γ(P1, . . . ,PC ) for “generic” degree-d polynomials {Pi}.• f2: small L2 norm.
• f3: uncorrelated with degree-d polynomials.
The pseudorandom parts f2 and f3 do not affect µf ,h much.⇒ we can focus on the structured part f1.
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 12 / 33
![Page 32: Higher-Order Fourier Analysis: Applications to Algebraic Property …research.nii.ac.jp/~yyoshida/resources/Talks/2016-CCC... · 2017. 7. 9. · Higher order Fourier analysis We look](https://reader035.vdocument.in/reader035/viewer/2022071022/5fd6614edb6a48475f1ccebd/html5/thumbnails/32.jpg)
One-sided error testable ≈Affine-subspace hereditary
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 13 / 33
![Page 33: Higher-Order Fourier Analysis: Applications to Algebraic Property …research.nii.ac.jp/~yyoshida/resources/Talks/2016-CCC... · 2017. 7. 9. · Higher order Fourier analysis We look](https://reader035.vdocument.in/reader035/viewer/2022071022/5fd6614edb6a48475f1ccebd/html5/thumbnails/33.jpg)
Affine-subspace hereditary
Definition
A property P is affine-subspace hereditary iff ∈ P ⇒ f |H ∈ P for any affine subspace H .
Ex.:
• degree-d polynomials, Fourier sparsity, odd-cycle-freeness
• f = gh for some polynomials g , h of degree ≤ d − 1.
• f = g 2 for some polynomial g of degree ≤ d − 1.
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 14 / 33
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Characterization of one-sided error testability
Conjecture ([BGS10])
P is testable with one-sided error by an oblivious tester⇔ P is (essentially) affine-subspace hereditary
⇒ is true [BGS10].
1
0
1
0
0
0
1
1
f |H 62 P1. Suppose f 2 P and
3. f is also rejected w.p.> 0, contradiction.
2. 9f |K , rejected
by the tester
Proof sketch:
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Characterization of one-sided error testability
Conjecture ([BGS10])
P is testable with one-sided error by an oblivious tester⇔ P is (essentially) affine-subspace hereditary
⇒ is true [BGS10].
1
0
1
0
0
0
1
1
f |H 62 P1. Suppose f 2 P and
3. f is also rejected w.p.> 0, contradiction.
2. 9f |K , rejected
by the tester
Proof sketch:
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 15 / 33
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Characterization of one-sided error testability
Conjecture ([BGS10])
P is testable with one-sided error by an oblivious tester⇔ P is (essentially) affine-subspace hereditary
⇒ is true [BGS10].
1
0
1
0
0
0
1
1
f |H 62 P1. Suppose f 2 P and
3. f is also rejected w.p.> 0, contradiction.
2. 9f |K , rejected
by the tester
Proof sketch:
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 15 / 33
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Alternative formulation via linear forms
Think of affine-triangle-freeness:No x , y1, y2 ∈ Fn
2 s.t. f (x + y1) = f (x + y2) = f (x + y1 + y2) = 1.
⇔ No x , y1, y2 ∈ Fn2 s.t.
f (L1(x , y1, y2)) = σ1 for L1(x , y1, y2) = x + y1 and σ1 = 1,
f (L2(x , y1, y2)) = σ2 for L2(x , y1, y2) = x + y2 and σ2 = 1,
f (L3(x , y1, y2)) = σ3 for L3(x , y1, y2) = x + y1 + y2 and σ3 = 1.
We call this (A = (L1, L2, L3), σ = (σ1, σ2, σ3))-freeness.
• A is called an affine system of linear forms.⇒ well studied in higher order Fourier analysis.
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Alternative formulation via linear forms
Think of affine-triangle-freeness:No x , y1, y2 ∈ Fn
2 s.t. f (x + y1) = f (x + y2) = f (x + y1 + y2) = 1.
⇔ No x , y1, y2 ∈ Fn2 s.t.
f (L1(x , y1, y2)) = σ1 for L1(x , y1, y2) = x + y1 and σ1 = 1,
f (L2(x , y1, y2)) = σ2 for L2(x , y1, y2) = x + y2 and σ2 = 1,
f (L3(x , y1, y2)) = σ3 for L3(x , y1, y2) = x + y1 + y2 and σ3 = 1.
