Algorithmic Questions in Higher-Order Fourier Analysis Madhur - - PowerPoint PPT Presentation

Algorithmic Questions in Higher-Order Fourier Analysis

f

1 2 − ǫ 1 2 − η

Madhur Tulsiani

TTI Chicago Based on joint works with Arnab Bhattacharyya, Eli Ben-Sasson, Pooya Hatami, Noga Ron-Zewi, Luca Trevisan, Salil Vadhan and Julia Wolf

Decomposition Theorems

Decomposition Theorems

Object of study Family of algorithms or functions

Decomposition Theorems

Object of study Family of algorithms or functions

=

Structured

+

No apparent structure (Pseudorandom)

Decomposition Theorems

Object of study Family of algorithms or functions

=

Structured

+

No apparent structure (Pseudorandom)

• Decompose an object in to structured and pseudorandom parts.

Decomposition Theorems

Object of study Family of algorithms or functions

=

Structured

+

No apparent structure (Pseudorandom)

• Decompose an object in to structured and pseudorandom parts.
• Can often ignore the pseudorandom part for many applications.

Structured part easier to study.

Fourier analysis

• Space of functions g : Fn

2 → R.

Fourier analysis

• Space of functions g : Fn

2 → R.

• Functions

χα(x) = (−1)α·x = (−1)

• j αjxj

form an orthonormal basis under the inner product f , g = Ex [f (x)g(x)].

Fourier analysis

• Space of functions g : Fn

2 → R.

• Functions

χα(x) = (−1)α·x = (−1)

• j αjxj

form an orthonormal basis under the inner product f , g = Ex [f (x)g(x)].

• Any function g can be written as

g =

• α∈Fn

2

ˆ g(α)χα .

Fourier analysis

• Space of functions g : Fn

2 → R.

• Functions

χα(x) = (−1)α·x = (−1)

• j αjxj

form an orthonormal basis under the inner product f , g = Ex [f (x)g(x)].

• Any function g can be written as

g =

• α∈Fn

2

ˆ g(α)χα .

• [Parseval]: g2 = g, g = Ex
• (g(x))2

=

α (ˆ

g(α))2.

A basic decomposition in Fourier analysis

g : Fn

2 → [−1, 1]

A basic decomposition in Fourier analysis

g : Fn

2 → [−1, 1]

χα(x) = (−1)α·x = (−1)

• i αjxj

α ∈ Fn

2

g =

• S
• g(α)χα

A basic decomposition in Fourier analysis

g : Fn

2 → [−1, 1]

χα(x) = (−1)α·x = (−1)

• i αjxj

α ∈ Fn

2

g =

• S
• g(α)χα =
• |

g(α)|>ǫ

• g(α)χα +
• |

g(α)|≤ǫ

• g(α)χα =

k

• i=1

ciχαi + f

A basic decomposition in Fourier analysis

g : Fn

2 → [−1, 1]

χα(x) = (−1)α·x = (−1)

• i αjxj

α ∈ Fn

2

g =

• S
• g(α)χα =
• |

g(α)|>ǫ

• g(α)χα +
• |

g(α)|≤ǫ

• g(α)χα =

k

• i=1

ciχαi + f

• k ≤ 1/ǫ2.

simple structure

A basic decomposition in Fourier analysis

g : Fn

2 → [−1, 1]

χα(x) = (−1)α·x = (−1)

• i αjxj

α ∈ Fn

2

g =

• S
• g(α)χα =
• |

g(α)|>ǫ

• g(α)χα +
• |

g(α)|≤ǫ

• g(α)χα =

k

• i=1

ciχαi + f

• k ≤ 1/ǫ2.

simple structure

• f has small correlation with linear functions.

pseudorandom ∀α, |f , χα| = |Ex [f (x)χα(x)]| ≤ ǫ

Getting high: Quadratic Fourier Analysis

[Gowers 98, Green 07]

• “Fourier pseudorandomness” often insufficient for many applications

(e.g. counting 4-term APs in a set).

Getting high: Quadratic Fourier Analysis

[Gowers 98, Green 07]

• “Fourier pseudorandomness” often insufficient for many applications

(e.g. counting 4-term APs in a set).

• [Gowers 98]: Defined uniformity norms (Gowers norms). “Right”

notion of pseudorandomness for many applications. f 4

U2 = Ex,y,z [f (x) · f (x + y) · f (x + z) · f (x + y + z)] .

Think f = indicator of a set. f U2 counts 2-dimensional "boxes" in set.

Getting high: Quadratic Fourier Analysis

[Gowers 98, Green 07]

• “Fourier pseudorandomness” often insufficient for many applications

(e.g. counting 4-term APs in a set).

