non-interactivesimulationand …computing sdp integrality gaps for csps [raghavendra-steurer ’09]...

Non-Interactive Simulation andDimension Reduction for Polynomials

Pritish Kamath

joint work with

BadihGhazi

PrasadRaghavendra

CCCUCSD

June 24,2018

1 / 9

Talk outline...

• Motivation

• Motivation

• Motivation

• “Dimension Reduction for Polynomials” lemma

• Summary & Open Directions!

2 / 9

Talk outline...

• Motivation

• Motivation

• Motivation

• “Dimension Reduction for Polynomials” lemma

• Summary & Open Directions!

•

2 / 9

Randomness Models in Distributed Tasks

randomnessrandomness randomnessrandomness randomness

0

1

0

1

(1−ε)2

(1−ε)2

ε/2

ε/2X Y

P(X, Y)In Information Theory . . .

▶ Common Information

[Gács-Körner ’73, Wyner ’75]

▶ Distributed Source Coding

[Slepian-Wolf ’73]

▶ · · ·

In Computer Science . . .

▶ Information Theoretic Crypto!

Key Agreement, Secure Computation, … ?

3 / 9


randomness

randomness randomnessrandomness randomness

0

1

0

1

(1−ε)2

(1−ε)2

ε/2

ε/2X Y






▶ · · ·




3 / 9


randomness

randomness randomness


0

1

0

1

(1−ε)2

(1−ε)2

ε/2

ε/2X Y






▶ · · ·




3 / 9


randomnessrandomness randomness


0

1

0

1

(1−ε)2

(1−ε)2

ε/2

ε/2X Y






▶ · · ·




3 / 9




0

1

0

1

(1−ε)2

(1−ε)2

ε/2

ε/2

X Y






▶ · · ·




3 / 9




0

1

0

1

(1−ε)2

(1−ε)2

ε/2

ε/2

X Y

P(X, Y)

In Information Theory . . .





▶ · · ·




3 / 9




0

1

0

1

(1−ε)2

(1−ε)2

ε/2

ε/2

X Y






▶ · · ·




3 / 9




0

1

0

1

(1−ε)2

(1−ε)2

ε/2

ε/2X Y

P(X, Y)

In Information Theory . . .





▶ · · ·




3 / 9




0

1

0

1

(1−ε)2

(1−ε)2

ε/2

ε/2

X Y






▶ · · ·




▶ Communication Complexity

[Bavarian-Gavinsky-Ito ’14]

[Canonne-Guruswami-Meka-Sudan ’15]

3 / 9




0

1

0

1

(1−ε)2

(1−ε)2

ε/2

ε/2

X Y






▶ · · ·




▶ Communication Complexity

[Bavarian-Gavinsky-Ito ’14]

[Canonne-Guruswami-Meka-Sudan ’15]

Abstract Goal:

Understand the power ofdifferent joint distributions!

3 / 9

Non-Interactive Simulation of Joint Distributions

X YP(X, Y)

a ba, b ∈ {0, 1}

(a, b) ∼ BSSδ

When can simulate BSSδ?

Main Question

How can a “constant-sized”problem be HARD?

4 / 9


X YP(X, Y)

BSSε

0

1

0

1

(1−ε)2

ε/2

ε/2(1−ε)

2

a ba, b ∈ {0, 1}

(a, b) ∼ BSSδ

1 0 0 1 0 1 0 0 1 0 1 1 · · ·

1 0 1 1 0 1 1 0 0 0 1 1 · · ·

When can simulate BSSδ?

Main Question


4 / 9


X YP(X, Y)

BSSε

0

1

0

1

(1−ε)2

ε/2

ε/2(1−ε)

2

a ba, b ∈ {0, 1}

(a, b) ∼ BSSδ

1 0 0 1 0 1 0 0 1 0 1 1 · · ·

1 0 1 1 0 1 1 0 0 0 1 1 · · ·

When can BSSε simulate BSSδ?