We call this (A = (L1, L2, L3), σ = (σ1, σ2, σ3))-freeness.
• A is called an affine system of linear forms.⇒ well studied in higher order Fourier analysis.
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Alternative formulation via linear forms
Think of affine-triangle-freeness:No x , y1, y2 ∈ Fn
2 s.t. f (x + y1) = f (x + y2) = f (x + y1 + y2) = 1.
⇔ No x , y1, y2 ∈ Fn2 s.t.
f (L1(x , y1, y2)) = σ1 for L1(x , y1, y2) = x + y1 and σ1 = 1,
f (L2(x , y1, y2)) = σ2 for L2(x , y1, y2) = x + y2 and σ2 = 1,
f (L3(x , y1, y2)) = σ3 for L3(x , y1, y2) = x + y1 + y2 and σ3 = 1.
We call this (A = (L1, L2, L3), σ = (σ1, σ2, σ3))-freeness.
• A is called an affine system of linear forms.⇒ well studied in higher order Fourier analysis.
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Alternative formulation via linear forms
Think of affine-triangle-freeness:No x , y1, y2 ∈ Fn
2 s.t. f (x + y1) = f (x + y2) = f (x + y1 + y2) = 1.
⇔ No x , y1, y2 ∈ Fn2 s.t.
f (L1(x , y1, y2)) = σ1 for L1(x , y1, y2) = x + y1 and σ1 = 1,
f (L2(x , y1, y2)) = σ2 for L2(x , y1, y2) = x + y2 and σ2 = 1,
f (L3(x , y1, y2)) = σ3 for L3(x , y1, y2) = x + y1 + y2 and σ3 = 1.
We call this (A = (L1, L2, L3), σ = (σ1, σ2, σ3))-freeness.
• A is called an affine system of linear forms.⇒ well studied in higher order Fourier analysis.
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Testability of subspace hereditary properties
Observation
The following are equivalent:
• P is affine-subspace hereditary.
• There exists a (possibly infinite) collection {(A1, σ1), . . .}s.t. f ∈ P ⇔ f is (Ai , σi)-free for each i .
Theorem ([BFH+13])
If each (Ai , σi) has bounded complexity, then the property is testablewith one-sided error.
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Testability of subspace hereditary properties
Observation
The following are equivalent:
• P is affine-subspace hereditary.
• There exists a (possibly infinite) collection {(A1, σ1), . . .}s.t. f ∈ P ⇔ f is (Ai , σi)-free for each i .
Theorem ([BFH+13])
If each (Ai , σi) has bounded complexity, then the property is testablewith one-sided error.
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Proof idea
Let’s focus on the case f = Γ(P1, . . . ,PC ) and P = affine 4-freeness.
f is ε-far from P⇒ There are x∗, y ∗1 , y
∗2 ∈ Fn
2 spanning an affine triangle.
Prx ,y1,y2
[f (x + y1) = f (x + y2) = f (x + y1 + y2) = 1]
≥ Prx ,y1,y2
[Pi(Lj(x , y1, y2)) = Pi(Lj(x∗, y ∗1 , y
∗2 )) ∀i ∈ [C ], j ∈ [3]],
which is non-negligibly high from the equidistribution theorem.⇒ Random sampling works.
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Proof idea
Let’s focus on the case f = Γ(P1, . . . ,PC ) and P = affine 4-freeness.
f is ε-far from P⇒ There are x∗, y ∗1 , y
∗2 ∈ Fn
2 spanning an affine triangle.
Prx ,y1,y2
[f (x + y1) = f (x + y2) = f (x + y1 + y2) = 1]
≥ Prx ,y1,y2
[Pi(Lj(x , y1, y2)) = Pi(Lj(x∗, y ∗1 , y
∗2 )) ∀i ∈ [C ], j ∈ [3]],
which is non-negligibly high from the equidistribution theorem.⇒ Random sampling works.
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Proof idea
Let’s focus on the case f = Γ(P1, . . . ,PC ) and P = affine 4-freeness.
f is ε-far from P⇒ There are x∗, y ∗1 , y
∗2 ∈ Fn
2 spanning an affine triangle.
