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Higher-Order Fourier Analysis:
Applications to Algebraic Property Testing
Yuichi Yoshida
National Institute of Informatics, andPreferred Infrastructure, Inc
May 17, 2016
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 1 / 37
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 17, 2016 2 / 37
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 17, 2016 2 / 37
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 17, 2016 2 / 37
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 17, 2016 3 / 37
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 17, 2016 3 / 37
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) ≤ 2/3.
• Query complexity is O(1/ε)⇒ constant!
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 4 / 37
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) ≤ 2/3.
• Query complexity is O(1/ε)⇒ constant!
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 4 / 37
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) ≤ 2/3.
• Query complexity is O(1/ε)⇒ constant!
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 4 / 37
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) ≤ 2/3.
• Query complexity is O(1/ε)⇒ constant!
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 4 / 37
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) ≤ 2/3.
• Query complexity is O(1/ε)⇒ constant!
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 4 / 37
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 17, 2016 5 / 37
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 17, 2016 5 / 37
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 17, 2016 5 / 37
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 17, 2016 5 / 37
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 17, 2016 6 / 37
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 17, 2016 6 / 37
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 17, 2016 7 / 37
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 17, 2016 8 / 37
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 17, 2016 8 / 37
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 17, 2016 8 / 37
Fourier analysis
For S ⊆ [n], define χS : Fn2 → {−1, 1} as χS(x) = (−1)
∑i∈S xi .
{χS} forms an orthonormal basis for functions: Fn2 → R:
Ex
[χS(x)χT (x)] = Ex
[(−1)∑
i∈S4T xi ] =
{1 if S = T ,
0 otherwise.
⇒ A function f : Fn2 → R can be uniquely decomposed as
f (x) =∑S⊆[n]
f (S)χS(x),
where f (S) = Ex [f (x)χS(x)] measures the correlation of f with χS .(Fourier coefficients)
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 9 / 37
Fourier analysis
For S ⊆ [n], define χS : Fn2 → {−1, 1} as χS(x) = (−1)
∑i∈S xi .
{χS} forms an orthonormal basis for functions: Fn2 → R:
Ex
[χS(x)χT (x)] = Ex
[(−1)∑
i∈S4T xi ] =
{1 if S = T ,
0 otherwise.
⇒ A function f : Fn2 → R can be uniquely decomposed as
f (x) =∑S⊆[n]
f (S)χS(x),
where f (S) = Ex [f (x)χS(x)] measures the correlation of f with χS .(Fourier coefficients)
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 9 / 37
Linearity testing
Input: a function f : Fn2 → F2 and ε > 0.
Goal: f (x) + f (y) ≡ f (x + y) (mod 2) for every x , y ∈ Fn2?
1: for i = 1 to Θ(1/ε) do2: Sample x , y ∈ Fn
2 uniformly at random.3: if f (x) + f (y) 6≡ f (x + y) (mod 2) then reject.4: Accept.
Theorem ([BLR93])
• If f is linear, always accepts. (one-sided error)
• If f is ε-far, rejects with probability at least 2/3.
• Query complexity is O(1/ε)
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 10 / 37
Linearity testing
Input: a function f : Fn2 → F2 and ε > 0.
Goal: f (x) + f (y) ≡ f (x + y) (mod 2) for every x , y ∈ Fn2?
1: for i = 1 to Θ(1/ε) do2: Sample x , y ∈ Fn
2 uniformly at random.3: if f (x) + f (y) 6≡ f (x + y) (mod 2) then reject.4: Accept.
Theorem ([BLR93])
• If f is linear, always accepts. (one-sided error)
• If f is ε-far, rejects with probability at least 2/3.
• Query complexity is O(1/ε)
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 10 / 37
Linearity testing
Input: a function f : Fn2 → F2 and ε > 0.
Goal: f (x) + f (y) ≡ f (x + y) (mod 2) for every x , y ∈ Fn2?
1: for i = 1 to Θ(1/ε) do2: Sample x , y ∈ Fn
2 uniformly at random.3: if f (x) + f (y) 6≡ f (x + y) (mod 2) then reject.4: Accept.
Theorem ([BLR93])
• If f is linear, always accepts. (one-sided error)
• If f is ε-far, rejects with probability at least 2/3.
• Query complexity is O(1/ε)
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 10 / 37
Observation
For any S ⊆ [n], we have
ε ≤ dLIN(f ) = Pr[f (x) 6≡∑i∈S
xi ] = Pr[(−1)f (x) 6= χS(x)]
= E[1− (−1)f (x)χS(x)
2
]=
1− g(S)
2(g := (−1)f ).
This fact can be used to analyze the rejection probability.
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 17, 2016 11 / 37
Observation
For any S ⊆ [n], we have
ε ≤ dLIN(f ) = Pr[f (x) 6≡∑i∈S
xi ] = Pr[(−1)f (x) 6= χS(x)]
= E[1− (−1)f (x)χS(x)
2
]=
1− g(S)
2(g := (−1)f ).