Main Question


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X YP(X, Y)

BSSε

0

1

0

1

(1−ε)2

ε/2

ε/2(1−ε)

2

a ba, b ∈ {0, 1}

(a, b) ∼ BSSδ

1 0 0 1 0 1 0 0 1 0 1 1 · · ·

1 0 1 1 0 1 1 0 0 0 1 1 · · ·

Answer:YES δ ≥ ε

NO δ < ε

When can BSSε simulate BSSδ?

Main Question


4 / 9


X YP(X, Y)

DISJ

0

1

0

1

1/31/3 1/3

a ba, b ∈ {0, 1}

(a, b) ∼ BSSδ

1 0 0 0 0 0 0 0 1 0 1 1 · · ·

0 1 0 0 0 1 0 1 0 1 0 0 · · ·

When can DISJ simulate BSSδ?

Main Question


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X YP(X, Y)

DISJ

0

1

0

1

1/31/3 1/3

a ba, b ∈ {0, 1}

(a, b) ∼ BSSδ

1 0 0 0 0 0 0 0 1 0 1 1 · · ·

0 1 0 0 0 1 0 1 0 1 0 0 · · ·

(Partial)Answer:

YES δ ≥ 38

OPEN δ ∈[

14 , 3

8

)NO δ < 1

4

When can DISJ simulate BSSδ?

Main Question


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X YP(X, Y)

a ba, b ∈ {0, 1}

(a, b) ∼ BSSδ

X1, X2, X3, X4, X5, . . .

Y1, Y2, Y3, Y4, Y5, . . .

When can P simulate BSSδ?

Main Question


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X YP(X, Y)

a ba, b ∈ [k]

(a, b) ∼ Q

X1, X2, X3, X4, X5, . . .

Y1, Y2, Y3, Y4, Y5, . . .

When can P simulate Q?

Main Question


4 / 9


X YP(X, Y)

a ba, b ∈ [k]

(a, b) ∼ Q

X1, X2, X3, X4, X5, . . .

Y1, Y2, Y3, Y4, Y5, . . .

When can P simulate Q?

Main Question

Analytically? OPEN in most cases!

Algorithmically decidable? Not obvious!


4 / 9

Tensor Power Problems

Non-interactive Simulation falls under the category of

“Tensor Power” problems.

5 / 9




In Information Theory,

▶ Zero-error Shannon capacity

▶ Zero-error Witsenhausen rate

5 / 9







In Computer Science,

▶ (Classical) Amortized value of 2-prover 1-round games

▶ (Quantum) Entangled value of 2-prover 1-round games

▶ (Quantum) Local State Transformation

▶ Computing SDP integrality gaps for CSPs

▶ Amortized communication complexity

5 / 9








▶ (Classical) Amortized value of 2-prover 1-round games

▶ (Quantum) Entangled value of 2-prover 1-round games

▶ (Quantum) Local State Transformation

▶ Computing SDP integrality gaps for CSPs [Raghavendra-Steurer ’09]

▶ Amortized communication complexity = Information complexity[Braverman-Rao ’11], [Braverman-Schneider ’15]

5 / 9





▶ Zero-error Shannon capacity [Open]

▶ Zero-error Witsenhausen rate [Open]


▶ (Classical) Amortized value of 2-prover 1-round games [Open]

▶ (Quantum) Entangled value of 2-prover 1-round games [Open]

▶ (Quantum) Local State Transformation [Open]

▶ Computing SDP integrality gaps for CSPs [Raghavendra-Steurer ’09]

▶ Amortized communication complexity = Information complexity[Braverman-Rao ’11], [Braverman-Schneider ’15]

5 / 9

Decidability via “Dimension Reduction”

Can P simulate Q?

Main Question

6 / 9


Xn Yn

f

[k]

g

[k]

Can P simulate Q?

Main Question

( f (Xn), g(Yn)) ∼ Q for (Xn, Yn) ∼ Pn

6 / 9


Xn Yn

f

[k]

g

[k]

Dimension Reduction

Xn0 Yn0

f

[k]

g

[k]

Can P simulate Q?

