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

FSTTCS ‘15 Workshop

Arnab Bhattacharyya Indian Institute of Science December 19, 2015

Higher-order Fourier analysis and an application

• Preliminaries and review of Fourier analysis
• What is “higher-order” Fourier analysis?
• An application to locally correctable codes

Roadmap

No historical account!

Some Preliminaries

𝔾 = finite field of fixed prime order

• For example, 𝔾 = 𝔾2 or 𝔾 = 𝔾97
• Theory can be extended to extensions of prime

fields [B.-Bhowmick ‘15]

Setting

Functions are always multivariate,

• n 𝑜 variables

𝑔: 𝔾𝑜 → ℂ ( 𝑔 ≤ 1) and 𝑄: 𝔾𝑜 → 𝔾

Functions

Current bounds aim to be efficient wrt n

Polynomial of degree 𝒆 is of the form: 𝑑𝑗1,…,𝑗𝑜𝑦1

𝑗1 ⋯ 𝑦𝑜 𝑗𝑜 𝑗1,…,𝑗𝑜

where each 𝑑𝑗1,…,𝑗𝑜 ∈ 𝔾 and 𝑗1 + ⋯ + 𝑗𝑜 ≤ 𝑒

Polynomial

Phase Polynomial

Phase polynomial of degree 𝒆 is a function 𝑔: 𝔾𝑜 → ℂ of the form 𝑔 𝑦 = e(𝑄 𝑦 ) where:

• 1. 𝑄: 𝔾𝑜 → 𝔾 is a polynomial of degree 𝑒
• 2. e 𝑙 = 𝑓2𝜌𝑗𝜌/|𝔾|

The inner product of two functions 𝑔, 𝑕: 𝔾𝑜 → ℂ is: 〈𝑔, 𝑕〉 = 𝔽𝑦∈𝔾𝑜 𝑔 𝑦 ⋅ 𝑕 𝑦

Inner Product

Magnitude captures correlation between 𝑔 and 𝑕

Additive derivative in direction ℎ ∈ 𝔾𝑜 of function 𝑄: 𝔾𝑜 → 𝔾 is: 𝐸ℎ𝑄 𝑦 = 𝑄 𝑦 + ℎ − 𝑄(𝑦)

Derivatives

Multiplicative derivative in direction ℎ ∈ 𝔾𝑜 of function 𝑔: 𝔾𝑜 → ℂ is: Δℎ𝑔 𝑦 = 𝑔 𝑦 + ℎ ⋅ 𝑔(𝑦)

Derivatives

Factor of degree 𝒆 and order m is a tuple of polynomials ℬ = (𝑄

1, 𝑄2, … , 𝑄 𝑛), each of degree 𝑒.

As shorthand, write: ℬ 𝑦 = (𝑄

1 𝑦 , … , 𝑄 𝑛 𝑦 )

Polynomial Factor

Fourier Analysis over 𝔾

Every function 𝑔: 𝔾𝑜 → ℂ is a linear combination of linear phases: 𝑔(𝑦) = 𝑔 ̂ 𝛽

𝛽∈𝔾𝑜

e 𝛽𝑗𝑦𝑗

𝑗

Fourier Representation

• The inner product of two linear phases is:

e 𝛽𝑗𝑦𝑗

𝑗

, 𝑓 𝛾𝑗𝑦𝑗

𝑗

= 𝔽𝑦 e 𝛽𝑗 − 𝛾𝑗 𝑦𝑗

𝑗

= 0

if 𝛽 ≠ 𝛾 and is 1 otherwise.

• So:

