Algorithmic Questions in Higher-Order Fourier Analysis · 2018-09-20 · Algorithmic Questions in...

Algorithmic Questions inHigher-Order Fourier Analysis

12 − ε1

2 − η

Madhur TulsianiTTI Chicago

Based on joint works withArnab Bhattacharyya, Eli

Ben-Sasson, Pooya Hatami,Noga Ron-Zewi, Luca

Trevisan, Salil Vadhan andJulia Wolf

Decomposition Theorems

Object of study Family ofalgorithms orfunctions

Structured

No apparentstructure

(Pseudorandom)

- Decompose an object in to structured and pseudorandom parts.

- Can often ignore the pseudorandom part for many applications.Structured part easier to study.

Structured

(Pseudorandom)

Structured

(Pseudorandom)

Structured

(Pseudorandom)

Structured

(Pseudorandom)

Fourier analysis- Space of functions g : Fn

2 → R.

- Functionsχα(x) = (−1)α·x = (−1)

∑jαj xj

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

g(α)χα .

- [Parseval]: ‖g‖2 = 〈g , g〉 = Ex[(g(x))2] =

∑α (g(α))2.

2 → R.

∑jαj xj

g =∑α∈Fn

g(α)χα .

∑α (g(α))2.

2 → R.

∑jαj xj

g =∑α∈Fn

g(α)χα .

∑α (g(α))2.

2 → R.

∑jαj xj

g =∑α∈Fn

g(α)χα .

∑α (g(α))2.

A basic decomposition in Fourier analysis

g : Fn2 → [−1, 1]

χα(x) = (−1)α·x = (−1)∑

iαj xj

α ∈ Fn2

g =∑

Sg(α)χα

=∑|g(α)|>ε

g(α)χα +∑|g(α)|≤ε

g(α)χα =k∑

i=1ciχαi + f

- k ≤ 1/ε2. simple structure

- f has small correlation with linear functions. pseudorandom

∀α, |〈f , χα〉| = |Ex [f (x)χα(x)]| ≤ ε

g : Fn2 → [−1, 1]

χα(x) = (−1)α·x = (−1)∑

iαj xj

α ∈ Fn2

g =∑

Sg(α)χα

=∑|g(α)|>ε

g(α)χα =k∑

i=1ciχαi + f

∀α, |〈f , χα〉| = |Ex [f (x)χα(x)]| ≤ ε

g : Fn2 → [−1, 1]

χα(x) = (−1)α·x = (−1)∑

iαj xj

α ∈ Fn2

g =∑

Sg(α)χα =

∑|g(α)|>ε

g(α)χα =k∑

i=1ciχαi + f

∀α, |〈f , χα〉| = |Ex [f (x)χα(x)]| ≤ ε

g : Fn2 → [−1, 1]

χα(x) = (−1)α·x = (−1)∑

iαj xj

α ∈ Fn2

g =∑

Sg(α)χα =

∑|g(α)|>ε

g(α)χα =k∑

i=1ciχαi + f

∀α, |〈f , χα〉| = |Ex [f (x)χα(x)]| ≤ ε

g : Fn2 → [−1, 1]

χα(x) = (−1)α·x = (−1)∑

iαj xj

α ∈ Fn2

g =∑

Sg(α)χα =

∑|g(α)|>ε

g(α)χα =k∑

i=1ciχαi + f

∀α, |〈f , χα〉| = |Ex [f (x)χα(x)]| ≤ ε

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 ‖4U2 = 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" inset.

- Can define higher norms similarly

‖f ‖8U3 = 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)

‖f ‖8U3 = Ex ,y ,z,w

Norms, Shnorms... so what?

- ‖f ‖U2 measures correlation with Fourier characters (linear phasefunctions). (

∣∣∣f (α)∣∣∣)4≤ ‖f ‖4

U2≤(maxα

∣∣∣f (α)∣∣∣)2

- [Green-Tao 05, Samorodnitsky 07]: Gowers U3 norm approximatelymeasures correlation with the set of quadratic phase functions.((−1)Q(x) for Q(x) = xT Ax + bT x + c).

