Higher-order Fourier analysis and an application FSTTCS 15 Workshop - PowerPoint PPT Presentation

Higher-order Fourier analysis and an application FSTTCS ‘15 Workshop Arnab Bhattacharyya Indian Institute of Science December 19, 2015

Roadmap • Preliminaries and review of Fourier analysis • What is “higher-order” Fourier analysis? • An application to locally correctable codes No historical account!

Some Preliminaries

Setting 𝔾 = finite field of fixed prime order • For example, 𝔾 = 𝔾 2 or 𝔾 = 𝔾 97 • Theory can be extended to extensions of prime fields [B.-Bhowmick ‘15]

Functions Functions are always multivariate, on 𝑜 variables 𝑔 : 𝔾 𝑜 → ℂ ( 𝑔 ≤ 1) and 𝑄 : 𝔾 𝑜 → 𝔾 Current bounds aim to be efficient wrt n

Polynomial Polynomial of degree 𝒆 is of the form: 𝑗 1 ⋯ 𝑦 𝑜 𝑗 𝑜 � 𝑑 𝑗 1 ,…, 𝑗 𝑜 𝑦 1 𝑗 1 ,…, 𝑗 𝑜 where each 𝑑 𝑗 1 ,…, 𝑗 𝑜 ∈ 𝔾 and 𝑗 1 + ⋯ + 𝑗 𝑜 ≤ 𝑒

Phase Polynomial Phase polynomial of degree 𝒆 is a function 𝑔 : 𝔾 𝑜 → ℂ of the form 𝑔 𝑦 = e ( 𝑄 𝑦 ) where: 1. 𝑄 : 𝔾 𝑜 → 𝔾 is a polynomial of degree 𝑒 2. e 𝑙 = 𝑓 2𝜌𝑗𝜌 /| 𝔾 |

Inner Product The inner product of two functions 𝑔 , 𝑕 : 𝔾 𝑜 → ℂ is: 〈𝑔 , 𝑕〉 = 𝔽 𝑦∈𝔾 𝑜 𝑔 𝑦 ⋅ 𝑕 𝑦 Magnitude captures correlation between 𝑔 and 𝑕

Derivatives Additive derivative in direction ℎ ∈ 𝔾 𝑜 of function 𝑄 : 𝔾 𝑜 → 𝔾 is: 𝐸 ℎ 𝑄 𝑦 = 𝑄 𝑦 + ℎ − 𝑄 ( 𝑦 )

Derivatives Multiplicative derivative in direction ℎ ∈ 𝔾 𝑜 of function 𝑔 : 𝔾 𝑜 → ℂ is: Δ ℎ 𝑔 𝑦 = 𝑔 𝑦 + ℎ ⋅ 𝑔 ( 𝑦 )

Polynomial Factor Factor of degree 𝒆 and order m is a tuple of polynomials ℬ = ( 𝑄 1 , 𝑄 2 , … , 𝑄 𝑛 ) , each of degree 𝑒 . As shorthand, write: ℬ 𝑦 = ( 𝑄 1 𝑦 , … , 𝑄 𝑛 𝑦 )

Fourier Analysis over 𝔾

Fourier Representation Every function 𝑔 : 𝔾 𝑜 → ℂ is a linear combination of linear phases: ̂ 𝛽 𝑔 ( 𝑦 ) = � 𝑔 e � 𝛽 𝑗 𝑦 𝑗 𝛽∈𝔾 𝑜 𝑗

Linear Phases • The inner product of two linear phases is: e � 𝛽 𝑗 𝑦 𝑗 , 𝑓 � 𝛾 𝑗 𝑦 𝑗 = 𝔽 𝑦 e � 𝛽 𝑗 − 𝛾 𝑗 𝑦 𝑗 = 0 𝑗 𝑗 𝑗 if 𝛽 ≠ 𝛾 and is 1 otherwise. • So: ̂ 𝛽 = 𝑔 , e ∑ 𝛽 𝑗 𝑦 𝑗 𝑔 = correlation with linear phase 𝑗