𝑔 ̂ 𝛽 = 𝑔, e ∑ 𝛽𝑗𝑦𝑗

𝑗

= correlation with linear phase

Linear Phases

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

Random functions

𝑔 𝑦 = 𝑕 𝑦 + ℎ 𝑦 where: 𝑕 𝑦 =

• 𝑔

̂ 𝛽 ⋅ e 𝛽𝑗𝑦𝑗

𝑗 𝛽: 𝑔 ̂ 𝛽 ≥𝜗

ℎ 𝑦 =

• 𝑔

̂ 𝛽 ⋅ e 𝛽𝑗𝑦𝑗

𝑗 𝛽: 𝑔 ̂ 𝛽 <𝜗

Decomposition Theorem

𝑕 𝑦 =

• 𝑔

̂ 𝛽 ⋅ e 𝛽𝑗𝑦𝑗

𝑗 𝛽: 𝑔 ̂ 𝛽 ≥𝜗

ℎ 𝑦 =

• 𝑔

̂ 𝛽 ⋅ e 𝛽𝑗𝑦𝑗

𝑗 𝛽: 𝑔 ̂ 𝛽 <𝜗

Decomposition Theorem

Every Fourier coefficient of ℎ is less than 𝜗, so ℎ is “pseudorandom”.

𝑕 𝑦 =

• 𝑔

̂ 𝛽 ⋅ e 𝛽𝑗𝑦𝑗

𝑗 𝛽: 𝑔 ̂ 𝛽 ≥𝜗

ℎ 𝑦 =

• 𝑔

̂ 𝛽 ⋅ e 𝛽𝑗𝑦𝑗

𝑗 𝛽: 𝑔 ̂ 𝛽 <𝜗

Decomposition Theorem

𝑕 has only 1/𝜗2 nonzero Fourier coefficients

𝑕 𝑦 =

• 𝑔

̂ 𝛽 ⋅ e 𝛽𝑗𝑦𝑗

𝑗 𝛽: 𝑔 ̂ 𝛽 ≥𝜗

ℎ 𝑦 =

• 𝑔

̂ 𝛽 ⋅ e 𝛽𝑗𝑦𝑗

𝑗 𝛽: 𝑔 ̂ 𝛽 <𝜗

Decomposition Theorem

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

For 𝑔: 𝔾𝑜 → ℂ, bias 𝑔 = |𝔽𝑦 𝑔 𝑦 | For 𝑄: 𝔾𝑜 → 𝔾, bias 𝑄 = |𝔽𝑦 e 𝑄 𝑦 |

Bias

[…, Naor-Naor ‘89, …]

𝜕0 𝜕1 𝜕2 𝜕3 𝜕 𝔾 −1

How well is 𝑸 equidistributed?

A factor ℬ = (𝑄

1, … , 𝑄𝜌) is 𝜷-

unbiased if every nonzero linear combination of 𝑄

1, … , 𝑄𝜌 has bias less

than 𝛽: bias ∑ 𝑑𝑗𝑄𝑗

𝜌 𝑗=1

< 𝛽 ∀ 𝑑1, … , 𝑑𝜌 ∈ 𝔾𝜌 ∖ {0}

Bias of Factor

Lemma: If ℬ is 𝛽-unbiased and of

• rder 𝑙, then for any 𝑑 ∈ 𝔾𝜌:

Pr ℬ 𝑦 = 𝑑 = 1 𝔾 𝜌 ± 𝛽

Bias implies equidistribution

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

Bias implies equidistribution

Corollary: If ℬ is 𝛽-unbiased and 𝛽 <

1 𝔾 𝑙, then ℬ maps onto 𝔾𝜌.

Gowers Norm

Given 𝑔: 𝔾𝑜 → ℂ, its Gowers norm of

• rder 𝒆 is:

𝑉𝑒 𝑔 = |𝔽𝑦,ℎ1,ℎ2,…,ℎ𝑒Δℎ1Δℎ2 ⋯ Δℎ𝑒𝑔 𝑦 |1/2𝑒

Gowers Norm

[Gowers ‘01]

Given 𝑔: 𝔾𝑜 → ℂ, its Gowers norm of order 𝒆 is: 𝑉𝑒 𝑔 = |𝔽𝑦,ℎ1,ℎ2,…,ℎ𝑒Δℎ1Δℎ2 ⋯ Δℎ𝑒𝑔 𝑦 |1/2𝑒

Gowers Norm

Observation: If 𝑔 = e(𝑄) is a phase poly, then:

𝑉𝑒 𝑔 = |𝔽𝑦,ℎ1,ℎ2,…,ℎ𝑒e 𝐸ℎ1𝐸ℎ2 ⋯ 𝐸ℎ𝑒𝑄 𝑦 |1/2𝑒

• If 𝑔 is a phase poly of degree 𝑒, then:

𝑉𝑒+1 𝑔 = 1

• Converse is true when 𝑒 < |𝔾|.