Prx ,y1,y2
[f (x + y1) = f (x + y2) = f (x + y1 + y2) = 1]
≥ Prx ,y1,y2
[Pi(Lj(x , y1, y2)) = Pi(Lj(x∗, y ∗1 , y
∗2 )) ∀i ∈ [C ], j ∈ [3]],
which is non-negligibly high from the equidistribution theorem.⇒ Random sampling works.
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Equidistribution theorem
The space Fn2 can be divided according to {Pi(Lj(x))}i∈[C ],j∈[3].
Theorem (Equidistribution theorem)
If Pi ’s are “generic” enough, then each cell has almost the same size.
Almost the same size
Fn2 =
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Testability ⇔ Estimability
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Testability ⇐ Estimability
Definition
P is estimable if we can estimate dP(·) to within δ with q(δ) queriesfor any δ > 0.
Trivial direction: P is estimable ⇒ P is testable.
Theorem ([HL13])
P is testable ⇒ P is estimable.
Algorithm:
1: H ← a random affine subspace of a constant dimension.2: return Output dP(f |H).
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Testability ⇐ Estimability
Definition
P is estimable if we can estimate dP(·) to within δ with q(δ) queriesfor any δ > 0.
Trivial direction: P is estimable ⇒ P is testable.
Theorem ([HL13])
P is testable ⇒ P is estimable.
Algorithm:
1: H ← a random affine subspace of a constant dimension.2: return Output dP(f |H).
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Testability ⇐ Estimability
Definition
P is estimable if we can estimate dP(·) to within δ with q(δ) queriesfor any δ > 0.
Trivial direction: P is estimable ⇒ P is testable.
Theorem ([HL13])
P is testable ⇒ P is estimable.
Algorithm:
1: H ← a random affine subspace of a constant dimension.2: return Output dP(f |H).
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Testability ⇐ Estimability
Definition
P is estimable if we can estimate dP(·) to within δ with q(δ) queriesfor any δ > 0.
Trivial direction: P is estimable ⇒ P is testable.
Theorem ([HL13])
P is testable ⇒ P is estimable.
Algorithm:
1: H ← a random affine subspace of a constant dimension.2: return Output dP(f |H).
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Intuition behind the proof
Why can we hope dP(f ) ≈ dP(f |H)?
(Oversimplified argument)
• Since P is testable, dP(f ) is determined by the distribution µf ,h.
• If f = Γ(P1, . . . ,PC ), then µf ,h is determined by Γ and degreesof P1, . . . ,PC (rather than Pi ’s themselves).
• f = Γ(P1, . . . ,PC ) and fH = Γ(P1|H , . . . ,PC |H) share the sameΓ and degrees.⇒ µf ,h ≈ µf |H ,h.⇒ dP(f ) ≈ dP(f |H).
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Intuition behind the proof
Why can we hope dP(f ) ≈ dP(f |H)?
(Oversimplified argument)
• Since P is testable, dP(f ) is determined by the distribution µf ,h.
• If f = Γ(P1, . . . ,PC ), then µf ,h is determined by Γ and degreesof P1, . . . ,PC (rather than Pi ’s themselves).
• f = Γ(P1, . . . ,PC ) and fH = Γ(P1|H , . . . ,PC |H) share the sameΓ and degrees.⇒ µf ,h ≈ µf |H ,h.⇒ dP(f ) ≈ dP(f |H).
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Intuition behind the proof
Why can we hope dP(f ) ≈ dP(f |H)?
(Oversimplified argument)
• Since P is testable, dP(f ) is determined by the distribution µf ,h.
• If f = Γ(P1, . . . ,PC ), then µf ,h is determined by Γ and degreesof P1, . . . ,PC (rather than Pi ’s themselves).
• f = Γ(P1, . . . ,PC ) and fH = Γ(P1|H , . . . ,PC |H) share the sameΓ and degrees.⇒ µf ,h ≈ µf |H ,h.⇒ dP(f ) ≈ dP(f |H).
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Intuition behind the proof
Why can we hope dP(f ) ≈ dP(f |H)?
(Oversimplified argument)
• Since P is testable, dP(f ) is determined by the distribution µf ,h.
• If f = Γ(P1, . . . ,PC ), then µf ,h is determined by Γ and degreesof P1, . . . ,PC (rather than Pi ’s themselves).
• f = Γ(P1, . . . ,PC ) and fH = Γ(P1|H , . . . ,PC |H) share the sameΓ and degrees.⇒ µf ,h ≈ µf |H ,h.⇒ dP(f ) ≈ dP(f |H).