• [Gowers 98]: Defined uniformity norms (Gowers norms). “Right”

notion of pseudorandomness for many applications. f 4

U2 = Ex,y,z [f (x) · f (x + y) · f (x + z) · f (x + y + z)] .

Think f = indicator of a set. f U2 counts 2-dimensional "boxes" in set.

• Can define higher norms similarly

f 8

U3 = Ex,y,z,w

• f(x) f(x+y) f(x+z) f(x+y+z)

f(x+w) f(x+y+w) f(x+z+w) f(x+y+z+w)

Norms, Shnorms... so what?

• f U2 measures correlation with Fourier characters (linear phase

functions).

• max

α

• ˆ

f (α)

• 4

≤ f 4

U2 ≤

• max

α

• ˆ

f (α)

• 2

Norms, Shnorms... so what?

• f U2 measures correlation with Fourier characters (linear phase

functions).

• max

α

• ˆ

f (α)

• 4

≤ f 4

U2 ≤

• max

α

• ˆ

f (α)

• 2
• [Green-Tao 05, Samorodnitsky 07]: Gowers U3 norm approximately

measures correlation with the set of quadratic phase functions. ((−1)Q(x) for Q(x) = xTAx + bTx + c).

Norms, Shnorms... so what?

• f U2 measures correlation with Fourier characters (linear phase

functions).

• max

α

• ˆ

f (α)

• 4

≤ f 4

U2 ≤

• max

α

• ˆ

f (α)

• 2
• [Green-Tao 05, Samorodnitsky 07]: Gowers U3 norm approximately

measures correlation with the set of quadratic phase functions. ((−1)Q(x) for Q(x) = xTAx + bTx + c). For f : Fn

2 → [−1, 1],

• f U3 ≤ ǫ

= ⇒ for all Q,

• f , (−1)Q

≤ ǫ.

Norms, Shnorms... so what?

• f U2 measures correlation with Fourier characters (linear phase

functions).

• max

α

• ˆ

f (α)

• 4

≤ f 4

U2 ≤

• max

α

• ˆ

f (α)

• 2
• [Green-Tao 05, Samorodnitsky 07]: Gowers U3 norm approximately

measures correlation with the set of quadratic phase functions. ((−1)Q(x) for Q(x) = xTAx + bTx + c). For f : Fn

2 → [−1, 1],

• f U3 ≤ ǫ

= ⇒ for all Q,

• f , (−1)Q

≤ ǫ.

• f U3 ≥ ǫ

= ⇒ for some Q,

• f , (−1)Q

≥ η(ǫ).

Decompositions in Quadratic Fourier Analysis

Theorem (Gowers-Wolf 09)

Given ǫ > 0, any g : Fn

2 → [−1, 1] can be decomposed as

g =

k

• i=1

ci(−1)Qi + f + e for quadratic functions Q1, . . . , Qk such that

Decompositions in Quadratic Fourier Analysis

Theorem (Gowers-Wolf 09)

Given ǫ > 0, any g : Fn

2 → [−1, 1] can be decomposed as

g =

k

• i=1

ci(−1)Qi + f + e for quadratic functions Q1, . . . , Qk such that

• f U3 ≤ ǫ, e1 ≤ ǫ

pseudorandom

Decompositions in Quadratic Fourier Analysis

Theorem (Gowers-Wolf 09)

Given ǫ > 0, any g : Fn

2 → [−1, 1] can be decomposed as

g =

k

• i=1

ci(−1)Qi + f + e for quadratic functions Q1, . . . , Qk such that

• f U3 ≤ ǫ, e1 ≤ ǫ

pseudorandom

i |ci| ≤ M(ǫ) for M(ǫ) = exp(1/ǫC).

structure

Decompositions in Quadratic Fourier Analysis

Theorem (Gowers-Wolf 09)

Given ǫ > 0, any g : Fn

2 → [−1, 1] can be decomposed as

g =

k

• i=1

ci(−1)Qi + f + e for quadratic functions Q1, . . . , Qk such that

• f U3 ≤ ǫ, e1 ≤ ǫ

pseudorandom

i |ci| ≤ M(ǫ) for M(ǫ) = exp(1/ǫC).

structure Similar to basic Fourier decomposition, where we get g =

k

• i=1

ciχαi(x) + f , with |f , χα| ≤ ǫ for all α and k ≤ 1/ǫ2 (also implies

i |ci| ≤ 1/ǫ).

