A Characterization of Locally Testable Affine-Invariant Properties via Decomposition Theorems Yuichi Yoshida National Institute of Informatics and Preferred Infrastructure, Inc June 1, 2014 Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties June 1, 2014 1 / 20
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 June 1, 2014 2 / 20
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 June 1, 2014 2 / 20
Local Testability and Affine-Invariance Definition P is locally testable if, for any ǫ > 0, there is an ǫ -tester with query complexity that only depends on ǫ . Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties June 1, 2014 3 / 20
Local Testability and Affine-Invariance Definition P is locally testable if, for any ǫ > 0, there is an ǫ -tester with query complexity that only depends on ǫ . 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 of locally testable affine-invariant properties: • d -degree Polynomials [AKK + 05, BKS + 10]. • Fourier sparsity [GOS + 11]. • Odd-cycle-freeness [BGRS12]. Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties June 1, 2014 3 / 20
The Question and Related Work Q. Characterization of locally testable affine-invariant properties? [KS08] • Locally testable with one-sided error ⇔ affine-subspace hereditary? [BGS10] Ex. 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 June 1, 2014 4 / 20
The Question and Related Work Q. Characterization of locally testable affine-invariant properties? [KS08] • Locally testable with one-sided error ⇔ affine-subspace hereditary? [BGS10] Ex. 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 June 1, 2014 4 / 20
The Question and Related Work Q. Characterization of locally testable affine-invariant properties? [KS08] • Locally testable with one-sided error ⇔ affine-subspace hereditary? [BGS10] Ex. 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 June 1, 2014 4 / 20
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 June 1, 2014 5 / 20
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 June 1, 2014 6 / 20
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 June 1, 2014 6 / 20
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 June 1, 2014 6 / 20
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 June 1, 2014 7 / 20
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 June 1, 2014 7 / 20
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 June 1, 2014 7 / 20
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 June 1, 2014 8 / 20
What is the Gowers Norm? Definition Let f : F n 2 → C . The d-th Gowers norm of 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. Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties June 1, 2014 9 / 20
What is the Gowers Norm? Definition Let f : F n 2 → C . The d-th Gowers norm of 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. • For any linear function L : F n 2 → F 2 , x , y 1 , y 2 ( − 1) L ( x )+ L ( x + y 1 )+ L ( x + y 2 )+ L ( x + y 1 + y 2 ) | = 1 . � ( − 1) L � U 2 = | E • For any polynomial P : F n 2 → F 2 of degree < d , � ( − 1) P � U d = 1. Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties June 1, 2014 9 / 20
Gowers Norm Measures Correlation with Non-Classical Polynomials Definition P : F n 2 → T is a non-classical polynomial of degree < d if � e 2 π i · P � U d = 1. 2 k +1 , . . . , 2 k +1 − 1 1 The range of P is { 0 , 2 k +1 } for some k (= depth ). Lemma f : F n 2 → C with � f � ∞ ≤ 1 . • � f � U d ≤ ǫ ⇒ � f , e 2 π i · P � ≤ ǫ for any non-classical polynomial P of degree < d. (Direct Theorem) • � f � U d ≥ ǫ ⇒ � f , e 2 π i · P � ≥ δ ( ǫ ) for some non-classical polynomial P of degree < d. (Inverse Theorem) Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties June 1, 2014 10 / 20
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 June 1, 2014 11 / 20
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