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Two-sided error testability ⇔Regular-reducibility
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Structured part
Recall that, for f = Γ(P1, . . . ,PC ) + f2 + f3,
µf ,h is determined by Γ and degrees of Pi ’s.
Let’s use them as a (constant-size) sketch of f .
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Regularity-instance (simplified)
Definition
A regularity-instance I is a tuple of
• a complexity parameter C ∈ N,
• a structure function Γ : FC2 → [0, 1],
• a degree-bound parameter d ∈ N,
• a degree parameter d = (d1, . . . , dC ) ∈ NC with di < d ,
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Satisfying a regularity-instance
Definition
Let I = (C , Γ, d ,d) be a regularity-instance.f satisfies I if it is of the form
f (x) = Γ(P1(x), . . . ,PC (x)) + Υ(x),
where
• Pi is a polynomial of degree di ,
• P1, . . . ,PC are “generic” enough.
• Υ is uncorrelated with degree-(d − 1) polynomials.
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Testing regularity-instances
Theorem ([Yos14a])
For any regularity-instance I , there is a tester for the property ofsatisfying I .
Algorithm:
1: H ← a random affine subspace of a constant dimension.2: if f |H is close to satisfying I then accept.3: else reject.
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Regular-reducibility
A property P is regular-reducible if for any δ > 0, there exists a setR of constant number of regularity-instances such that:
f 2 P �
� ✏� � g : ✏-far from P� ✏� �
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Characterization of two-sided error testability
Theorem
An affine-invariant property P is testablem
P is regular-reducible.
Proof sketch:
• Regular-reducible ⇒ testableRegularity-instances are testable, and testability impliesestimability [HL13]. Hence, we can estimate the distance to R.
• Testable ⇒ regular-reducibleThe behavior of a tester depends only on µf ,h. Since Γ and ddetermines the distribution, we can find R using the tester.
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Characterization of two-sided error testability
Theorem
An affine-invariant property P is testablem
P is regular-reducible.
Proof sketch:
• Regular-reducible ⇒ testableRegularity-instances are testable, and testability impliesestimability [HL13]. Hence, we can estimate the distance to R.
• Testable ⇒ regular-reducibleThe behavior of a tester depends only on µf ,h. Since Γ and ddetermines the distribution, we can find R using the tester.
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 29 / 33
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Notes
• We need to deal with “non-classical” polynomials instead ofpolynomials.
• Another characterization of testability was shown by introducing“functions limits” [Yos14b].
• Applications of the characterizations:• Low-degree polynomials.• Having a low spectral norm
∑S |f̂ (S)|.
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 30 / 33
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Summary
Higher order Fourier analysis is useful for studying property testing as
• we care about the distribution µf ,h for h = O(1),
• which is determined by the structured part given by thedecomposition theorem.
We are almost done, qualitatively .
• one-sided error testability ≈ affine-subspace hereditary (ofbounded complexity)
• two-sided error testability ⇔ regular-reducibility.
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 31 / 33
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Summary
Higher order Fourier analysis is useful for studying property testing as
• we care about the distribution µf ,h for h = O(1),
• which is determined by the structured part given by thedecomposition theorem.
We are almost done, qualitatively .
• one-sided error testability ≈ affine-subspace hereditary (ofbounded complexity)
• two-sided error testability ⇔ regular-reducibility.
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 31 / 33
![Page 67: Higher-Order Fourier Analysis: Applications to Algebraic Property …research.nii.ac.jp/~yyoshida/resources/Talks/2016-CCC... · 2017. 7. 9. · Higher order Fourier analysis We look](https://reader035.vdocument.in/reader035/viewer/2022071022/5fd6614edb6a48475f1ccebd/html5/thumbnails/67.jpg)
Future direction
Property Testing
• Other groups:• Abelian ⇒ higher order Fourier analysis exists [Sze12].• Non-Abelian ⇒ representation theory? [OY16]
• Why is affine invariance easier to deal with than permutationinvariance?
Other applications of higher order Fourier analysis.
• Coding theory [BG16, BL15a].
• Learning theory [BHT15].
• Complexity theory [BL15b].
• Algorithms for polynomials [Bha14].
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 28, 2016 32 / 33