Decompositions in Higher-Order Fourier Analysis

Theorem (Gowers-Wolf 10)

Given ǫ > 0 and p > d, there exists M(ǫ, p) such that any g : Fn

p → [−1, 1] can be decomposed as

g =

k

• i=1

ci · ωPi + f + e for P1, . . . , Pk ∈ Pd (polynomials of degree at most d) such that

• f Ud+1 ≤ ǫ, e1 ≤ ǫ

i |ci| ≤ M(ǫ, p).

Decompositions in Higher-Order Fourier Analysis

Theorem (Gowers-Wolf 10)

Given ǫ > 0 and p > d, there exists M(ǫ, p) such that any g : Fn

p → [−1, 1] can be decomposed as

g =

k

• i=1

ci · ωPi + f + e for P1, . . . , Pk ∈ Pd (polynomials of degree at most d) such that

• f Ud+1 ≤ ǫ, e1 ≤ ǫ

i |ci| ≤ M(ǫ, p).

• Stronger decomposition theorems proved by [HL 11] and

[BFL 12].

• Decomposition theorems for the case when p ≤ d require

non-classical polynomials.

Q1: Can we compute these decompositions efficiently?

Algorithmic version of the basic Fourier decomposition

Theorem (Goldreich-Levin 89)

There is a randomized algorithm, which given ǫ, δ > 0 and oracle access to g : Fn

2 → [−1, 1], runs in time O

• n2 log n · (1/ǫ2) · log(1/δ)
• and
• utputs a decomposition

g =

k

• i=1

ci · χαi + f such that

Algorithmic version of the basic Fourier decomposition

Theorem (Goldreich-Levin 89)

There is a randomized algorithm, which given ǫ, δ > 0 and oracle access to g : Fn

2 → [−1, 1], runs in time O

• n2 log n · (1/ǫ2) · log(1/δ)
• and
• utputs a decomposition

g =

k

• i=1

ci · χαi + f such that

• k = O(1/ǫ2)

Algorithmic version of the basic Fourier decomposition

Theorem (Goldreich-Levin 89)

There is a randomized algorithm, which given ǫ, δ > 0 and oracle access to g : Fn

2 → [−1, 1], runs in time O

• n2 log n · (1/ǫ2) · log(1/δ)
• and
• utputs a decomposition

g =

k

• i=1

ci · χαi + f such that

• k = O(1/ǫ2)
• P[∃i such that |ci −

g(αi)| ≥ ǫ] ≤ δ

Algorithmic version of the basic Fourier decomposition

Theorem (Goldreich-Levin 89)

There is a randomized algorithm, which given ǫ, δ > 0 and oracle access to g : Fn

2 → [−1, 1], runs in time O

• n2 log n · (1/ǫ2) · log(1/δ)
• and
• utputs a decomposition

g =

k

• i=1

ci · χαi + f such that

• k = O(1/ǫ2)
• P[∃i such that |ci −

g(αi)| ≥ ǫ] ≤ δ

• P[∃α such that
• f (α)
• ≥ ǫ] ≤ δ
• Finding large Fourier coefficients has many applications.

What’s so different about quadratics?

• Set of quadratic phase functions ((−1)Q) is not an orthonormal
• basis. No Parseval’s identity.

What’s so different about quadratics?

• Set of quadratic phase functions ((−1)Q) is not an orthonormal
• basis. No Parseval’s identity.
• Proof of decomposition by Gowers and Wolf is non-constructive

(using the Hahn-Banach theorem).

What’s so different about quadratics?

• Set of quadratic phase functions ((−1)Q) is not an orthonormal
• basis. No Parseval’s identity.
• Proof of decomposition by Gowers and Wolf is non-constructive

(using the Hahn-Banach theorem). ci(−1)Qi + f

s.t.

i |ci| ≤ M(ǫ), f U3 ≤ ǫ

What’s so different about quadratics?

• Set of quadratic phase functions ((−1)Q) is not an orthonormal
• basis. No Parseval’s identity.
• Proof of decomposition by Gowers and Wolf is non-constructive

(using the Hahn-Banach theorem). ci(−1)Qi + f

s.t.

i |ci| ≤ M(ǫ), f U3 ≤ ǫ

g

What’s so different about quadratics?

• Set of quadratic phase functions ((−1)Q) is not an orthonormal
• basis. No Parseval’s identity.
• Proof of decomposition by Gowers and Wolf is non-constructive

(using the Hahn-Banach theorem). ci(−1)Qi + f

s.t.

i |ci| ≤ M(ǫ), f U3 ≤ ǫ

g

What’s so different about quadratics?