For f : Fn2 → [−1, 1],

- ‖f ‖U3 ≤ ε =⇒ for all Q,∣∣⟨f , (−1)Q⟩∣∣ ≤ ε.

- ‖f ‖U3 ≥ ε =⇒ for some Q,∣∣⟨f , (−1)Q⟩∣∣ ≥ η(ε).

∣∣∣f (α)∣∣∣)4≤ ‖f ‖4

U2≤(maxα

∣∣∣f (α)∣∣∣)2

- [Green-Tao 05, Samorodnitsky 07]: Gowers U3 norm approximatelymeasures correlation with the set of quadratic phase functions.((−1)Q(x) for Q(x) = xT Ax + bT x + c).

For f : Fn2 → [−1, 1],

- ‖f ‖U3 ≤ ε =⇒ for all Q,∣∣⟨f , (−1)Q⟩∣∣ ≤ ε.

∣∣∣f (α)∣∣∣)4≤ ‖f ‖4

U2≤(maxα

∣∣∣f (α)∣∣∣)2

- [Green-Tao 05, Samorodnitsky 07]: Gowers U3 norm approximatelymeasures correlation with the set of quadratic phase functions.((−1)Q(x) for Q(x) = xT Ax + bT x + c). For f : Fn

2 → [−1, 1],- ‖f ‖U3 ≤ ε =⇒ for all Q,

∣∣⟨f , (−1)Q⟩∣∣ ≤ ε.

∣∣∣f (α)∣∣∣)4≤ ‖f ‖4

U2≤(maxα

∣∣∣f (α)∣∣∣)2

- [Green-Tao 05, Samorodnitsky 07]: Gowers U3 norm approximatelymeasures correlation with the set of quadratic phase functions.((−1)Q(x) for Q(x) = xT Ax + bT x + c). For f : Fn

2 → [−1, 1],- ‖f ‖U3 ≤ ε =⇒ for all Q,

∣∣⟨f , (−1)Q⟩∣∣ ≤ ε.- ‖f ‖U3 ≥ ε =⇒ for some Q,

∣∣⟨f , (−1)Q⟩∣∣ ≥ η(ε).

Decompositions in Quadratic Fourier Analysis

Theorem (Gowers-Wolf 09)Given ε > 0, any g : Fn

2 → [−1, 1] can be decomposed as

g =k∑

i=1ci (−1)Qi + f + e

for quadratic functions Q1, . . . ,Qk such that

- ‖f ‖U3 ≤ ε, ‖e‖1 ≤ ε pseudorandom

i |ci | ≤ M(ε) for M(ε) = exp(1/εC ). structure

Similar to basic Fourier decomposition, where we get

g =k∑

i=1ciχαi (x) + f ,

with |〈f , χα〉| ≤ ε for all α and k ≤ 1/ε2 (also implies∑

i |ci | ≤ 1/ε).

g =k∑

i=1ci (−1)Qi + f + e

g =k∑

i |ci | ≤ 1/ε).

g =k∑

i=1ci (−1)Qi + f + e

g =k∑

i |ci | ≤ 1/ε).

g =k∑

i=1ci (−1)Qi + f + e

g =k∑

i |ci | ≤ 1/ε).

Decompositions in Higher-Order Fourier AnalysisTheorem (Gowers-Wolf 10)Given ε > 0 and p > d, there exists M(ε, p) such that anyg : Fn

p → [−1, 1] can be decomposed as

g =k∑

i=1ci · ωPi + f + e

for P1, . . . ,Pk ∈ Pd (polynomials of degree at most d) such that

- ‖f ‖Ud+1 ≤ ε, ‖e‖1 ≤ ε

i |ci | ≤ M(ε, p).

- Stronger decomposition theorems proved by [HL 11] and[BFL 12].

- Decomposition theorems for the case when p ≤ d requirenon-classical polynomials.

