A Characterization of Locally Testable Affine-Invariant Properties via Decomposition Theorems Yuichi Yoshida National Institute of Informatics and Preferred Infrastructure, Inc November 29, 2013 Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties November 29, 2013 1 / 25
Property Testing Definition f : { 0 , 1 } n → { 0 , 1 } is ǫ -far from P if, for any g : { 0 , 1 } n → { 0 , 1 } satisfying P , x [ f ( x ) � = g ( x )] ≥ ǫ. Pr Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties November 29, 2013 2 / 25
Property Testing Definition f : { 0 , 1 } n → { 0 , 1 } is ǫ -far from P if, for any g : { 0 , 1 } n → { 0 , 1 } satisfying P , x [ f ( x ) � = g ( x )] ≥ ǫ. Pr ǫ -tester for a property P : • Given f : { 0 , 1 } n → { 0 , 1 } P Accept w.p. 2/3 as a query access. • Proximity parameter ǫ > 0. ε -far Reject w.p. 2/3 Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties November 29, 2013 2 / 25
Local Testability Definition P is locally testable if, for any ǫ > 0, there is an ǫ -tester with query complexity that only depends on ǫ . Examples of locally testable properties: • Linearity: O (1 /ǫ ) [BLR93] • d -degree Polynomials: O (2 d + 1 /ǫ ) [AKK + 05, BKS + 10] • Fourier sparsity [GOS + 11] • Odd-cycle-freeness: O (1 /ǫ 2 ) [BGRS12] � ∃ odd k and x 1 , . . . , x k such that � i x i = 0, f ( x i ) = 1 for all i . • k -Juntas: O ( k /ǫ + k log k ) [FKR + 04, Bla09]. Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties November 29, 2013 3 / 25
Affine-Invariant Properties Definition P is affine-invariant if a function f : F n 2 → { 0 , 1 } satisfies P , then f ◦ A satisfies P for any bijective affine transformation A : F n 2 → F n 2 . Examples: Linearity, low-degree polynomials, Fourier sparsity, odd-cycle-freeness. Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties November 29, 2013 4 / 25
Affine-Invariant Properties Definition P is affine-invariant if a function f : F n 2 → { 0 , 1 } satisfies P , then f ◦ A satisfies P for any bijective affine transformation A : F n 2 → F n 2 . Examples: Linearity, low-degree polynomials, Fourier sparsity, odd-cycle-freeness. Q. Characterization of locally testable affine-invariant properties? [KS08] Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties November 29, 2013 4 / 25
Related Work • Locally testable with one-sided error ⇔ affine-subspace hereditary? [BGS10] Ex. Linearity, low-degree polynomials, odd-cycle-freeness. • ⇒ is true. [BGS10] • ⇐ is true (if the property has bounded complexity). [BFH + 13]. Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties November 29, 2013 5 / 25
Related Work • Locally testable with one-sided error ⇔ affine-subspace hereditary? [BGS10] Ex. Linearity, low-degree polynomials, odd-cycle-freeness. • ⇒ is true. [BGS10] • ⇐ is true (if the property has bounded complexity). [BFH + 13]. • P is locally testable ⇒ distance to P is estimable. [HL13] Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties November 29, 2013 5 / 25
Related Work • Locally testable with one-sided error ⇔ affine-subspace hereditary? [BGS10] Ex. Linearity, low-degree polynomials, odd-cycle-freeness. • ⇒ is true. [BGS10] • ⇐ is true (if the property has bounded complexity). [BFH + 13]. • P is locally testable ⇒ distance to P is estimable. [HL13] • P is locally testable ⇔ regular-reducible. [This work] Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties November 29, 2013 5 / 25
Graph Property Testing Definition A graph G = ( V , E ) is ǫ -far from a property P if we must add or remove at least ǫ | V | 2 edges to make G satisfy P . Examples of locally testable properties: • 3-Colorability [GGR98] • H -freeness [AFKS00] • Monotone properties [AS08b] • Hereditary properties [AS08a] Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties November 29, 2013 6 / 25
V1 V2 V4 V3 A Characterization of Locally Testable Graph Properties η 12 Szemer´ edi’s regularity lemma: Every graph can be partitioned into a constant number of parts so η 13 η 14 that each pair of parts looks random. Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties November 29, 2013 7 / 25
V1 V2 V4 V3 A Characterization of Locally Testable Graph Properties η 12 Szemer´ edi’s regularity lemma: Every graph can be partitioned into a constant number of parts so η 13 η 14 that each pair of parts looks random. Theorem ([AFNS09]) A graph property P is locally testable ⇔ whether P holds is determined only by the set of densities { η ij } i , j . Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties November 29, 2013 7 / 25