Gowers norm for phase polys

• 𝑉1 𝑔 =

𝔽 𝑔

2 = bias 𝑔

• 𝑉2 𝑔 =

∑ 𝑔 ̂4(𝛽)

𝛽

4

• 𝑉1 𝑔 ≤ 𝑉2 𝑔 ≤ 𝑉3 𝑔 ≤ ⋯

(C.-S.)

Other Observations

• For random 𝑔: 𝔾𝑜 → ℂ and fixed 𝑒,

𝑉𝑒 𝑔 → 0

• By monotonicity, low Gowers norm

implies low bias and low Fourier coefficients.

Pseudorandomness

Lemma: 𝑉𝑒+1 𝑔 ≥ max | 𝑔, e 𝑄 | where max is over all polynomials 𝑄 of degree 𝑒.

Correlation with Polynomials

Proof: For any poly 𝑄 of degree 𝑒: 𝔽 𝑔 𝑦 ⋅ e −𝑄 𝑦 = 𝑉1 𝑔 ⋅ e −𝑄 ≤ 𝑉𝑒+1 𝑔 ⋅ e −𝑄 = 𝑉𝑒+1(𝑔)

Theorem: If 𝑒 < |𝔾|, for all 𝜗 > 0, there exists 𝜀 = 𝜀(𝜗, 𝑒, 𝔾) such that if 𝑉𝑒+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.

Gowers Inverse Theorem

Consider 𝑔: 𝔾2

1 → ℂ with:

𝑔 0 = 1 𝑔 1 = 𝑗 𝑔 not a phase poly but 𝑉3 𝑔 = 1!

Small Fields

Consider 𝑔 = e(𝑄) where 𝑄: 𝔾2

𝑜 → 𝔾2

is symmetric polynomial of degree 4. 𝑉4 𝑔 = Ω(1) but: 𝑔, e 𝐷 = exp −𝑜 for all cubic poly 𝐷.

[Lovett-Meshulam-Samorodnitsky ’08, Green-Tao ‘09]

Small fields: worse news

Just define non-classical phase polynomials of degree 𝒆 to be functions 𝑔: 𝔾𝑜 → ℂ such that 𝑔 = 1 and Δℎ1Δℎ2 ⋯ Δℎ𝑒+1𝑔 𝑦 = 1 for all 𝑦, ℎ1, … , ℎ𝑒+1 ∈ 𝔾𝑜

Nevertheless…

Inverse Theorem for small fields

Theorem: For all 𝜗 > 0, there exists 𝜀 = 𝜀(𝜗, 𝑒, 𝔾) such that if 𝑉𝑒+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.

Theorem: If 𝑀1, … , 𝑀𝑛 are 𝑛 linear forms (𝑀𝑘 𝑌1, … , 𝑌𝜌 = ∑ ℓ𝑗,𝑘𝑌𝑗

𝜌 𝑗=1

), then: 𝔽𝑌1,…,𝑌𝑙∈𝔾𝑜 𝑔(𝑀𝑘 𝑌1, … , 𝑌𝜌

𝑛 𝑘=1

≤ 𝑉𝑢(𝑔) if 𝑔: 𝔾𝑜 → ℂ and 𝑢 is the complexity of the linear forms 𝑀1, … , 𝑀𝑛.

Pseudorandomness & Counting

[Gowers-Wolf ‘10]

• Given 𝑔: 𝔾𝑜 → ℝ and we want to “count” the

number of 3-term AP’s: 𝔽𝑌,𝑍 𝑔 𝑌 ⋅ 𝑔 𝑌 + 𝑍 ⋅ 𝑔 𝑌 + 2𝑍 ≤ 𝑔 ̂3(𝛽)

𝛽

• Similarly, number of 4-term AP’s controlled by 3rd
• rder Gowers norm of 𝑔.