Decompositions in Higher-Order Fourier AnalysisTheorem (Gowers-Wolf 10)Given ε > 0 and p > d, there exists M(ε, p) such that anyg : Fn

p → [−1, 1] can be decomposed as

g =k∑

i=1ci · ωPi + f + e

for P1, . . . ,Pk ∈ Pd (polynomials of degree at most d) such that

- ‖f ‖Ud+1 ≤ ε, ‖e‖1 ≤ ε

i |ci | ≤ M(ε, p).

- Stronger decomposition theorems proved by [HL 11] and[BFL 12].

- Decomposition theorems for the case when p ≤ d requirenon-classical polynomials.

Q1: Can we compute these decompositionsefficiently?

Algorithmic version of the basic Fourier decomposition

Theorem (Goldreich-Levin 89)There is a randomized algorithm, which given ε, δ > 0 and oracle accessto g : Fn

2 → [−1, 1], runs in time O(n2 log n · (1/ε2) · log(1/δ)

outputs a decomposition

g =k∑

i=1ci · χαi + f

such that

- k = O(1/ε2)

- P[∃i such that |ci − g(αi )| ≥ ε] ≤ δ

- P[∃α such that∣∣∣f (α)

∣∣∣ ≥ ε] ≤ δ- Finding large Fourier coefficients has many applications.

g =k∑

i=1ci · χαi + f

such that

- k = O(1/ε2)

g =k∑

i=1ci · χαi + f

such that

- k = O(1/ε2)

g =k∑

i=1ci · χαi + f

such that

- k = O(1/ε2)

What’s so different about quadratics?

- Set of quadratic phase functions ((−1)Q) is not an orthonormalbasis. 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(ε), ‖f ‖U3 ≤ ε

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

∑ci (−1)Qi + f

s.t.∑

i |ci | ≤ M(ε), ‖f ‖U3 ≤ ε

∑ci (−1)Qi + f

s.t.∑

i |ci | ≤ M(ε), ‖f ‖U3 ≤ ε

∑ci (−1)Qi + f

s.t.∑

i |ci | ≤ M(ε), ‖f ‖U3 ≤ ε

∑ci (−1)Qi + f

s.t.∑

i |ci | ≤ M(ε), ‖f ‖U3 ≤ ε

∑ci (−1)Qi + f

s.t.∑

i |ci | ≤ M(ε), ‖f ‖U3 ≤ ε

A quadratic Goldreich-Levin Theorem

Theorem (T, Wolf 11)For M(ε) = exp(1/εC ), can compute in time poly(n,M(ε), log(1/δ)), adecomposition

g =k∑

i=1ci (−1)Qi + f + e

such that

- with probability 1− δ, ‖f ‖U3 ≤ ε and ‖e‖1 ≤ ε.

i |ci | ≤ M(ε) and k ≤ (M(ε))2.

Improved quadratic Goldreich-Levin Theorem

Theorem (BRTW 12)For M(ε) = O(exp(log4(1/ε))), can compute in timepoly(n,M(ε), log(1/δ)), a decomposition

g =k∑

i=1ci (−1)Qi + f + e

such that

- with probability 1− δ, ‖f ‖U3 ≤ ε and ‖e‖1 ≤ ε.

i |ci | ≤ M(ε) and k ≤ (M(ε))2.

A constructive proof of decomposition [TTV 09]

Goal: Given g : Fn2 → [−1, 1], find a decomposition g =

∑i ci (−1)Qi + f

such that ‖f ‖U3 ≤ ε.

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−1‖2 − ‖ft‖2 = 2η⟨ft−1, (−1)Qt

⟩− η2 ≥ η2.

[Samorodnitsky 07]: ∀Q∣∣⟨(−1)Q , f

⟩∣∣ ≤ η(ε) =⇒ ‖f ‖U3 ≤ ε.

∑i ci (−1)Qi + f

Algorithm:

- h0 = 0, f0 = g − h0, t = 1.

⟩> η

- ht = ht−1 + η · (−1)Qt =∑t

r=1 η · (−1)Qr

- ft = g − ht- t = t + 1

- return ht

⟩− η2 ≥ η2.

⟩∣∣ ≤ η(ε) =⇒ ‖f ‖U3 ≤ ε.

