Higher-order Fourier Analysis over Finite Fields, and Applications. Pooya Hatami

Transcription

1 Higher-order Fourier Analysis over Finite Fields, and Applications Pooya Hatami

2 Coding Theory: Task: Reliably transmit a message through an unreliable channel. m F k 2 c m F N 2

3 Coding Theory: Task: Reliably transmit a message through an unreliable channel. m F k 2 c m c m δ F N 2

4 Coding Theory: Task: Reliably transmit a message through an unreliable channel. m F k 2 c m c m δ F N 2 Good code: (i) Large minimum distance δ while having large rate k N.

5 Coding Theory: Task: Reliably transmit a message through an unreliable channel. m F k 2 c m c m δ F N 2 Good code: (i) Large minimum distance δ while having large rate k N. (ii) Efficiently testable and decodable.

6 Hadamard Codes: 1 2 m c m m F k 2 c m F 2k 2 c m : evaluation vector of m 1 x 1 + m 2 x m k x k F 2 [x 1,..., x k ]

7 Reed Muller codes: Degree d polynomials in F 2 [x 1,..., x k ]

8 Reed Muller codes: δ Problem 1. How many degree d polynomials in F 2 [x 1,..., x n ] are there in B δ (f )?

9 Polynomial Decompositions: Degree 4 polynomial P F 2 [x 1,..., x n ].

10 Polynomial Decompositions: Degree 4 polynomial P F 2 [x 1,..., x n ]. Is P(x) = Q 1 (x)q 2 (x) + Q 3 (x)q 4 (x), for some degree 3 polynomials Q 1,..., Q 4?

11 Polynomial Decompositions: Degree 4 polynomial P F 2 [x 1,..., x n ]. Is P(x) = Q 1 (x)q 2 (x) + Q 3 (x)q 4 (x), for some degree 3 polynomials Q 1,..., Q 4? Problem 2. Given a degree d polynomial P and a prescribed decomposition. Find such a decomposition of P or say it is not possible.

12 Polynomial Decompositions: Degree 4 polynomial P F 2 [x 1,..., x n ]. Is P(x) = Q 1 (x)q 2 (x) + Q 3 (x)q 4 (x), for some degree 3 polynomials Q 1,..., Q 4? Problem 2. Given a degree d polynomial P and a prescribed decomposition, Efficiently find such a decomposition of P or say it is not possible.

13 Algebraic Property Testing: Is f : F n 2 F 2 a degree d polynomial? f P d (F n 2) δ d (f)

14 Algebraic Property Testing: Is f : F n 2 F 2 a degree d polynomial? f P d (F n 2) δ d (f) [AKKLR 05] Query f only on constant number of inputs. 1. Always accept if deg(f ) d. 2. Reject w.h.p. if δ d (f ) > ɛ.

15 Algebraic Property Testing: Is f : F n 2 F 2 a degree d polynomial? f P d (F n 2) δ d (f) [AKKLR 05] Query f only on constant number of inputs. 1. Always accept if deg(f ) d. 2. Reject w.h.p. if δ d (f ) > ɛ. Problem 3. Which algebraic properties are testable?

16 Higher-order Fourier analysis over finite fields, which is an extension of Fourier analysis. [Bergelson, Green, Kaufman, Gowers, Lovett, Meshulam, Samorodnitsky, Tao, Viola, Wolf... ]

17 Fourier Analysis over F n p Study f : F n p R by looking at how it correlates with linear phases. f(x)= σ F n p f (σ) ω σi x i

18 Fourier Analysis over F n p Study f : F n p R by looking at how it correlates with linear phases. f(x)= σ: f (σ) ɛ f (σ) ω σ i x i + f psd

19 Higher-order Fourier Analysis over F n p Study f : F n p R by looking at how it correlates with higher degree polynomial phases, ω P(x 1,...,x n). Not orthogonal, no unique expansion. f = C i=1 λ iω P i (x) + f psd?

20 Higher-order Fourier Analysis over F n p Study f : F n p R by looking at how it correlates with higher degree polynomial phases, ω P(x 1,...,x n). Not orthogonal, no unique expansion. f = C i=1 λ iω P i (x) + f psd? [Bergelson, Green, Tao, Ziegler] establish such decomposition theorems via inverse theorems for certain norms called Gowers norms.