V1 V2 V4 V3 A Characterization of Locally Testable Graph Properties η 12 Szemer´ edi’s regularity lemma: Every graph can be partitioned into a constant number of parts so η 13 η 14 that each pair of parts looks random. Theorem ([AFNS09]) A graph property P is locally testable ⇔ whether P holds is determined only by the set of densities { η ij } i , j . Q. How can we extract such constant-size sketches from functions? Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties November 29, 2013 7 / 25
Constant Sketch for Functions Theorem (Decomposition Theorem [BFH + 13]) For any γ > 0 , d ≥ 1 , and r : N → N , there exists C such that: 2 → { 0 , 1 } can be decomposed as f = f ′ + f ′′ , any function f : F n where Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties November 29, 2013 8 / 25
Constant Sketch for Functions Theorem (Decomposition Theorem [BFH + 13]) For any γ > 0 , d ≥ 1 , and r : N → N , there exists C such that: 2 → { 0 , 1 } can be decomposed as f = f ′ + f ′′ , any function f : F n where • a structured part f ′ : F n 2 → [0 , 1] , where • f ′ = Γ( P 1 , . . . , P C ) with C ≤ C, • P 1 , . . . , P C are “non-classical” polynomials of degree < d and rank ≥ r ( C ) . • Γ : T C → [0 , 1] is a function. Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties November 29, 2013 8 / 25
Constant Sketch for Functions Theorem (Decomposition Theorem [BFH + 13]) For any γ > 0 , d ≥ 1 , and r : N → N , there exists C such that: 2 → { 0 , 1 } can be decomposed as f = f ′ + f ′′ , any function f : F n where • a structured part f ′ : F n 2 → [0 , 1] , where • f ′ = Γ( P 1 , . . . , P C ) with C ≤ C, • P 1 , . . . , P C are “non-classical” polynomials of degree < d and rank ≥ r ( C ) . • Γ : T C → [0 , 1] is a function. • a pseudo-random part f ′′ : F n 2 → [ − 1 , 1] • The Gowers norm � f ′′ � U d is at most γ . Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties November 29, 2013 8 / 25
Factors Polynomial sequence ( P 1 , . . . , P C ) partitions F n 2 into atoms F n 2 = { x | P 1 ( x ) = b 1 , . . . , P C ( x ) = b C } . Almost the same size The decomposition theorem says: f = +Υ Γ ( P 1 , . . . , P C ) Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties November 29, 2013 9 / 25
What is the Gowers Norm? Definition Let f : F n 2 → C . The Gowers norm of order d for f is 1 / 2 d � � J | I | f ( x + � f � U d := y i ) , E x , y 1 ,..., y d I ⊆{ 1 ,..., d } i ∈ I where J denotes complex conjugation. • � f � U 1 = | E x f ( x ) | • � f � U 1 ≤ � f � U 2 ≤ � f � U 3 ≤ · · · • � f � U d measures correlation with polynomials of degree < d . Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties November 29, 2013 10 / 25
Correlation with Polynomials of Degree < d Proposition For any polynomial P : F n 2 → { 0 , 1 } of degree < d, � ( − 1) P � U d = 1 . Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties November 29, 2013 11 / 25
Correlation with Polynomials of Degree < d Proposition For any polynomial P : F n 2 → { 0 , 1 } of degree < d, � ( − 1) P � U d = 1 . However, the converse does not hold when d ≥ 4... Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties November 29, 2013 11 / 25
Correlation with Polynomials of Degree < d Proposition For any polynomial P : F n 2 → { 0 , 1 } of degree < d, � ( − 1) P � U d = 1 . However, the converse does not hold when d ≥ 4... Definition P : F n 2 → T is a non-classical polynomial of degree < d if � exp(2 π i · f ) � U d = 1. 2 k +1 , . . . , 2 k +1 − 1 1 It turns out that the range of P is U k +1 := { 0 , 2 k +1 } for some k (= depth ). Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties November 29, 2013 11 / 25
Is This Really a Constant-size Sketch? • Structured part: f ′ = Γ( P 1 , . . . , P C ). • Γ indeed has a constant-size representation, but P 1 , . . . , P C may not have (even if we just want to specify the coset { P ◦ A } ). • The rank of ( P 1 , . . . , P C ) is high ⇒ Their degrees and depths determine almost everything. Ex. the distribution of the restriction of f to a random affine subspace. Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties November 29, 2013 12 / 25
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