• Set of quadratic phase functions ((−1)Q) is not an orthonormal
• basis. No Parseval’s identity.
• Proof of decomposition by Gowers and Wolf is non-constructive

(using the Hahn-Banach theorem). ci(−1)Qi + f

s.t.

i |ci| ≤ M(ǫ), f U3 ≤ ǫ

g

• Use inverse theorem for Gowers norm to get a contradiction.

A quadratic Goldreich-Levin Theorem

Theorem (T, Wolf 11)

For M(ǫ) = exp(1/ǫC), can compute in time poly(n, M(ǫ), log(1/δ)), a decomposition g =

k

• i=1

ci(−1)Qi + f + e such that

• with probability 1 − δ, f U3 ≤ ǫ and e1 ≤ ǫ.

i |ci| ≤ M(ǫ) and k ≤ (M(ǫ))2.

Improved quadratic Goldreich-Levin Theorem

Theorem (BRTW 12)

For M(ǫ) = O(exp(log4(1/ǫ))), can compute in time poly(n, M(ǫ), log(1/δ)), a decomposition g =

k

• i=1

ci(−1)Qi + f + e such that

• with probability 1 − δ, f U3 ≤ ǫ and e1 ≤ ǫ.

i |ci| ≤ M(ǫ) and k ≤ (M(ǫ))2.

A constructive proof of decomposition [TTV 09]

Goal: Given g : Fn

2 → [−1, 1], find a decomposition g = i ci(−1)Qi + f

such that f U3 ≤ ǫ.

A constructive proof of decomposition [TTV 09]

Goal: Given g : Fn

2 → [−1, 1], find a decomposition g = i ci(−1)Qi + f

such that f U3 ≤ ǫ. Algorithm:

• h0 = 0, f0 = g − h0, t = 1.

A constructive proof of decomposition [TTV 09]

Goal: Given g : Fn

2 → [−1, 1], find a decomposition g = i ci(−1)Qi + f

such that f U3 ≤ ǫ. Algorithm:

• h0 = 0, f0 = g − h0, t = 1.
• while there is a quadratic function Qt such that
• ft−1, (−1)Qt

> η

A constructive proof of decomposition [TTV 09]

Goal: Given g : Fn

2 → [−1, 1], find a decomposition g = i ci(−1)Qi + f

such that f U3 ≤ ǫ. Algorithm:

• h0 = 0, f0 = g − h0, t = 1.
• while there is a quadratic function Qt such that
• ft−1, (−1)Qt

> η

• ht = ht−1 + η · (−1)Qt = t

r=1 η · (−1)Qr

• ft = g − ht
• t = t + 1

A constructive proof of decomposition [TTV 09]

Goal: Given g : Fn

2 → [−1, 1], find a decomposition g = i ci(−1)Qi + f

such that f U3 ≤ ǫ. Algorithm:

• h0 = 0, f0 = g − h0, t = 1.
• while there is a quadratic function Qt such that
• ft−1, (−1)Qt

> η

• ht = ht−1 + η · (−1)Qt = t

r=1 η · (−1)Qr

• ft = g − ht
• t = t + 1
• return ht

A constructive proof of decomposition [TTV 09]

Goal: Given g : Fn

2 → [−1, 1], find a decomposition g = i ci(−1)Qi + f

such that f U3 ≤ ǫ. Algorithm:

• h0 = 0, f0 = g − h0, t = 1.
• while there is a quadratic function Qt such that
• ft−1, (−1)Qt

> η

• ht = ht−1 + η · (−1)Qt = t

r=1 η · (−1)Qr

• ft = g − ht
• t = t + 1
• return ht

Convergence: ft−12 − ft2 = 2η

• ft−1, (−1)Qt

− η2 ≥ η2.

A constructive proof of decomposition [TTV 09]

Goal: Given g : Fn

2 → [−1, 1], find a decomposition g = i ci(−1)Qi + f

such that f U3 ≤ ǫ. Algorithm:

• h0 = 0, f0 = g − h0, t = 1.
• while there is a quadratic function Qt such that
• ft−1, (−1)Qt

> η

• ht = ht−1 + η · (−1)Qt = t

r=1 η · (−1)Qr

• ft = g − ht
• t = t + 1
• return ht

Convergence: ft−12 − ft2 = 2η

• ft−1, (−1)Qt

− η2 ≥ η2. [Samorodnitsky 07]: ∀Q

• (−1)Q, f
• ≤ η(ǫ) =

⇒ f U3 ≤ ǫ.