Examples

Rank

Given a polynomial 𝑄: 𝔾𝑜 → 𝔾 of degree 𝑒, its rank is the smallest integer 𝑠 such that: 𝑄 𝑦 = Γ 𝑅1 𝑦 , … , 𝑅𝑠 𝑦 ∀𝑦 ∈ 𝔾𝑜 where 𝑅1, … , 𝑅𝑠 are polys of degree 𝑒 − 1 and Γ: 𝔾𝑠 → 𝔾 is arbitrary.

Rank

• For random poly 𝑄 of fixed degree 𝑒,

rank 𝑄 = 𝜕(1)

• High rank is pseudorandom behavior

Pseudorandomness

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

Rank & Gowers Norm

Lemma: If 𝑄(𝑦) = Γ 𝑅1(𝑦), … , 𝑅𝜌(𝑦) where 𝑅1, … , 𝑅𝜌 are polys of deg 𝑒 − 1, then 𝑉𝑒 e 𝑄 ≥

1 𝔾 𝑙/2.

Low rank implies large Gowers norm

Lemma: If 𝑄(𝑦) = Γ 𝑅1(𝑦), … , 𝑅𝜌(𝑦) where 𝑅1, … , 𝑅𝜌 are polys of deg 𝑒 − 1, then 𝑉𝑒 e 𝑄 ≥

1 𝔾 𝑙/2.

Low rank implies large Gowers norm

Proof: By (linear) Fourier analysis: e 𝑄 𝑦 = Γ 𝛽 ⋅ e 𝛽𝑗 ⋅ 𝑅𝑗 𝑦

𝑗 𝛽

Therefore: |𝔽𝑦 Γ 𝛽 ⋅ e 𝛽𝑗 ⋅ 𝑅𝑗 𝑦 − 𝑄 𝑦

𝑗

| = 1

𝛽

Then, there’s an 𝛽 such that e 𝑄 , e ∑ 𝛽𝑗𝑅𝑗

𝑗

≥ 𝔾 −𝜌/2.

Inverse theorem for polys

Theorem: For all 𝜗 and 𝑒, there exists 𝑆 = 𝑆 𝜗, 𝑒, 𝔾 such that if 𝑄 is a poly

• f degree 𝑒 and 𝑉𝑒 e 𝑄

> 𝜗, then rank(𝑄)< 𝑆.

[Tao-Ziegler ‘11]

Bias-rank theorem

Theorem: For all 𝜗 and 𝑒, there exists 𝑆 = 𝑆 𝜗, 𝑒, 𝔾 such that if 𝑄 is a poly

• f degree 𝑒 and bias(𝑄) > 𝜗, then

rank(𝑄)< 𝑆.

[Green-Tao ‘09, Kaufman-Lovett ‘08]

Decomposition Theorem

For any 𝜗 > 0 and integer 𝑠 > 1, there is a 𝑙 so that any bounded 𝑔: 𝔾𝑜 → ℂ has a decomposition: 𝑔 = 𝑕 + ℎ where 𝑕 = Γ(𝑄

1, … , 𝑄𝜌) for degree < 𝑠

non-classical polynomials 𝑄

1, … , 𝑄𝜌 and

𝑉𝑠 ℎ < 𝜗.

An Application: Locally Correctable Codes

[B.-Gopi ‘15]

Tackling Adversarial Errors

Message Encoding Corrupted Encoding

≤ 𝜺 fraction

Locally Decodable Codes

Message Encoding

Corrupted Encoding

≤ 𝜺 fraction

𝒋

𝒓, 𝜺 -Locally Decodable Codes

Message Encoding

Corrupted Encoding

≤ 𝜺 fraction

𝒋 𝒓

Locally Correctable Codes

Message Encoding

Corrupted Encoding

≤ 𝜺 fraction

Correction Local

𝒓, 𝜺 -Locally Correctable Code

Message Encoding

Corrupted Encoding

≤ 𝜺 fraction

Correction Local 𝒓

Locally Correctable Code (LCC)

Any encoding 𝑧 ∈ Σ𝑜 by a (𝒓, 𝜺)-LCC has the property that for every 𝜀- corruption 𝑧𝑧 of 𝑧 and for every 𝑗 ∈ [𝑜], with probability at least 90%,

• ne can recover 𝑧[𝑗] by looking at 𝑟

symbols in 𝑧𝑧.