∑i ci (−1)Qi + f

Algorithm:

- h0 = 0, f0 = g − h0, t = 1.

⟩> η

- ht = ht−1 + η · (−1)Qt =∑t

r=1 η · (−1)Qr

- ft = g − ht- t = t + 1

- return ht

⟩− η2 ≥ η2.

⟩∣∣ ≤ η(ε) =⇒ ‖f ‖U3 ≤ ε.

∑i ci (−1)Qi + f

Algorithm:

- h0 = 0, f0 = g − h0, t = 1.

⟩> η

- ht = ht−1 + η · (−1)Qt =∑t

r=1 η · (−1)Qr

- ft = g − ht- t = t + 1

- return ht

⟩− η2 ≥ η2.

⟩∣∣ ≤ η(ε) =⇒ ‖f ‖U3 ≤ ε.

∑i ci (−1)Qi + f

Algorithm:

- h0 = 0, f0 = g − h0, t = 1.

⟩> η

- ht = ht−1 + η · (−1)Qt =∑t

r=1 η · (−1)Qr

- ft = g − ht- t = t + 1

- return ht

⟩− η2 ≥ η2.

⟩∣∣ ≤ η(ε) =⇒ ‖f ‖U3 ≤ ε.

∑i ci (−1)Qi + f

Algorithm:

- h0 = 0, f0 = g − h0, t = 1.

⟩> η

- ht = ht−1 + η · (−1)Qt =∑t

r=1 η · (−1)Qr

- ft = g − ht- t = t + 1

- return ht

⟩− η2 ≥ η2.

⟩∣∣ ≤ η(ε) =⇒ ‖f ‖U3 ≤ ε.

∑i ci (−1)Qi + f

Algorithm:

- h0 = 0, f0 = g − h0, t = 1.

⟩> η

- ht = ht−1 + η · (−1)Qt =∑t

r=1 η · (−1)Qr

- ft = g − ht- t = t + 1

- return ht

⟩− η2 ≥ η2.

⟩∣∣ ≤ η(ε) =⇒ ‖f ‖U3 ≤ ε.

∑i ci (−1)Qi + f

Algorithm:

- h0 = 0, f0 = g − h0, t = 1.

⟩> η

- ht = ht−1 + η · (−1)Qt =∑t

r=1 η · (−1)Qr

- ft = g − ht- t = t + 1

- return ht

⟩− η2 ≥ η2.

⟩∣∣ ≤ η(ε) =⇒ ‖f ‖U3 ≤ ε.

The algorithmic problem

Question: Given f : Fn2 → {−1, 1}, does there exist Q such that⟨

f , (−1)Q⟩ ≥ ε? 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− ε/2 around f (if oneexists).

(−1)q

≤ 12 −

Truth-tables of functions (−1)Q form the Reed-Muller code of order 2.

Want a codeword inside a ball of distance 1/2− ε/2 around f (if oneexists).

(−1)q

≤ 12 −

Truth-tables of functions (−1)Q form the Reed-Muller code of order 2.Want a codeword inside a ball of distance 1/2− ε/2 around f (if oneexists).

(−1)q

≤ 12 −

Q2: Decoding beyond the list-decoding radius

Finding codewords at large distances

2 − ε

- List decoding radius is 14 .

[GKZ 08, Gopalan 10, BL 14]

- Number of codewords withindistance 1

2 − ε may be exponential.

- But we only need to find onecodeword! In time poly(n)(polylogarithmic in code length).

2 − ε

12 − ε

2 − ε

Finding codewords at large distances- Given (the coefficients of) a degree-d polynomial P : Fn

p → Fp, theReed-Muller encoding of P is of length pn and is given by the tableof values {P(x)}x∈Fn

- Problem: Given F : Fnp → Fp, if there exists P ∈ Pd such that

∆(F ,P) ≤ 1− 1p − ε

find a P ′ ∈ Pd such that

∆(F ,P ′) ≤ 1− 1p − η

- If there exists a Reed-Muller codeword within a ball of radius1− 1

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

∆(F ,P) ≤ 1− 1p − ε

∆(F ,P ′) ≤ 1− 1p − η

∆(F ,P) ≤ 1− 1p − ε

∆(F ,P ′) ≤ 1− 1p − η

Finding a single codeword: the quadratic case

12 − ε

12 − η

- [Samorodnitsky 07]: Approximate solutionto testing problem using Gowers norm.