A constructive proof of decomposition [TTV 09]

Goal: Given g : Fn

2 → [−1, 1], find a decomposition g = i ci(−1)Qi + f

such that f U3 ≤ ǫ. Algorithm:

• h0 = 0, f0 = g − h0, t = 1.
• while there is a quadratic function Qt such that
• ft−1, (−1)Qt

> η

• ht = ht−1 + η · (−1)Qt = t

r=1 η · (−1)Qr

• ft = g − ht
• t = t + 1
• return ht

Convergence: ft−12 − ft2 = 2η

• ft−1, (−1)Qt

− η2 ≥ η2. [Samorodnitsky 07]: ∀Q

• (−1)Q, f
• ≤ η(ǫ) =

⇒ f U3 ≤ ǫ.

The algorithmic problem

Question: Given f : Fn

2 → {−1, 1}, does there exist Q such that

• f , (−1)Q

≥ ǫ? If yes, find one.

The algorithmic problem

Question: Given f : Fn

2 → {−1, 1}, does there exist Q such that

• f , (−1)Q

≥ ǫ? If yes, find one. Truth-tables of functions (−1)Q form the Reed-Muller code of order 2.

The algorithmic problem

Question: Given f : Fn

2 → {−1, 1}, does there exist Q such that

• f , (−1)Q

≥ ǫ? If yes, find one. Truth-tables of functions (−1)Q form the Reed-Muller code of order 2. Want a codeword inside a ball of distance 1/2 − ǫ/2 around f (if one exists). f (−1)q ≤ 1

2 − ǫ 2

Q2: Decoding beyond the list-decoding radius

Finding codewords at large distances

f

Finding codewords at large distances

f

1 8

Finding codewords at large distances

f

1 8 1 4

• List decoding radius is 1

4.

[GKZ 08, Gopalan 10, BL 14]

Finding codewords at large distances

f

1 8 1 4 1 2 − ǫ

• List decoding radius is 1

4.

[GKZ 08, Gopalan 10, BL 14]

• Number of codewords within

distance 1

2 − ǫ may be exponential.

Finding codewords at large distances

f

1 8 1 4 1 2 − ǫ

• List decoding radius is 1

4.

[GKZ 08, Gopalan 10, BL 14]

• Number of codewords within

distance 1

2 − ǫ may be exponential.

• But we only need to find one

codeword! In time poly(n) (polylogarithmic in code length).

Finding codewords at large distances

• Given (the coefficients of) a degree-d polynomial P : Fn

p → Fp, the

Reed-Muller encoding of P is of length pn and is given by the table

• f values {P(x)}x∈Fn

p.

Finding codewords at large distances

• Given (the coefficients of) a degree-d polynomial P : Fn

p → Fp, the

Reed-Muller encoding of P is of length pn and is given by the table

• f values {P(x)}x∈Fn

p.

• Problem: Given F : Fn

p → Fp, if there exists P ∈ Pd such that

∆(F, P) ≤ 1 − 1 p − ǫ find a P′ ∈ Pd such that ∆(F, P′) ≤ 1 − 1 p − η

Finding codewords at large distances

• Given (the coefficients of) a degree-d polynomial P : Fn

p → Fp, the

Reed-Muller encoding of P is of length pn and is given by the table

• f values {P(x)}x∈Fn

p.

• Problem: Given F : Fn

p → Fp, if there exists P ∈ Pd such that

∆(F, P) ≤ 1 − 1 p − ǫ find a P′ ∈ Pd such that ∆(F, P′) ≤ 1 − 1 p − η

• If there exists a Reed-Muller codeword within a ball of radius

1 − 1

p − ǫ, find one within a ball of radius 1 − 1 p − η.

Finding a single codeword: the quadratic case

f

1 2 − ǫ

Finding a single codeword: the quadratic case

f

1 2 − ǫ

• [Samorodnitsky 07]: Approximate solution

to testing problem using Gowers norm.

Finding a single codeword: the quadratic case

f

1 2 − ǫ

• [Samorodnitsky 07]: Approximate solution

to testing problem using Gowers norm. − ∃q

• f , (−1)Q

≥ ǫ = ⇒ f U3 ≥ ǫ

Finding a single codeword: the quadratic case

f

1 2 − ǫ 1 2 − η

• [Samorodnitsky 07]: Approximate solution

to testing problem using Gowers norm. − ∃q

• f , (−1)Q

≥ ǫ = ⇒ f U3 ≥ ǫ − f U3 ≥ ǫ = ⇒ ∃Q

• f , (−1)Q

≥ η(ǫ)

Finding a single codeword: the quadratic case

f

1 2 − ǫ 1 2 − η

• [Samorodnitsky 07]: Approximate solution

to testing problem using Gowers norm. − ∃q

• f , (−1)Q

≥ ǫ = ⇒ f U3 ≥ ǫ − f U3 ≥ ǫ = ⇒ ∃Q

• f , (−1)Q

≥ η(ǫ)

• [TW 11] convert Samorodnitsky’s proof

into an algorithm. Find codeword within distance 1

2 − η if there is one within 1 2 − ǫ.