Main Question


( f (Xn0 ), g(Yn0 )) ≈ε ( f (Xn), g(Yn))

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Xn Yn

f

[k]

g

[k]

Dimension Reduction

Xn0 Yn0

f

[k]

g

[k]

Can P simulate Q?

Main Question


If P can simulate Q . . .

( f (Xn0 ), g(Yn0 )) ≈ε ( f (Xn), g(Yn))

. . . then P can ε-approximately simulate Qwith only n0 samples

6 / 9


Xn Yn

f

[k]

g

[k]

Dimension Reduction

Xn0 Yn0

f

[k]

g

[k]

Can P simulate Q?

Main Question



( f (Xn0 ), g(Yn0 )) ≈ε ( f (Xn), g(Yn))


Key point: n0 = n0(ε, P, k) is

explicit & does not depend on n.

6 / 9


Xn Yn

f

[k]

g

[k]

Dimension Reduction

Xn0 Yn0

f

[k]

g

[k]

Can P simulate Q?

Main Question



( f (Xn0 ), g(Yn0 )) ≈ε ( f (Xn), g(Yn))




DECIDABLE

6 / 9


Xn Yn

f

[k]

g

[k]

Dimension Reduction

Xn0 Yn0

f

[k]

g

[k]

Can P simulate Q?

Main Question



( f (Xn0 ), g(Yn0 )) ≈ε ( f (Xn), g(Yn))




DECIDABLE

Any distributed task performed with unbounded amounts of correlated randomness . . .. . . can also be approximately performed with an explicitly bounded number of samples!

Main “take-away” Theorem

6 / 9


Xn Yn

f

[k]

g

[k]

Dimension Reduction

Xn0 Yn0

f

[k]

g

[k]

Can P simulate Q?

Main Question



( f (Xn0 ), g(Yn0 )) ≈ε ( f (Xn), g(Yn))




DECIDABLE


(extends to interactive settings even with inputs!)


6 / 9

Non-Interactive Agreement Distillation

Xn YnP⊗n

f (Xn) g(Yn)

f (Xn), g(Yn) ∈ [k]

supn, f ,g

Pr[ f (Xn) = g(Yn)] ?

E[ f ] = ( 1k , . . . , 1

k ) and E[g] = ( 1k , . . . , 1

k ).

Max Agreement Distillation

For convenience, f : Xn 7→ Rk

where, i ∈ [k] corresponds to ei .

Borell’s Theorem [Bor85]

10

f (Xn) = sign(X1)

10

g(Yn) = sign(Y1)

“Halfspaces are most Noise Stable”

“Majority is Stablest” [MOO’04, Mos’10]

0

1

2

f (X1, X2)

0

1

2

g(Y1, Y2)

“Peace Sign Conjecture”

“Plurality is Stablest” [KKMO’04, IM’12]

7 / 9

Non-Interactive Agreement from Correlated Gaussians

Xn YnG⊗n

ρ

N([

00

],

[1 ρ

ρ 1

])

f (Xn) g(Yn)

f (Xn), g(Yn) ∈ [k]

supn, f ,g

Pr[ f (Xn) = g(Yn)] ?

E[ f ] = ( 1k , . . . , 1

k ) and E[g] = ( 1k , . . . , 1

k ).


For convenience, f : Xn 7→ Rk

where, i ∈ [k] corresponds to ei .


10

f (Xn) = sign(X1)

10

g(Yn) = sign(Y1)



0

1

2

f (X1, X2)

0

1

2

g(Y1, Y2)



7 / 9


Xn YnG⊗n

ρ

N([

00

],

[1 ρ

ρ 1

])

f (Xn) g(Yn)

f (Xn), g(Yn) ∈ [k]

k = 2

supn, f ,g

Pr[ f (Xn) = g(Yn)] ?

E[ f ] = ( 12 , 1

2 ) and E[g] = ( 12 , 1

2 ).



10

f (Xn) = sign(X1)

10

g(Yn) = sign(Y1)



0

1

2

f (X1, X2)

0

1

2

g(Y1, Y2)



7 / 9


Xn YnG⊗n

ρ

N([

00

],

[1 ρ

ρ 1

])

f (Xn) g(Yn)

f (Xn), g(Yn) ∈ [k]

k = 2

supn, f ,g

Pr[ f (Xn) = g(Yn)] ?