21 Higher-order Fourier Analysis over F n p Study f : F n p R by looking at how it correlates with higher degree polynomial phases, ω P(x 1,...,x n). Not orthogonal, no unique expansion. f = C i=1 λ iω P i (x) + f psd? Theorem[Trevisan-Tulsiani-Vadhan, Gowers] For any collection G of functions g : X D the following holds. Every function f can be written as f = F(g 1,..., g C ) + f psd

22 Higher-order Fourier Analysis over F n p Study f : F n p R by looking at how it correlates with higher degree polynomial phases, ω P(x 1,...,x n). Not orthogonal, no unique expansion. f = C i=1 λ iω P i (x) + f psd? Theorem[Trevisan-Tulsiani-Vadhan, Gowers] For any collection G of functions g : F n p D the following holds. Every function f can be written as f = σ F n p F(σ)ω P i σ i g i + f psd

23 Higher-order Fourier Analysis over F n p Study f : F n p R by looking at how it correlates with higher degree polynomial phases, ω P(x 1,...,x n). f = C i=1 λ iω P i (x) + f psd

24 Higher-order Fourier Analysis over F n p Study f : F n p R by looking at how it correlates with higher degree polynomial phases, ω P(x 1,...,x n). f = C i=1 λ iω P i (x) + f psd Need to understand the joint distribution of a collection of degree d polynomials.

25 Higher-order Fourier Analysis over F n p Study f : F n p R by looking at how it correlates with higher degree polynomial phases, ω P(x 1,...,x n). f = C i=1 λ iω P i (x) + f psd Problem: P 1,..., P C P d (F n p). X, Y F n p uniform. Characterize the distribution of P 1 (X)... P 10 (X) P 1 (X + Y )... P 10 (X + Y ). P 1 (X + 4Y )... P 10 (X + 4Y )

26 Higher-order Fourier Analysis over F n p Study f : F n p R by looking at how it correlates with higher degree polynomial phases, ω P(x 1,...,x n). f = C i=1 λ iω P i (x) + f psd Problem: P 1,..., P C P d (F n p). X, Y F n p uniform. Characterize the distribution of P 1 (X)... P 10 (X) P 1 (X + Y )... P 10 (X + Y ). P 1 (X + 4Y )... P 10 (X + 4Y ) [HHL 15 (general case), BFHHL 13 (affine linear forms)] [KL 08 and GT 09]: Distribution of (P 1 (X),..., P C (X)).

27 Problem 1. [KLP 10, BL 15] Number of degree d polynomials in F p [x 1,..., x n ] in hamming ball of radius δ e ɛ is 2 O(nd e).

28 Problem 1. [KLP 10, BL 15] Number of degree d polynomials in F p [x 1,..., x n ] in hamming ball of radius δ e ɛ is 2 O(nd e). Problem 2. [BHL 15] Polynomial time algorithm for finding prescribed polynomial decompositions.

29 Problem 1. [KLP 10, BL 15] Number of degree d polynomials in F p [x 1,..., x n ] in hamming ball of radius δ e ɛ is 2 O(nd e). Problem 2. [BHL 15] Polynomial time algorithm for finding prescribed polynomial decompositions. Problem 3. [BFL 12, BFHHL 13] Characterization of testable algebraic (i.e. affine invariant) properties.

30 Problem 1. [KLP 10, BL 15] Number of degree d polynomials in F p [x 1,..., x n ] in hamming ball of radius δ e ɛ is 2 O(nd e). Problem 2. [BHL 15] Polynomial time algorithm for finding prescribed polynomial decompositions. Problem 3. [BFL 12, BFHHL 13] Characterization of testable algebraic (i.e. affine invariant) properties. Problem 4. Is there a constant query tester that given f : F n 2 F 2 distinguishes between the following? f is ɛ-correlated to some cubic, or f is δ(ɛ)-correlated to all cubics, where 0 < δ(ɛ) ɛ.

31 Problem 1. [KLP 10, BL 15] Number of degree d polynomials in F p [x 1,..., x n ] in hamming ball of radius δ e ɛ is 2 O(nd e). Problem 2. [BHL 15] Polynomial time algorithm for finding prescribed polynomial decompositions. Problem 3. [BFL 12, BFHHL 13] Characterization of testable algebraic (i.e. affine invariant) properties. Open Problem. Is there a constant query tester that given f : F n 2 F 2 distinguishes between the following? f is ɛ-correlated to some cubic, or f is δ(ɛ)-correlated to all cubics, where 0 < δ(ɛ) ɛ.

