A Characterization of Locally Testable Affine-Invariant Properties via Decomposition Theorems

Transcription

1 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, / 20

2 Property Testing Definition f : {0, 1} n {0, 1} is ɛ-far from P if, for any g : {0, 1} n {0, 1} satisfying P, Pr x [f (x) g(x)] ɛ. Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties June 1, / 20

3 Property Testing Definition f : {0, 1} n {0, 1} is ɛ-far from P if, for any g : {0, 1} n {0, 1} satisfying P, Pr x [f (x) g(x)] ɛ. ɛ-tester for a property P: Given f : {0, 1} n {0, 1} as a query access. Proximity parameter ɛ > 0. P Accept w.p. 2/3 ε-far Reject w.p. 2/3 Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties June 1, / 20

4 Local Testability and Affine-Invariance Definition P is locally testable if, for any ɛ > 0, there is an ɛ-tester with query complexity that only depends on ɛ. Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties June 1, / 20

5 Local Testability and Affine-Invariance Definition P is locally testable if, for any ɛ > 0, there is an ɛ-tester with query complexity that only depends on ɛ. 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, / 20

6 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, / 20

7 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, / 20

8 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, / 20

9 Graph Property Testing Definition A graph G = (V, E) is ɛ-far from a property P if we must add or remove at least ɛ 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, / 20

10 A Characterization of Locally Testable Graph Properties Szemerédi s regularity lemma: Every graph can be partitioned into a constant number of parts so that each pair of parts looks random. η 12 η 13 η 14 Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties June 1, / 20

11 A Characterization of Locally Testable Graph Properties Szemerédi s regularity lemma: Every graph can be partitioned into a constant number of parts so that each pair of parts looks random. η 12 η 13 η 14 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, / 20

12 A Characterization of Locally Testable Graph Properties Szemerédi s regularity lemma: Every graph can be partitioned into a constant number of parts so that each pair of parts looks random. η 12 η 13 η 14 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, / 20

13 Constant Sketch for Functions Theorem (Decomposition Theorem [BFH + 13]) For any γ > 0, d 1, and r : N N, there exists C such that: any function f : F n 2 {0, 1} can be decomposed as f = f + f, where Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties June 1, / 20

14 Constant Sketch for Functions Theorem (Decomposition Theorem [BFH + 13]) For any γ > 0, d 1, and r : N N, there exists C such that: any function f : F n 2 {0, 1} can be decomposed as f = f + f, 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, / 20

15 Constant Sketch for Functions Theorem (Decomposition Theorem [BFH + 13]) For any γ > 0, d 1, and r : N N, there exists C such that: any function f : F n 2 {0, 1} can be decomposed as f = f + f, 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, / 20

16 Factors Polynomial sequence (P 1,..., P C ) partitions F n 2 into atoms {x P 1 (x) = b 1,..., P C (x) = b C }. F n 2 = Almost the same size The decomposition theorem says: f = + (P 1,...,P C ) Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties June 1, / 20

17 What is the Gowers Norm? Definition Let f : F n 2 C. The d-th Gowers norm of f is ( f U d := E x,y 1,...,y d I {1,...,d} J I f (x + i I ) 1/2 d y i ), where J denotes complex conjugation. Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties June 1, / 20

18 What is the Gowers Norm? Definition Let f : F n 2 C. The d-th Gowers norm of f is ( f U d := E x,y 1,...,y d I {1,...,d} J I f (x + i I ) 1/2 d y i ), where J denotes complex conjugation. For any linear function L : F n 2 F 2, ( 1) L U 2 = E ( 1) L(x)+L(x+y 1)+L(x+y 2 )+L(x+y 1 +y 2 ) = 1. x,y 1,y 2 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, / 20

19 Gowers Norm Measures Correlation with Non-Classical Polynomials Definition P : F n 2 T is a non-classical polynomial of degree < d if e 2πi P U d = 1. The range of P is {0, 1 2 k+1,..., 2k k+1 } for some k (= depth). Lemma f : F n 2 C with f 1. f U d ɛ f, e 2πi P ɛ for any non-classical polynomial P of degree < d. (Direct Theorem) f U d ɛ f, e 2πi P δ(ɛ) for some non-classical polynomial P of degree < d. (Inverse Theorem) Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties June 1, / 20

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, / 20

21 Regularity-Instance Formalize f has some specific structured part. Definition A regularity-instance I is a tuple of an error parameter γ > 0, a structure function Γ : T C [0, 1], a complexity parameter C N, a degree-bound parameter d N, a degree parameter d = (d 1,..., d C ) N C with d i < d, a depth parameter h = (h 1,..., h C ) N C with h i < d i p 1, and a rank parameter r N. Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties June 1, / 20

22 Satisfying a Regularity-Instance Definition Let I = (γ, Γ, C, d, d, h, r) be a regularity-instance. f satisfies I if it is of the form where f (x) = Γ(P 1 (x),..., P C (x)) + Υ(x), P i is a polynomial of degree d i and depth h i, (P 1,..., P C ) has rank at least r, Υ U d γ. Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties June 1, / 20

23 Testing the Property of Satisfying a Regularity-Instance Theorem The property of satisfying a regularity-instance is locally testable (if the rank parameter is sufficiently large). Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties June 1, / 20

24 Testing the Property of Satisfying a Regularity-Instance Theorem The property of satisfying a regularity-instance is locally testable (if the rank parameter is sufficiently large). Algorithm: 1: Set δ small enough and m large enough. 2: Take a random affine embedding A : F m 2 F n 2. 3: if f A is δ-close to satisfying I then accept. 4: else reject. Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties June 1, / 20

25 Regular-Reducibility A property P is regular-reducible if for any δ > 0, there exists a set R of constant number of high-rank regularity-instances with constant parameters such that: f 2 P apple g : -far from P Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties June 1, / 20

26 Our Characterization Theorem An affine-invariant property P is locally testable P is regular-reducible. Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties June 1, / 20

27 Proof Intuition Regular-reducible Locally testable Combining the testability of regularity-instances and [HL13], we can estimate the distance to R. Locally testable Regular-reducible The behavior of a tester depends only on the distribution of the restriction to a random affine subspace. Since Γ, d, and h determines the distribution, we can find R using the tester. Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties June 1, / 20

28 Conclusions Obtained a characterization of locally testable affine-invariant properties. Easily extendable to F p for a prime p. Query complexity is actually unknown due to the Gowers inverse theorem. Other parts involve Ackermann-like functions. Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties June 1, / 20

29 Open Problems Characterization based on function (ultra)limits? Characterization of linear-invariant properties? Study other groups? Abelian Higher order Fourier analysis developed by [Sze12]. Non-Abelian Representation theory. Why is affine invariance easier to deal with than permutation invariance? Yuichi Yoshida (NII and PFI) Characterizing Locally Testable Properties June 1, / 20