Random functions With high probability, a random function 𝑔 : 𝔾 𝑜 → ℂ with | 𝑔 | = 1 has ̂ 𝛽 → 0 . each 𝑔

Decomposition Theorem 𝑔 𝑦 = 𝑕 𝑦 + ℎ 𝑦 where: ̂ 𝛽 ⋅ e � 𝛽 𝑗 𝑦 𝑗 𝑕 𝑦 = � 𝑔 ̂ 𝛽 ≥𝜗 𝑗 𝛽 : 𝑔 ̂ 𝛽 ⋅ e � 𝛽 𝑗 𝑦 𝑗 ℎ 𝑦 = � 𝑔 ̂ 𝛽 <𝜗 𝑗 𝛽 : 𝑔

Decomposition Theorem ̂ 𝛽 ⋅ e � 𝛽 𝑗 𝑦 𝑗 𝑕 𝑦 = � 𝑔 ̂ 𝛽 ≥𝜗 𝑗 𝛽 : 𝑔 ̂ 𝛽 ⋅ e � 𝛽 𝑗 𝑦 𝑗 ℎ 𝑦 = � 𝑔 ̂ 𝛽 <𝜗 𝑗 𝛽 : 𝑔 Every Fourier coefficient of ℎ is less than 𝜗 , so ℎ is “pseudorandom”.

Decomposition Theorem ̂ 𝛽 ⋅ e � 𝛽 𝑗 𝑦 𝑗 𝑕 𝑦 = � 𝑔 ̂ 𝛽 ≥𝜗 𝑗 𝛽 : 𝑔 ̂ 𝛽 ⋅ e � 𝛽 𝑗 𝑦 𝑗 ℎ 𝑦 = � 𝑔 ̂ 𝛽 <𝜗 𝑗 𝛽 : 𝑔 𝑕 has only 1/ 𝜗 2 nonzero Fourier coefficients

Decomposition Theorem ̂ 𝛽 ⋅ e � 𝛽 𝑗 𝑦 𝑗 𝑕 𝑦 = � 𝑔 ̂ 𝛽 ≥𝜗 𝑗 𝛽 : 𝑔 ̂ 𝛽 ⋅ e � 𝛽 𝑗 𝑦 𝑗 ℎ 𝑦 = � 𝑔 ̂ 𝛽 <𝜗 𝑗 𝛽 : 𝑔 The nonzero Fourier coefficients of 𝑕 can be found in poly time [Goldreich-Levin ‘89]

Elements of Higher-Order Fourier Analysis

Higher-order Fourier analysis is the interplay between three different notions of pseudorandomness for functions and factors. 1. Bias 2. Gowers norm 3. Rank

Bias

Bias For 𝑔 : 𝔾 𝑜 → ℂ , bias 𝑔 = | 𝔽 𝑦 𝑔 𝑦 | For 𝑄 : 𝔾 𝑜 → 𝔾 , bias 𝑄 = | 𝔽 𝑦 e 𝑄 𝑦 | […, Naor-Naor ‘89, …]

𝜕 3 𝜕 2 𝜕 1 𝜕 0 𝜕 𝔾 −1 How well is 𝑸 equidistributed?

Bias of Factor A factor ℬ = ( 𝑄 1 , … , 𝑄 𝜌 ) is 𝜷 - unbiased if every nonzero linear combination of 𝑄 1 , … , 𝑄 𝜌 has bias less than 𝛽 : 𝜌 bias ∑ 𝑑 𝑗 𝑄 𝑗 < 𝛽 𝑗=1 ∀ 𝑑 1 , … , 𝑑 𝜌 ∈ 𝔾 𝜌 ∖ {0}

Bias implies equidistribution Lemma : If ℬ is 𝛽 -unbiased and of order 𝑙 , then for any 𝑑 ∈ 𝔾 𝜌 : 1 Pr ℬ 𝑦 = 𝑑 = 𝔾 𝜌 ± 𝛽