E[ f ] = (0.3, 0.7) and E[g] = (0.6, 0.4).



10

f (Xn) = sign(X1 − α)

10

g(Yn) = sign(Y1 − β)



Generalizes to non-uniform marginals!

0

1

2

f (X1, X2)

0

1

2

g(Y1, Y2)



7 / 9


Xn YnG⊗n

ρ

N([

00

],

[1 ρ

ρ 1

])

f (Xn) g(Yn)

f (Xn), g(Yn) ∈ [k]

k = 3

supn, f ,g

Pr[ f (Xn) = g(Yn)] ?

E[ f ] = ( 13 , 1

3 , 13 ) and E[g] = ( 1

3 , 13 , 1

3 ).



10

f (Xn) = sign(X1)

10

g(Yn) = sign(Y1)




0

1

2

f (X1, X2)

0

1

2

g(Y1, Y2)



7 / 9


Xn YnG⊗n

ρ

N([

00

],

[1 ρ

ρ 1

])

f (Xn) g(Yn)

f (Xn), g(Yn) ∈ [k]

k = 3

supn, f ,g

Pr[ f (Xn) = g(Yn)] ?

E[ f ] = ( 13 , 1

3 , 13 ) and E[g] = ( 1

3 , 13 , 1

3 ).



10

f (Xn) = sign(X1)

10

g(Yn) = sign(Y1)




0

1

2

f (X1, X2)

0

1

2

g(Y1, Y2)



Generalize to non-uniform marginals? FALSE![HMN’16]

7 / 9


Xn YnG⊗n

ρ

N([

00

],

[1 ρ

ρ 1

])

f (Xn) g(Yn)

f (Xn), g(Yn) ∈ [k]

k = 3

supn, f ,g

Pr[ f (Xn) = g(Yn)] ?

E[ f ] = ( 13 , 1

3 , 13 ) and E[g] = ( 1

3 , 13 , 1

3 ).



10

f (Xn) = sign(X1)

10

g(Yn) = sign(Y1)




0

1

2

f (X1, X2)

0

1

2

g(Y1, Y2)




■ “Peace Sign Conjecture” implies optimal

strategy exists with 2 samples.

Q1. Does optimal strategy even exist with

some finite #samples?

Q2. How many samples n0(ε) needed to

get ε-close to optimal agreement?

Can we obtain an “explicit” bound?

[De-Mossel-Neeman’17]

7 / 9


Xn YnG⊗n

ρ

N([

00

],

[1 ρ

ρ 1

])

f (Xn) g(Yn)

f (Xn), g(Yn) ∈ [k]

k = 3

supn, f ,g

Pr[ f (Xn) = g(Yn)] ?

E[ f ] = ( 13 , 1

3 , 13 ) and E[g] = ( 1

3 , 13 , 1

3 ).



10

f (Xn) = sign(X1)

10

g(Yn) = sign(Y1)




0

1

2

f (X1, X2)

0

1

2

g(Y1, Y2)




■ “Peace Sign Conjecture” implies optimal

strategy exists with 2 samples.

Q1. Does optimal strategy even exist with

some finite #samples?

Q2. How many samples n0(ε) needed to

get ε-close to optimal agreement?


[De-Mossel-Neeman’17]

Any distributed task performed with unbounded amounts of correlated randomness . . .

. . . can also be approximately performed with an explicitly bounded number of samples!


7 / 9


Xn YnG⊗n

ρ

N([

00

],

[1 ρ

ρ 1

])

f (Xn) g(Yn)

f (Xn), g(Yn) ∈ [k]

( f (Xn0 ), g(Yn0 )) ≈ε ( f (Xn), g(Yn))

Xn Yn

f

[k]

g

[k]

Dimension Reduction

Xn0 Yn0

f

[k]

g

[k]

supn, f ,g

Pr[ f (Xn) = g(Yn)] ?