− ∃q⟨f , (−1)Q⟩ ≥ ε =⇒ ‖f ‖U3 ≥ ε

− ‖f ‖U3 ≥ ε =⇒ ∃Q⟨f , (−1)Q⟩ ≥ η(ε)

- [TW 11] convert Samorodnitsky’s proofinto an algorithm. Find codeword withindistance 1

2 − η if there is one within 12 − ε.

- First example of any kind of decodingbeyond the list decoding radius.

12 − ε

12 − η

- [Samorodnitsky 07]: Approximate solutionto testing problem using Gowers norm.

− ∃q⟨f , (−1)Q⟩ ≥ ε =⇒ ‖f ‖U3 ≥ ε

− ‖f ‖U3 ≥ ε =⇒ ∃Q⟨f , (−1)Q⟩ ≥ η(ε)

12 − ε

12 − η

- [Samorodnitsky 07]: Approximate solutionto testing problem using Gowers norm.− ∃q

⟨f , (−1)Q⟩ ≥ ε =⇒ ‖f ‖U3 ≥ ε

− ‖f ‖U3 ≥ ε =⇒ ∃Q⟨f , (−1)Q⟩ ≥ η(ε)

12 − ε1

2 − η

⟨f , (−1)Q⟩ ≥ ε =⇒ ‖f ‖U3 ≥ ε

− ‖f ‖U3 ≥ ε =⇒ ∃Q⟨f , (−1)Q⟩ ≥ η(ε)

12 − ε1

2 − η

⟨f , (−1)Q⟩ ≥ ε =⇒ ‖f ‖U3 ≥ ε

− ‖f ‖U3 ≥ ε =⇒ ∃Q⟨f , (−1)Q⟩ ≥ η(ε)

12 − ε1

2 − η

⟨f , (−1)Q⟩ ≥ ε =⇒ ‖f ‖U3 ≥ ε

− ‖f ‖U3 ≥ ε =⇒ ∃Q⟨f , (−1)Q⟩ ≥ η(ε)

Algorithmic versions of combinatorial theorems

- Samorodnitsky’s proof applies various combinatorial theorems (e.g.Balog-Szemerédi-Gowers) to “nice” subsets of Fn

- [BSG]: If S ⊆ Fn2 satisfies Px ,y∈S [x + y ∈ S] ≥ ε, then there exists

A ⊆ S with certain additive properties.

- S and A are exponential in size. Need to work with randomizedmembership oracles. Gives a noisy version of the set S.

- Modify proofs of combinatorial theorems to go from algorithms inthe hypothesis to algorithms in conclusion.

- Statements of the form: “Given (approximate) membership oraclefor S, it can be converted to an oracle A whose output issandwiched 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 inS ⊆ Fn

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

|A| ≥ εO(1)|S| and |A + A| ≤ ε−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 “almostperiods” i.e., there is a large set X ⊆ Fn

1A ∗ 1A(·) ≈ 1A ∗ 1A(·+ x) ∀x ∈ X

1A ∗ 1A(·) ≈ distribution of sum of two random elements from A.

|A| ≥ εO(1)|S| and |A + A| ≤ ε−O(1)|A|.

1A ∗ 1A(·) ≈ 1A ∗ 1A(·+ x) ∀x ∈ X

|A| ≥ εO(1)|S| and |A + A| ≤ ε−O(1)|A|.

1A ∗ 1A(·) ≈ 1A ∗ 1A(·+ x) ∀x ∈ X

- [Sanders 10]: Stronger inverse theorem for U3-norm using almostperiodicity from [CS 09].

- [BRTW 14]: Sampling-based proof of [CS 09]. Improved quadraticGoldreich-Levin.