Bias implies equidistribution Lemma : If ℬ is 𝛽 -unbiased and of order 𝑙 , then for any 𝑑 ∈ 𝔾 𝜌 : 1 Pr ℬ 𝑦 = 𝑑 = 𝔾 𝜌 ± 𝛽 Corollary : If ℬ is 𝛽 -unbiased and 1 𝔾 𝑙 , then ℬ maps onto 𝔾 𝜌 . 𝛽 <

Gowers Norm

Gowers Norm Given 𝑔 : 𝔾 𝑜 → ℂ , its Gowers norm of order 𝒆 is: 𝑉 𝑒 𝑔 = | 𝔽 𝑦 , ℎ 1 , ℎ 2 ,…, ℎ 𝑒 Δ ℎ 1 Δ ℎ 2 ⋯ Δ ℎ 𝑒 𝑔 𝑦 | 1 / 2 𝑒 [Gowers ‘01]

Gowers Norm Given 𝑔 : 𝔾 𝑜 → ℂ , its Gowers norm of order 𝒆 is: 𝑉 𝑒 𝑔 = | 𝔽 𝑦 , ℎ 1 , ℎ 2 ,…, ℎ 𝑒 Δ ℎ 1 Δ ℎ 2 ⋯ Δ ℎ 𝑒 𝑔 𝑦 | 1 / 2 𝑒 Observation : If 𝑔 = e ( 𝑄 ) is a phase poly, then: 𝑉 𝑒 𝑔 = | 𝔽 𝑦 , ℎ 1 , ℎ 2 ,…, ℎ 𝑒 e 𝐸 ℎ 1 𝐸 ℎ 2 ⋯ 𝐸 ℎ 𝑒 𝑄 𝑦 | 1 / 2 𝑒

Gowers norm for phase polys • If 𝑔 is a phase poly of degree 𝑒 , then: 𝑉 𝑒+1 𝑔 = 1 • Converse is true when 𝑒 < | 𝔾 | .

Other Observations • 𝑉 1 𝑔 = 2 = bias 𝑔 𝔽 𝑔 • 𝑉 2 𝑔 = 4 ̂ 4 ( 𝛽 ) ∑ 𝑔 𝛽 • 𝑉 1 𝑔 ≤ 𝑉 2 𝑔 ≤ 𝑉 3 𝑔 ≤ ⋯ (C.-S.)

Pseudorandomness • For random 𝑔 : 𝔾 𝑜 → ℂ and fixed 𝑒 , 𝑉 𝑒 𝑔 → 0 • By monotonicity, low Gowers norm implies low bias and low Fourier coefficients.

Correlation with Polynomials Lemma : 𝑉 𝑒+1 𝑔 ≥ max | 𝑔 , e 𝑄 | where max is over all polynomials 𝑄 of degree 𝑒 . Proof : For any poly 𝑄 of degree 𝑒 : = 𝑉 1 𝑔 ⋅ e −𝑄 𝔽 𝑔 𝑦 ⋅ e −𝑄 𝑦 ≤ 𝑉 𝑒+1 𝑔 ⋅ e −𝑄 = 𝑉 𝑒+1 ( 𝑔 )

Gowers Inverse Theorem Theorem : If 𝑒 < | 𝔾 | , for all 𝜗 > 0 , there exists 𝜀 = 𝜀 ( 𝜗 , 𝑒 , 𝔾 ) such that if 𝑉 𝑒+1 𝑔 > 𝜗 , then 𝑔 , e 𝑄 > 𝜀 for some poly 𝑄 of degree 𝑒 . Proof : • [Green-Tao ‘09] Combinatorial for phase poly 𝑔 (c.f. Madhur’s talk later). • [Bergelson-Tao-Ziegler ‘10] Ergodic theoretic proof for arbitrary 𝑔 .

Small Fields 1 → ℂ with: Consider 𝑔 : 𝔾 2 𝑔 0 = 1 𝑔 1 = 𝑗 𝑔 not a phase poly but 𝑉 3 𝑔 = 1 !