32 Theorem. [BHT] There is a poly(n)-time deterministic algorithm that given a polynomial P, and Γ : F l p F p, and d 1,..., d l 1, either outputs P 1,..., P r of degrees d 1,..., d l, s.t. P = Γ(P 1,..., P d ), or correctly outputs NOT POSSIBLE.

33 Proof illustration: Find P 1, P 2 of degree d 1 such that P = P 1 P 2

34 Proof illustration: Find P 1, P 2 of degree d 1 such that P = P 1 P 2 Algorithmic Regularity Lemma for Polynomials [BHT]: P = Λ(Q 1,..., Q r )

35 Proof illustration: Find P 1, P 2 of degree d 1 such that P = P 1 P 2 Algorithmic Regularity Lemma for Polynomials [BHT]: P = Λ(Q 1,..., Q r ) x j s.t. for all i, deg(q i ) = deg(q i xj =0). P xj =0 = Λ(Q 1 xj =0,..., Q r xj =0)

36 Proof illustration: Find P 1, P 2 of degree d 1 such that P = P 1 P 2 Algorithmic Regularity Lemma for Polynomials [BHT]: P = Λ(Q 1,..., Q r ) x j s.t. for all i, deg(q i ) = deg(q i xj =0). P xj =0 = Λ(Q 1 xj =0,..., Q r xj =0) Recurse on P xj =0. If NOT POSSIBLE, then output NOT POSSIBLE. Otherwise we find P 1, P 2 such that P x j =0 = P 1 P 2.

37 Proof illustration: Find P 1, P 2 of degree d 1 such that P = P 1 P 2 Algorithmic Regularity Lemma for Polynomials [BHT]: P = Λ(Q 1,..., Q r ) x j s.t. for all i, deg(q i ) = deg(q i xj =0). P xj =0 = Λ(Q 1 xj =0,..., Q r xj =0) Recurse on P xj =0. If NOT POSSIBLE, then output NOT POSSIBLE. Otherwise we find P 1, P 2 such that P x j =0 = P 1 P 2. Λ(Q 1,..., Q r) = P 1P 2

38 Proof illustration: Find P 1, P 2 of degree d 1 such that P = P 1 P 2 Algorithmic Regularity Lemma for Polynomials [BHT]: P = Λ(Q 1,..., Q r ) x j s.t. for all i, deg(q i ) = deg(q i xj =0). P xj =0 = Λ(Q 1 xj =0,..., Q r xj =0) Recurse on P xj =0. If NOT POSSIBLE, then output NOT POSSIBLE. Otherwise we find P 1, P 2 such that P x j =0 = P 1 P 2. Λ(Q 1,..., Q r) = G 1 (Q 1,..., Q r, R 1,..., R C ) G 2 (Q 1,..., Q r, R 1,..., R C )

39 Proof illustration: Find P 1, P 2 of degree d 1 such that P = P 1 P 2 Algorithmic Regularity Lemma for Polynomials [BHT]: P = Λ(Q 1,..., Q r ) x j s.t. for all i, deg(q i ) = deg(q i xj =0). P xj =0 = Λ(Q 1 xj =0,..., Q r xj =0) Recurse on P xj =0. If NOT POSSIBLE, then output NOT POSSIBLE. Otherwise we find P 1, P 2 such that P x j =0 = P 1 P 2. Λ(a 1,..., a r ) = G 1 (a 1,..., a r, 0,..., 0) G 2 (a 1,..., a r, 0,..., 0)

40 Proof illustration: Find P 1, P 2 of degree d 1 such that P = P 1 P 2 Algorithmic Regularity Lemma for Polynomials [BHT]: P = Λ(Q 1,..., Q r ) x j s.t. for all i, deg(q i ) = deg(q i xj =0). P xj =0 = Λ(Q 1 xj =0,..., Q r xj =0) Recurse on P xj =0. If NOT POSSIBLE, then output NOT POSSIBLE. Otherwise we find P 1, P 2 such that P x j =0 = P 1 P 2. P = Λ(Q 1,..., Q r ) = G 1 (Q 1,..., Q r, 0,..., 0) G 2 (Q 1,..., Q r, 0,..., 0)