- Question: Can sampling based proofs be used to find bettersubspace structure?

Decompositions for higher-degrees

- Question: Given F : Fnp → Fp, does there exist a polynomial P ∈ Pd

such that∣∣⟨ωF , ωP⟩∣∣ ≥ ε? If yes, find one.

≤ p−1p − ε

- Can be solved for the special case when F ∈ Pk and p > k, inversetheorem by [GT 09].

≤ p−1p − ε

Decomposition Theorems and Regularity- [GT 09]: Actually prove a decomposition theorem when F ∈ Pk :

ωF = Γ(P1, . . . ,Pm) + f2

where P1, . . . ,Pm ∈ Pd and ‖f2‖Ud+1 ≤ ε.

- Here, Γ : Fmp → R. By (linear) Fourier analysis

Γ(P1, . . . ,Pm) =∑

c1,...,cm

Γ(c1, . . . , cm) · ω∑

ici Pi

which gives decomposition in the required form.

- Proof by [GT 09] and many other applications require the factorB = {P1, . . . ,Pm} to satisfy certain “regularity” properties.Obtaining regularity is the main challenge in converting their proofto an algorithm.

ωF = Γ(P1, . . . ,Pm) + f2

Γ(P1, . . . ,Pm) =∑

c1,...,cm

Γ(c1, . . . , cm) · ω∑

ici Pi

ωF = Γ(P1, . . . ,Pm) + f2

Γ(P1, . . . ,Pm) =∑

c1,...,cm

Γ(c1, . . . , cm) · ω∑

ici Pi

Polynomial Regularity Lemmas

- Regulariy lemmas for polynomials are useful for several applicationsof higher-order Fourier analysis.

- Analogues of Szemerédi regularity lemma. Regular partition a graphis 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) ∈ Fmp \ {0m},

rankd−1(c1P1 + · · ·+ cmPm) ≥ Λ(m).- [KL 08]: For all (c1, . . . , cm) ∈ Fm

p \ {0m},∑

ciPi and it’sderivatiives have high-rank.

- Polynomial Regularity Lemmas: Given B = {P1, . . . ,Pm}, it can berefined to B′ = {P ′1, . . . ,P ′m′} which is regular.

- Like Szemerédi’s regularity lemma, proofs find a certificate ofnon-regularity and make progress by local modification.

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

rankd−1(c1P1 + · · ·+ cmPm) ≥ Λ(m).- [KL 08]: For all (c1, . . . , cm) ∈ Fm

p \ {0m},∑

ciPi and it’sderivatiives have high-rank.

- 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) ∈ Fmp \ {0m},

∑ciPi and it’s

derivatiives have high-rank.

∑ciPi and it’s

Q3: Algorithmic Regularity Lemmas

Algorithmic notions of regularity

- Algorithmic step in the regularity lemma is finding a certificate ofnon-regularity.

- [BHT 15]: Slightly modified notions of regularity (equivalent up tosome loss of parameters) and corresponding algorithmic lemmas.

‖c1P1 + · · ·+ cmPm‖Ud ≤ δ(m).

∑ciPi and it’s

derivatiives have small-bias.

- Show these notions provide required equidistribution for variousknown applications.

‖c1P1 + · · ·+ cmPm‖Ud ≤ δ(m).

∑ciPi and it’s

‖c1P1 + · · ·+ cmPm‖Ud ≤ δ(m).

∑ciPi and it’s

‖c1P1 + · · ·+ cmPm‖Ud ≤ δ(m).

∑ciPi and it’s

‖c1P1 + · · ·+ cmPm‖Ud ≤ δ(m).

∑ciPi and it’s

Further questions

- Higher-degree decomposition theorems.

- (Approximate) Decoding beyond the list decoding radius for othercodes. 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 guaranteedby the proof?

- Regularity lemmas give terrible quantitative bounds. Is there a wayto use weaker regularity properties and obtain better bounds?

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Algorithmic Questions in Higher-Order Fourier Analysis · 2018-09-20 · Algorithmic Questions in...

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