E[ f ] = ( 1k , . . . , 1

k ) and E[g] = ( 1k , . . . , 1

k ).



10

f (Xn) = sign(X1)

10

g(Yn) = sign(Y1)




0

1

2

f (X1, X2)

0

1

2

g(Y1, Y2)




How many samples n0(ε) needed toget ε-close to the optimal agreement?


7 / 9


Xn YnG⊗n

ρ

N([

00

],

[1 ρ

ρ 1

])

f (Xn) g(Yn)

f (Xn), g(Yn) ∈ [k]

( f (Xn0 ), g(Yn0 )) ≈ε ( f (Xn), g(Yn))

Xn Yn

f

[k]

g

[k]

Dimension Reduction

Xn0 Yn0

f

[k]

g

[k]

supn, f ,g

Pr[ f (Xn) = g(Yn)] ?

E[ f ] = ( 1k , . . . , 1

k ) and E[g] = ( 1k , . . . , 1

k ).



10

f (Xn) = sign(X1)

10

g(Yn) = sign(Y1)




0

1

2

f (X1, X2)

0

1

2

g(Y1, Y2)






Gaussian Case: (X, Y) ∼ Gρ

[De-Mossel-Neeman’17, ’18] n0 = Ackermann(?)

[This Work!] n0 = exp(

k, 1ε , 1

1−ρ

)

7 / 9


Xn YnG⊗n

ρ

N([

00

],

[1 ρ

ρ 1

])

f (Xn) g(Yn)

f (Xn), g(Yn) ∈ [k]

( f (Xn0 ), g(Yn0 )) ≈ε ( f (Xn), g(Yn))

Xn Yn

f

[k]

g

[k]

Dimension Reduction

Xn0 Yn0

f

[k]

g

[k]

supn, f ,g

Pr[ f (Xn) = g(Yn)] ?

E[ f ] = ( 1k , . . . , 1

k ) and E[g] = ( 1k , . . . , 1

k ).



10

f (Xn) = sign(X1)

10

g(Yn) = sign(Y1)




0

1

2

f (X1, X2)

0

1

2

g(Y1, Y2)









k, 1ε , 1

1−ρ

)General Case: (X, Y) ∼ P[Ghazi-K-Sudan’16] : Reduces∗ to Gaussian case!Using Regularity Lemma and Invariance Principle.

Solved k = 2 case, due to Borell’s theorem!

7 / 9


Xn YnG⊗n

ρ

N([

00

],

[1 ρ

ρ 1

])

f (Xn) g(Yn)

f (Xn), g(Yn) ∈ [k]

( f (Xn0 ), g(Yn0 )) ≈ε ( f (Xn), g(Yn))

Xn Yn

f

[k]

g

[k]

Dimension Reduction

Xn0 Yn0

f

[k]

g

[k]

supn, f ,g

Pr[ f (Xn) = g(Yn)] ?

E[ f ] = ( 1k , . . . , 1

k ) and E[g] = ( 1k , . . . , 1

k ).



10

f (Xn) = sign(X1)

10

g(Yn) = sign(Y1)




0

1

2

f (X1, X2)

0

1

2

g(Y1, Y2)









k, 1ε , 1

1−ρ

)General Case: (X, Y) ∼ P[Ghazi-K-Sudan’16] : Reduces∗ to Gaussian case![De-Mossel-Neeman’18] n0 = Ackermann(?)[This Work!] n0 = exp

(k, 1

ε , 11−ρ , log 1

α

)

7 / 9

Main Technique!

