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Algorithmic Questions inHigher-Order Fourier Analysis
f
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.
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.
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.
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.
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.
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
2
g(α)χα .
- [Parseval]: ‖g‖2 = 〈g , g〉 = Ex[(g(x))2] =
∑α (g(α))2.
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
2
g(α)χα .
- [Parseval]: ‖g‖2 = 〈g , g〉 = Ex[(g(x))2] =
∑α (g(α))2.
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
2
g(α)χα .
- [Parseval]: ‖g‖2 = 〈g , g〉 = Ex[(g(x))2] =
∑α (g(α))2.
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
2
g(α)χα .
- [Parseval]: ‖g‖2 = 〈g , g〉 = Ex[(g(x))2] =
∑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
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
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
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
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
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)
]
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)
]
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)
]
Norms, Shnorms... so what?
- ‖f ‖U2 measures correlation with Fourier characters (linear phasefunctions). (
maxα
∣∣∣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).
Norms, Shnorms... so what?
- ‖f ‖U2 measures correlation with Fourier characters (linear phasefunctions). (
maxα
∣∣∣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).
Norms, Shnorms... so what?
- ‖f ‖U2 measures correlation with Fourier characters (linear phasefunctions). (
maxα
∣∣∣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).
Norms, Shnorms... so what?
- ‖f ‖U2 measures correlation with Fourier characters (linear phasefunctions). (
maxα
∣∣∣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).
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).
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).
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).
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)
)and
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.
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)
)and
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.
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)
)and
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.
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)
)and
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.
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
g
- Use inverse theorem for Gowers norm to get a contradiction.
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
g
- Use inverse theorem for Gowers norm to get a contradiction.
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
g
- Use inverse theorem for Gowers norm to get a contradiction.
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
g
- Use inverse theorem for Gowers norm to get a contradiction.
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
g
- Use inverse theorem for Gowers norm to get a contradiction.
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
g
- Use inverse theorem for Gowers norm to get a contradiction.
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.
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.
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.
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.
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.
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.
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.
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.
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).
f
(−1)q
≤ 12 −
ε2
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).
f
(−1)q
≤ 12 −
ε2
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).
f
(−1)q
≤ 12 −
ε2
Q2: Decoding beyond the list-decoding radius
Finding codewords at large distances
f
18
14 1
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).
Finding codewords at large distances
f18
14 1
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).
Finding codewords at large distances
f18
14
12 − 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).
Finding codewords at large distances
f18
14 1
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).
Finding codewords at large distances
f18
14 1
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).
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
p.
- 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.
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
p.
- 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.
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
p.
- 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.
Finding a single codeword: the quadratic case
f
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.
Finding a single codeword: the quadratic case
f
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.
Finding a single codeword: the quadratic case
f
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.
Finding a single codeword: the quadratic case
f
12 − ε1
2 − 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.
Finding a single codeword: the quadratic case
f
12 − ε1
2 − 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.
Finding a single codeword: the quadratic case
f
12 − ε1
2 − 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.
Algorithmic versions of combinatorial theorems
Fn2S
A
- Samorodnitsky’s proof applies various combinatorial theorems (e.g.Balog-Szemerédi-Gowers) to “nice” subsets of Fn
2.
- [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.
Algorithmic versions of combinatorial theorems
Fn2S
A
- Samorodnitsky’s proof applies various combinatorial theorems (e.g.Balog-Szemerédi-Gowers) to “nice” subsets of Fn
2.
- [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.
Algorithmic versions of combinatorial theorems
Fn2S
A
- Samorodnitsky’s proof applies various combinatorial theorems (e.g.Balog-Szemerédi-Gowers) to “nice” subsets of Fn
2.
- [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.
Algorithmic versions of combinatorial theorems
Fn2S
A
- Samorodnitsky’s proof applies various combinatorial theorems (e.g.Balog-Szemerédi-Gowers) to “nice” subsets of Fn
2.
- [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.
Algorithmic versions of combinatorial theorems
Fn2S
A1
A2
- 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.
Algorithmic versions of combinatorial theorems
Fn2S
A1
A2
- 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.
Algorithmic versions of combinatorial theorems
Fn2S
A1
A2
- 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
2.
- [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
2 s.t
1A ∗ 1A(·) ≈ 1A ∗ 1A(·+ x) ∀x ∈ X
1A ∗ 1A(·) ≈ distribution of sum of two random elements from A.
Finding subspace structure
Most combinatorial results used here find and refine subspace structure inS ⊆ Fn
2.
- [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
2 s.t
1A ∗ 1A(·) ≈ 1A ∗ 1A(·+ x) ∀x ∈ X
1A ∗ 1A(·) ≈ distribution of sum of two random elements from A.
Finding subspace structure
Most combinatorial results used here find and refine subspace structure inS ⊆ Fn
2.
- [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
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 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?
Finding subspace structure
- [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?
Finding subspace structure
- [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.
f
P
≤ p−1p − ε
- Can be solved for the special case when F ∈ Pk and p > k, inversetheorem by [GT 09].
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.
f
P
≤ p−1p − ε
- Can be solved for the special case when F ∈ Pk and p > k, inversetheorem by [GT 09].
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.
f
P
≤ p−1p − ε
- Can be solved for the special case when F ∈ Pk and p > k, inversetheorem 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 ‖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.
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.
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.
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.
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.
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) ∈ 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.
- 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.
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) ∈ 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.
- 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.
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) ∈ 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.
- 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.
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.
- [GT 09]: For all (c1, . . . , cm) ∈ Fmp \ {0m},
‖c1P1 + · · ·+ cmPm‖Ud ≤ δ(m).
- [KL 08]: For all (c1, . . . , cm) ∈ Fmp \ {0m},
∑ciPi and it’s
derivatiives have small-bias.
- Show these notions provide required equidistribution for variousknown applications.
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.
- [GT 09]: For all (c1, . . . , cm) ∈ Fmp \ {0m},
‖c1P1 + · · ·+ cmPm‖Ud ≤ δ(m).
- [KL 08]: For all (c1, . . . , cm) ∈ Fmp \ {0m},
∑ciPi and it’s
derivatiives have small-bias.
- Show these notions provide required equidistribution for variousknown applications.
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.
- [GT 09]: For all (c1, . . . , cm) ∈ Fmp \ {0m},
‖c1P1 + · · ·+ cmPm‖Ud ≤ δ(m).
- [KL 08]: For all (c1, . . . , cm) ∈ Fmp \ {0m},
∑ciPi and it’s
derivatiives have small-bias.
- Show these notions provide required equidistribution for variousknown applications.
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.
- [GT 09]: For all (c1, . . . , cm) ∈ Fmp \ {0m},
‖c1P1 + · · ·+ cmPm‖Ud ≤ δ(m).
- [KL 08]: For all (c1, . . . , cm) ∈ Fmp \ {0m},
∑ciPi and it’s
derivatiives have small-bias.
- Show these notions provide required equidistribution for variousknown applications.
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.
- [GT 09]: For all (c1, . . . , cm) ∈ Fmp \ {0m},
‖c1P1 + · · ·+ cmPm‖Ud ≤ δ(m).
- [KL 08]: For all (c1, . . . , cm) ∈ Fmp \ {0m},
∑ciPi and it’s
derivatiives have small-bias.
- Show these notions provide required equidistribution for variousknown applications.
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?
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?
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?
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?
Thank You
Questions?