Finding a single codeword: the quadratic case

f

1 2 − ǫ 1 2 − η

• [Samorodnitsky 07]: Approximate solution

to testing problem using Gowers norm. − ∃q

• f , (−1)Q

≥ ǫ = ⇒ f U3 ≥ ǫ − f U3 ≥ ǫ = ⇒ ∃Q

• f , (−1)Q

≥ η(ǫ)

• [TW 11] convert Samorodnitsky’s proof

into an algorithm. Find codeword within distance 1

2 − η if there is one within 1 2 − ǫ.

• First example of any kind of decoding

beyond the list decoding radius.

Algorithmic versions of combinatorial theorems

Fn

2

S

• Samorodnitsky’s proof applies various combinatorial theorems (e.g.

Balog-Szemerédi-Gowers) to “nice” subsets of Fn

2.

Algorithmic versions of combinatorial theorems

Fn

2

S A

• Samorodnitsky’s proof applies various combinatorial theorems (e.g.

Balog-Szemerédi-Gowers) to “nice” subsets of Fn

2.

• [BSG]: If S ⊆ Fn

2 satisfies Px,y∈S [x + y ∈ S] ≥ ǫ, then there exists

A ⊆ S with certain additive properties.

Algorithmic versions of combinatorial theorems

Fn

2

S A

• Samorodnitsky’s proof applies various combinatorial theorems (e.g.

Balog-Szemerédi-Gowers) to “nice” subsets of Fn

2.

• [BSG]: If S ⊆ Fn

2 satisfies Px,y∈S [x + y ∈ S] ≥ ǫ, then there exists

A ⊆ S with certain additive properties.

• S and A are exponential in size. Need to work with randomized

membership oracles. Gives a noisy version of the set S.

Algorithmic versions of combinatorial theorems

Fn

2

S

• Samorodnitsky’s proof applies various combinatorial theorems (e.g.

Balog-Szemerédi-Gowers) to “nice” subsets of Fn

2.

• [BSG]: If S ⊆ Fn

2 satisfies Px,y∈S [x + y ∈ S] ≥ ǫ, then there exists

A ⊆ S with certain additive properties.

• S and A are exponential in size. Need to work with randomized

membership oracles. Gives a noisy version of the set S.

Algorithmic versions of combinatorial theorems

Fn

2

S

• Modify proofs of combinatorial theorems to go from algorithms in

the hypothesis to algorithms in conclusion.

Algorithmic versions of combinatorial theorems

Fn

2

S A1 A2

• Modify proofs of combinatorial theorems to go from algorithms in

the hypothesis to algorithms in conclusion.

• Statements of the form: “Given (approximate) membership oracle

for S, it can be converted to an oracle A whose output is sandwiched between A1 and A2 with certain additive properties.”

Algorithmic versions of combinatorial theorems

Fn

2

S A1 A2

• Modify proofs of combinatorial theorems to go from algorithms in

the hypothesis to algorithms in conclusion.

• Statements of the form: “Given (approximate) membership oracle

for S, it can be converted to an oracle A whose output is sandwiched between A1 and A2 with certain additive properties.”

• Prove “robust” versions of theorems from additive combinatorics.

Finding subspace structure

Most combinatorial results used here find and refine subspace structure in S ⊆ Fn

2.

• [BSG]: If Px,y∈S [x + y ∈ S] ≥ ǫ then ∃A ⊆ S s.t.

|A| ≥ ǫO(1)|S| and |A + A| ≤ ǫ−O(1)|A|.

Finding subspace structure

Most combinatorial results used here find and refine subspace structure in S ⊆ Fn

2.

• [BSG]: If Px,y∈S [x + y ∈ S] ≥ ǫ then ∃A ⊆ S s.t.

|A| ≥ ǫO(1)|S| and |A + A| ≤ ǫ−O(1)|A|.

• [Freiman-Ruzsa]: |A + A| ≤ K · |A| =

⇒ Span(A) ≤ 2O(K) · |A|.

Finding subspace structure

Most combinatorial results used here find and refine subspace structure in S ⊆ Fn

2.

• [BSG]: If Px,y∈S [x + y ∈ S] ≥ ǫ then ∃A ⊆ S s.t.

|A| ≥ ǫO(1)|S| and |A + A| ≤ ǫ−O(1)|A|.