8 / 9

Main Technique!

fi : Rn → R gi : Rn → R

fi gj

{ f1, f2, f3} {g1, g2, g3}

Dimension Reduction

fi : Rn0 → R gi : Rn0 → R

fi gj

{f1, f2, f3

}{g1, g2, g3}

⟨fi, gj

⟩G⊗n

ρ

≈ε

⟨fi, gj

⟩G⊗n0

ρ

⟨fi, gj

⟩G⊗n

ρ

:= EX,Y∼G⊗n

ρ

[ fi(X)gj(Y)]

8 / 9

Main Technique!


fi gj

{ f1, f2, f3} {g1, g2, g3}

Dimension Reduction


fi gj

{f1, f2, f3

}{g1, g2, g3}

⟨fi, gj

⟩G⊗n

ρ

≈ε

⟨fi, gj

⟩G⊗n0

ρ

ui : [n]→ R vj : [n]→ R

{u1, u2, u3} {v1, v2, v3}

Johnson-Lindenstrauss

ui : [n0 ]→ R vj : [n0 ]→ R

{u1, u2, u3} {v1, v2, v3}

⟨ui, vj

⟩Rn≈ε

⟨ui, vj

⟩Rn0

8 / 9

Main Technique!


fi gj

{ f1, f2, f3} {g1, g2, g3}

Dimension Reduction


fi gj

{f1, f2, f3

}{g1, g2, g3}

⟨fi, gj

⟩G⊗n

ρ

≈ε

⟨fi, gj

⟩G⊗n0

ρ

ui : [n]→ R vj : [n]→ R

{u1, u2, u3} {v1, v2, v3}

M ∼ N (0, 1)n0×n

ui ← Mui√n0

vj ←Mvj√

n0

ui : [n0 ]→ R vj : [n0 ]→ R

{u1, u2, u3} {v1, v2, v3}

⟨ui, vj

⟩Rn≈ε

⟨ui, vj

⟩Rn0

8 / 9

Main Technique!


fi gj

{ f1, f2, f3} {g1, g2, g3}

Dimension Reduction


fi gj

{f1, f2, f3

}{g1, g2, g3}

⟨fi, gj

⟩G⊗n

ρ

≈ε

⟨fi, gj

⟩G⊗n0

ρ

ui : [n]→ R vj : [n]→ R

{u1, u2, u3} {v1, v2, v3}

M ∼ N (0, 1)n0×n

ui ← Mui√n0

vj ←Mvj√

n0

EM[⟨

ui , vj⟩]

=⟨ui , vj

⟩VarM

(⟨ui , vj

⟩)< ε2

n0 = O( 1ε2 )⟨

ui, vj

⟩Rn≈ε

⟨ui, vj

⟩Rn0

8 / 9

Main Technique!


fi gj

{ f1, f2, f3} {g1, g2, g3}

M ∼ N (0, 1)n×n0

fi(a)← fi

(Ma√

n0

)gj(b)← gj

(Mb√

n0

)


fi gj

{f1, f2, f3

}{g1, g2, g3}

⟨fi, gj

⟩G⊗n

ρ

≈ε

⟨fi, gj

⟩G⊗n0

ρ

ui : [n]→ R vj : [n]→ R

{u1, u2, u3} {v1, v2, v3}

M ∼ N (0, 1)n0×n

ui ← Mui√n0

vj ←Mvj√

n0

EM[⟨

ui , vj⟩]

=⟨ui , vj

⟩VarM

(⟨ui , vj

⟩)< ε2

n0 = O( 1ε2 )⟨

ui, vj

⟩Rn≈ε

⟨ui, vj

⟩Rn0

8 / 9

Main Technique!


fi gj

{ f1, f2, f3} {g1, g2, g3}

M ∼ N (0, 1)n×n0

fi(a)← fi

(Ma√

n0

)gj(b)← gj

(Mb√

n0

)

EM

[⟨fi , gj

⟩]=⟨

fi , gj⟩

VarM

(⟨fi , gj

⟩)< ε2

?