LDC/LCC Applications

• Private Information Retrieval (PIR) schemes
• Secure Multiparty Computation
• Complexity theoretic applications:

– Arithmetic circuit lower bounds, Average-case complexity, Derandomization

• Hadamard code 𝐼 ⊆ 0,1 2𝑜
• Interpret 𝑜-bit message 𝑏1, … , 𝑏𝑜 as

linear form 𝐼 𝑦 = ∑ 𝑏𝑗𝑦𝑗

𝑗

and write evaluations of 𝐼 on all 0,1 𝑜

• To recover 𝐼(𝑦), choose random 𝑧 and
• utput 𝐼 𝑦 + 𝑧 − 𝐼(𝑧)

LCC example

• If 𝑟 is a constant, current shortest

LCC is Reed-Muller code of order 𝑟 − 1 (evaluation table of a polynomial of degree 𝑟 − 1 on a field of size > 𝑟)

– To recover 𝑄(𝑦), pass line ℓ in random direction thru 𝑦, evaluate on 𝑟 points on line to interpolate 𝑄ℓ and evaluate 𝑄ℓ(𝑦)

Current Status: Construction

• Same length also achieved by the

“lifted codes” of [Guo-Kopparty- Sudan ‘13].

Current Status: Construction

• Hadamard code known to be
• ptimal for 2 queries (for constant

alphabet)

• For larger number of queries, only

very weak bounds known

Current Status: Lower bounds

• Reed-Muller (and [GKS ‘13]) optimal

𝑟-query LCC among affine-invariant codes

Our Result

• For a codeword 𝑥 ∈ Σ𝔾𝑜, we can

view it as a function 𝑥: 𝔾𝑜 → Σ

• Code 𝐷 is affine-invariant if for any

𝑥 ∈ 𝐷, 𝑥 ∘ 𝐵 ∈ 𝐷 for any affine transformation 𝐵: 𝔾𝑜 → 𝔾𝑜.

Affine invariance

• “Generic” way to introduce many constraints

among codeword positions.

• Affine-invariance natural for algebraically

defined error-correcting codes

• Study of connection between correctability

and invariances formally initiated by [Kaufman-Sudan ‘08].

Why affine invariance?

• [Ben Sasson-Sudan ‘11] showed

that Reed-Muller is optimal among all linear, affine-invariant codes.

–Their result does not assume fixed field size as ours does

Previous work

The metric induced by the || ⋅ ||𝑉𝑟-norm on the space

• f all bounded functions has an 𝜗-net of size

exp (𝑃 𝑜𝑟−1 ).

Key Lemma

• Net consists of all functions of the form Γ(𝑄

1, … , 𝑄𝜌)

where 𝑄

1, … , 𝑄𝜌 are degree < 𝑟, non-classical

polynomials, 𝑙 is a constant, and Γ arbitrary.

• By decomposition theorem, such a function

approximates given 𝑔!

• Can discretize Γ without affecting error too much.

Proof of Key Lemma

• Take two codewords 𝑔 and 𝑕.
• If decoder runs on 𝑔 ∘ 𝐵 for random position 𝑧 and any

affine map 𝐵, it must with good prob give different answer than 𝑕(𝐵 𝑧 ).

• On the other hand, if 𝑔 and 𝑕 close in 𝑉𝑟 norm, then for

any 𝑧 and queried positions 𝑧1, … , 𝑧𝑟, 𝔽[ 𝑔 ∘ 𝐵 𝑧 − 𝑕 ∘ 𝐵 𝑧 , 𝒠 𝑔 ∘ 𝐵 𝑧1 , … , 𝑔 ∘ 𝐵 𝑧𝑟 is small over random 𝐵.

• Contradiction, so 𝑔 and 𝑕 lie in different cells of 𝜗-net for

𝑉𝑟-norm.

Proof of Theorem

Uses counting lemma

• List-decoding radius for Reed-Muller codes

[Bhowmick-Lovett ‘14, ‘15]

• New algorithms for factoring and decomposing

polynomials [B. ‘14]

• New testers for algebraic properties [B.-Fischer-

Hatami-Hatami-Lovett ‘13]

• …?

More applications

Thanks!