• [Freiman-Ruzsa]: |A + A| ≤ K · |A| =

⇒ Span(A) ≤ 2O(K) · |A|.

• [CS 09]: If |A + A| ≤ K · |A|, then 1A ∗ 1A has a large set of “almost

periods” i.e., there is a large set X ⊆ Fn

2 s.t

1A ∗ 1A(·) ≈ 1A ∗ 1A(· + x) ∀x ∈ X 1A ∗ 1A(·) ≈ distribution of sum of two random elements from A.

Finding subspace structure

• [Sanders 10]: Stronger inverse theorem for U3-norm using almost

periodicity from [CS 09].

Finding subspace structure

• [Sanders 10]: Stronger inverse theorem for U3-norm using almost

periodicity from [CS 09].

• [BRTW 14]: Sampling-based proof of [CS 09]. Improved quadratic

Goldreich-Levin.

Finding subspace structure

• [Sanders 10]: Stronger inverse theorem for U3-norm using almost

periodicity from [CS 09].

• [BRTW 14]: Sampling-based proof of [CS 09]. Improved quadratic

Goldreich-Levin.

• Question: Can sampling based proofs be used to find better

subspace structure?

Decompositions for higher-degrees

• Question: Given F : Fn

p → Fp, does there exist a polynomial P ∈ Pd

such that

• ωF, ωP

≥ ǫ? If yes, find one.

Decompositions for higher-degrees

• Question: Given F : Fn

p → Fp, does there exist a polynomial P ∈ Pd

such that

• ωF, ωP

≥ ǫ? If yes, find one. f P ≤ p−1

p

− ǫ

Decompositions for higher-degrees

• Question: Given F : Fn

p → Fp, does there exist a polynomial P ∈ Pd

such that

• ωF, ωP

≥ ǫ? If yes, find one. f P ≤ p−1

p

− ǫ

• Can be solved for the special case when F ∈ Pk and p > k, inverse

theorem by [GT 09].

Decomposition Theorems and Regularity

• [GT 09]: Actually prove a decomposition theorem when F ∈ Pk:

ωF = Γ(P1, . . . , Pm) + f2 where P1, . . . , Pm ∈ Pd and f2Ud+1 ≤ ǫ.

Decomposition Theorems and Regularity

• [GT 09]: Actually prove a decomposition theorem when F ∈ Pk:

ωF = Γ(P1, . . . , Pm) + f2 where P1, . . . , Pm ∈ Pd and f2Ud+1 ≤ ǫ.

• Here, Γ : Fm

p → R. By (linear) Fourier analysis

Γ(P1, . . . , Pm) =

• c1,...,cm
• Γ(c1, . . . , cm) · ω
• i ciPi

which gives decomposition in the required form.

Decomposition Theorems and Regularity

• [GT 09]: Actually prove a decomposition theorem when F ∈ Pk:

ωF = Γ(P1, . . . , Pm) + f2 where P1, . . . , Pm ∈ Pd and f2Ud+1 ≤ ǫ.

• Here, Γ : Fm

p → R. By (linear) Fourier analysis

Γ(P1, . . . , Pm) =

• c1,...,cm
• Γ(c1, . . . , cm) · ω
• i ciPi

which gives decomposition in the required form.

• Proof by [GT 09] and many other applications require the factor

B = {P1, . . . , Pm} to satisfy certain “regularity” properties. Obtaining regularity is the main challenge in converting their proof to an algorithm.

Polynomial Regularity Lemmas

• Regulariy lemmas for polynomials are useful for several applications
• f higher-order Fourier analysis.
• Analogues of Szemerédi regularity lemma. Regular partition a graph

is highly structured. So is a regular collection of polynomials.

Polynomial Regularity Lemmas

• Regulariy lemmas for polynomials are useful for several applications
• f higher-order Fourier analysis.
• Analogues of Szemerédi regularity lemma. Regular partition a graph

is highly structured. So is a regular collection of polynomials.

• Different notions of regulariy for a factor B = {P1, . . . , Pm}:

Polynomial Regularity Lemmas

• Regulariy lemmas for polynomials are useful for several applications
• f higher-order Fourier analysis.
• Analogues of Szemerédi regularity lemma. Regular partition a graph

is highly structured. So is a regular collection of polynomials.

• Different notions of regulariy for a factor B = {P1, . . . , Pm}:
• [GT 09]: For all (c1, . . . , cm) ∈ Fm

p \ {0m},

rankd−1(c1P1 + · · · + cmPm) ≥ Λ(m).