This fact can be used to analyze the rejection probability.
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 17, 2016 11 / 37
Observation
For any S ⊆ [n], we have
ε ≤ dLIN(f ) = Pr[f (x) 6≡∑i∈S
xi ] = Pr[(−1)f (x) 6= χS(x)]
= E[1− (−1)f (x)χS(x)
2
]=
1− g(S)
2(g := (−1)f ).
This fact can be used to analyze the rejection probability.
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 17, 2016 11 / 37
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 17, 2016 12 / 37
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 17, 2016 12 / 37
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 17, 2016 12 / 37
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 17, 2016 13 / 37
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 17, 2016 14 / 37
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 17, 2016 14 / 37
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 17, 2016 14 / 37
Decomposition theorem for Fourier analysis
Any function f : F2 → {−1, 1} can be decomposed as:
f =∑
S⊆[n]:|f (S)|>ε
f (S)χS(x) +∑
S⊆[n]:|f (S)|≤ε
f (S)χS(x).
• By Perseval’s identity∑
S f (S)2 = 1⇒ the former depends only on O(1/ε2) many linear functions.
• The latter has negligible Fourier coefficients.
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 15 / 37
Decomposition theorem for Fourier analysis
Any function f : F2 → {−1, 1} can be decomposed as:
f =∑
S⊆[n]:|f (S)|>ε
f (S)χS(x) +∑
S⊆[n]:|f (S)|≤ε
f (S)χS(x).
• By Perseval’s identity∑
S f (S)2 = 1
⇒ the former depends only on O(1/ε2) many linear functions.
• The latter has negligible Fourier coefficients.
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 15 / 37
Decomposition theorem for Fourier analysis
Any function f : F2 → {−1, 1} can be decomposed as:
f =∑
S⊆[n]:|f (S)|>ε
f (S)χS(x) +∑
S⊆[n]:|f (S)|≤ε
f (S)χS(x).
• By Perseval’s identity∑
S f (S)2 = 1⇒ the former depends only on O(1/ε2) many linear functions.
• The latter has negligible Fourier coefficients.
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 15 / 37
Decomposition theorem for Fourier analysis
Any function f : F2 → {−1, 1} can be decomposed as:
f =∑
S⊆[n]:|f (S)|>ε
f (S)χS(x) +∑
S⊆[n]:|f (S)|≤ε
f (S)χS(x).
• By Perseval’s identity∑
S f (S)2 = 1⇒ the former depends only on O(1/ε2) many linear functions.
• The latter has negligible Fourier coefficients.
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 15 / 37
One-sided error testable ≈Affine-subspace hereditary
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 16 / 37
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 17, 2016 17 / 37
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 17, 2016 18 / 37
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 17, 2016 18 / 37
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 17, 2016 18 / 37
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.
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 19 / 37
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.
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 19 / 37
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.
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 19 / 37
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.
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 19 / 37
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.
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 20 / 37
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.
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 20 / 37
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.
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 21 / 37
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.
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 21 / 37
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.
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 21 / 37
Equidistribution theorem
The space Fn2 can be divided according to {Pi(Lj(x))}i∈[C ],j∈[3].
Theorem (Equidistribution theorem)
If each Pi is “generic” enough, then each cell has almost the samesize.
Almost the same size
Fn2 =
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 22 / 37
Equidistribution theorem for Fourier analysis
Let P1, . . . ,PC : Fn2 → F2 be linear functions. If Pi(x) is of the form∑
j∈Si xj , and 1S1 , . . . , 1SC are linearly independent, then for anyσ1, . . . , σC ,
Prx
[Pi(x) = σi ∀i ∈ [C ]] =#{x ∈ Fn
2 |∑
j∈Si xj = σi ∀i ∈ [C ]}2n
= 2−|C |.
⇒ Equidistributed.
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 23 / 37
Testability ⇔ Estimability
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 24 / 37
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).
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 25 / 37
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).
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 25 / 37
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).
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 25 / 37
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).
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 25 / 37
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).
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 26 / 37
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).
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 26 / 37
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).
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 26 / 37
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).
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 26 / 37
Two-sided error testability ⇔Regular-reducibility
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 27 / 37
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 .
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 28 / 37
Regularity-instance (simplified)
Definition
A regularity-instance I is a tuple of
• a structure function Γ : FC2 → [0, 1],
• a complexity parameter C ∈ N,
• a degree-bound parameter d ∈ N,
• a degree parameter d = (d1, . . . , dC ) ∈ NC with di < d ,
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 29 / 37
Satisfying a regularity-instance
Definition
Let I = (γ, Γ,C , 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.
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 30 / 37
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.
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 31 / 37
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� ✏� �
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 32 / 37
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 17, 2016 33 / 37
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 17, 2016 33 / 37
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 |f (S)|.
Yuichi Yoshida (NII and PFI) Applications to algebraic property testing May 17, 2016 34 / 37
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 17, 2016 35 / 37
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 17, 2016 35 / 37
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 17, 2016 36 / 37