Small fields: worse news 𝑜 → 𝔾 2 Consider 𝑔 = e ( 𝑄 ) where 𝑄 : 𝔾 2 is symmetric polynomial of degree 4 . 𝑉 4 𝑔 = Ω (1) but: 𝑔 , e 𝐷 = exp −𝑜 for all cubic poly 𝐷 . [Lovett-Meshulam-Samorodnitsky ’08, Green-Tao ‘09]

Nevertheless… Just define non-classical phase polynomials of degree 𝒆 to be functions 𝑔 : 𝔾 𝑜 → ℂ such that 𝑔 = 1 and Δ ℎ 1 Δ ℎ 2 ⋯ Δ ℎ 𝑒+1 𝑔 𝑦 = 1 for all 𝑦 , ℎ 1 , … , ℎ 𝑒+1 ∈ 𝔾 𝑜

Inverse Theorem for small fields Theorem : For all 𝜗 > 0 , there exists 𝜀 = 𝜀 ( 𝜗 , 𝑒 , 𝔾 ) such that if 𝑉 𝑒+1 𝑔 > 𝜗 , then 𝑔 , 𝑕 > 𝜀 for some non- classical phase poly 𝑕 of degree 𝑒 . Proof : • [Tao-Ziegler] Combinatorial for phase poly 𝑔 . • [Tao-Ziegler] Nonstandard proof for arbitrary 𝑔 .

Pseudorandomness & Counting Theorem: If 𝑀 1 , … , 𝑀 𝑛 are 𝑛 linear forms 𝜌 ( 𝑀 𝑘 𝑌 1 , … , 𝑌 𝜌 = ∑ ℓ 𝑗 , 𝑘 𝑌 𝑗 ), then: 𝑗=1 𝑛 ≤ 𝑉 𝑢 ( 𝑔 ) 𝔽 𝑌 1 ,…, 𝑌 𝑙 ∈𝔾 𝑜 � 𝑔 ( 𝑀 𝑘 𝑌 1 , … , 𝑌 𝜌 𝑘=1 if 𝑔 : 𝔾 𝑜 → ℂ and 𝑢 is the complexity of the linear forms 𝑀 1 , … , 𝑀 𝑛 . [Gowers-Wolf ‘10]

Examples • Given 𝑔 : 𝔾 𝑜 → ℝ and we want to “count” the number of 3-term AP’s: ̂ 3 ( 𝛽 ) 𝔽 𝑌 , 𝑍 𝑔 𝑌 ⋅ 𝑔 𝑌 + 𝑍 ⋅ 𝑔 𝑌 + 2𝑍 ≤ � 𝑔 𝛽 • Similarly, number of 4-term AP’s controlled by 3rd order Gowers norm of 𝑔 .

Rank

Rank Given a polynomial 𝑄 : 𝔾 𝑜 → 𝔾 of degree 𝑒 , its rank is the smallest integer 𝑠 such that : ∀𝑦 ∈ 𝔾 𝑜 𝑄 𝑦 = Γ 𝑅 1 𝑦 , … , 𝑅 𝑠 𝑦 where 𝑅 1 , … , 𝑅 𝑠 are polys of degree 𝑒 − 1 and Γ : 𝔾 𝑠 → 𝔾 is arbitrary.

Pseudorandomness • For random poly 𝑄 of fixed degree 𝑒 , rank 𝑄 = 𝜕 (1) • High rank is pseudorandom behavior

Rank & Gowers Norm If 𝑄 : 𝔾 𝑜 → 𝔾 is a poly of degree 𝑒 , 𝑄 has high rank if and only if e ( 𝑄 ) has low Gowers norm of order 𝑒 !

Low rank implies large Gowers norm Lemma : If 𝑄 ( 𝑦 ) = Γ 𝑅 1 ( 𝑦 ), … , 𝑅 𝜌 ( 𝑦 ) where 𝑅 1 , … , 𝑅 𝜌 are polys of deg 𝑒 − 1 , then 1 𝑉 𝑒 e 𝑄 ≥ 𝔾 𝑙 / 2 .