Polynomial Regularity Lemmas

• Regulariy lemmas for polynomials are useful for several applications
• f higher-order Fourier analysis.
• Analogues of Szemerédi regularity lemma. Regular partition a graph

is highly structured. So is a regular collection of polynomials.

• Different notions of regulariy for a factor B = {P1, . . . , Pm}:
• [GT 09]: For all (c1, . . . , cm) ∈ Fm

p \ {0m},

rankd−1(c1P1 + · · · + cmPm) ≥ Λ(m).

• [KL 08]: For all (c1, . . . , cm) ∈ Fm

p \ {0m}, ciPi and it’s

derivatiives have high-rank.

• Polynomial Regularity Lemmas: Given B = {P1, . . . , Pm}, it can be

refined to B′ = {P′

1, . . . , P′ m′} which is regular.

Polynomial Regularity Lemmas

• Regulariy lemmas for polynomials are useful for several applications
• f higher-order Fourier analysis.
• Analogues of Szemerédi regularity lemma. Regular partition a graph

is highly structured. So is a regular collection of polynomials.

• Different notions of regulariy for a factor B = {P1, . . . , Pm}:
• [GT 09]: For all (c1, . . . , cm) ∈ Fm

p \ {0m},

rankd−1(c1P1 + · · · + cmPm) ≥ Λ(m).

• [KL 08]: For all (c1, . . . , cm) ∈ Fm

p \ {0m}, ciPi and it’s

derivatiives have high-rank.

• Polynomial Regularity Lemmas: Given B = {P1, . . . , Pm}, it can be

refined to B′ = {P′

1, . . . , P′ m′} which is regular.

• Like Szemerédi’s regularity lemma, proofs find a certificate of

non-regularity and make progress by local modification.

Q3: Algorithmic Regularity Lemmas

Algorithmic notions of regularity

• Algorithmic step in the regularity lemma is finding a certificate of

non-regularity.

Algorithmic notions of regularity

• Algorithmic step in the regularity lemma is finding a certificate of

non-regularity.

• [BHT 15]: Slightly modified notions of regularity (equivalent up to

some loss of parameters) and corresponding algorithmic lemmas.

Algorithmic notions of regularity

• Algorithmic step in the regularity lemma is finding a certificate of

non-regularity.

• [BHT 15]: Slightly modified notions of regularity (equivalent up to

some loss of parameters) and corresponding algorithmic lemmas.

• [GT 09]: For all (c1, . . . , cm) ∈ Fm

p \ {0m},

c1P1 + · · · + cmPmUd ≤ δ(m).

Algorithmic notions of regularity

• Algorithmic step in the regularity lemma is finding a certificate of

non-regularity.

• [BHT 15]: Slightly modified notions of regularity (equivalent up to

some loss of parameters) and corresponding algorithmic lemmas.

• [GT 09]: For all (c1, . . . , cm) ∈ Fm

p \ {0m},

c1P1 + · · · + cmPmUd ≤ δ(m).

• [KL 08]: For all (c1, . . . , cm) ∈ Fm

p \ {0m}, ciPi and it’s

derivatiives have small-bias.

Algorithmic notions of regularity

• Algorithmic step in the regularity lemma is finding a certificate of

non-regularity.

• [BHT 15]: Slightly modified notions of regularity (equivalent up to

some loss of parameters) and corresponding algorithmic lemmas.

• [GT 09]: For all (c1, . . . , cm) ∈ Fm

p \ {0m},

c1P1 + · · · + cmPmUd ≤ δ(m).

• [KL 08]: For all (c1, . . . , cm) ∈ Fm

p \ {0m}, ciPi and it’s

derivatiives have small-bias.

• Show these notions provide required equidistribution for various

known applications.

Further questions

• Higher-degree decomposition theorems.

Further questions

• Higher-degree decomposition theorems.
• (Approximate) Decoding beyond the list decoding radius for other
• codes. Even for distances slightly beyond the list-decoding radius.

Further questions

• Higher-degree decomposition theorems.
• (Approximate) Decoding beyond the list decoding radius for other
• codes. Even for distances slightly beyond the list-decoding radius.
• Do algorithms really need to be derived from proofs of existence?

Can there be a simpler algorithm for which a solution is guaranteed by the proof?

Further questions

• Higher-degree decomposition theorems.
• (Approximate) Decoding beyond the list decoding radius for other
• codes. Even for distances slightly beyond the list-decoding radius.
• Do algorithms really need to be derived from proofs of existence?

Can there be a simpler algorithm for which a solution is guaranteed by the proof?

• Regularity lemmas give terrible quantitative bounds. Is there a way

to use weaker regularity properties and obtain better bounds?

Thank You Questions?