⟨fi, gj

⟩G⊗n

ρ

≈ε

⟨fi, gj

⟩G⊗n0

ρ

ui : [n]→ R vj : [n]→ R

{u1, u2, u3} {v1, v2, v3}

M ∼ N (0, 1)n0×n

ui ← Mui√n0

vj ←Mvj√

n0

EM[⟨

ui , vj⟩]

=⟨ui , vj

⟩VarM

(⟨ui , vj

⟩)< ε2

n0 = O( 1ε2 )⟨

ui, vj

⟩Rn≈ε

⟨ui, vj

⟩Rn0

8 / 9

Main Technique!


fi gj

{ f1, f2, f3} {g1, g2, g3}

M ∼ N (0, 1)n×n0

fi(a)← fi

(Ma√

n0

)gj(b)← gj

(Mb√

n0

)

EM

[⟨fi , gj

⟩]≈ε

⟨fi , gj

⟩VarM

(⟨fi , gj

⟩)< ε2

If f and g are multilinear, degree d,

n0 = dO(d)

ε2⟨fi, gj

⟩G⊗n

ρ

≈ε

⟨fi, gj

⟩G⊗n0

ρ

ui : [n]→ R vj : [n]→ R

{u1, u2, u3} {v1, v2, v3}

M ∼ N (0, 1)n0×n

ui ← Mui√n0

vj ←Mvj√

n0

EM[⟨

ui , vj⟩]

=⟨ui , vj

⟩VarM

(⟨ui , vj

⟩)< ε2

n0 = O( 1ε2 )⟨

ui, vj

⟩Rn≈ε

⟨ui, vj

⟩Rn0

8 / 9

Main Technique!


fi gj

{ f1, f2, f3} {g1, g2, g3}

M ∼ N (0, 1)n×n0

fi(a)← fi

(Ma∥a∥2

)gj(b)← gj

(Mb∥b∥2

)

EM

[⟨fi , gj

⟩]≈ε

⟨fi , gj

⟩VarM

(⟨fi , gj

⟩)< ε2


n0 = dO(d)

ε2⟨fi, gj

⟩G⊗n

ρ

≈ε

⟨fi, gj

⟩G⊗n0

ρ

ui : [n]→ R vj : [n]→ R

{u1, u2, u3} {v1, v2, v3}

M ∼ N (0, 1)n0×n

ui ← Mui√n0

vj ←Mvj√

n0

EM[⟨

ui , vj⟩]

=⟨ui , vj

⟩VarM

(⟨ui , vj

⟩)< ε2

n0 = O( 1ε2 )⟨

ui, vj

⟩Rn≈ε

⟨ui, vj

⟩Rn0

8 / 9

Main Technique: Dimension Reduction for Polynomials!


fi gj

{ f1, f2, f3} {g1, g2, g3}

M ∼ N (0, 1)n×n0

fi(a)← fi

(Ma∥a∥2

)gj(b)← gj

(Mb∥b∥2

)

EM

[⟨fi , gj

⟩]≈ε

⟨fi , gj

⟩VarM

(⟨fi , gj

⟩)< ε2


n0 = dO(d)

ε2⟨fi, gj

⟩G⊗n

ρ

≈ε

⟨fi, gj

⟩G⊗n0

ρ

ui : [n]→ R vj : [n]→ R

{u1, u2, u3} {v1, v2, v3}

M ∼ N (0, 1)n0×n

ui ← Mui√n0

vj ←Mvj√

n0

EM[⟨

ui , vj⟩]

=⟨ui , vj

⟩VarM

(⟨ui , vj

⟩)< ε2

n0 = O( 1ε2 )⟨

ui, vj

⟩Rn≈ε

⟨ui, vj

⟩Rn0

8 / 9

Main Technique: Dimension Reduction for Polynomials!


fi gj

{ f1, f2, f3} {g1, g2, g3}

M ∼ N (0, 1)n×n0

fi(a)← fi

(Ma∥a∥2

)gj(b)← gj

(Mb∥b∥2

)

EM

[⟨fi , gj

⟩]≈ε

⟨fi , gj

⟩VarM

(⟨fi , gj

⟩)< ε2


n0 = dO(d)

ε2⟨fi, gj

⟩G⊗n

ρ

≈ε

⟨fi, gj

⟩G⊗n0

ρ

▶ Don’t care about seed length of M.

▶ Crucially, preserves other statisticalproperties!

Ma∥a∥2∼ N (0, 1)⊗n

Comparison with [Kane-Rao ’18]

Thanks to Sankeerth Rao

& Mitali Bafna!

8 / 9

Summary &Open Questions . . .

▶ Lower bounds on randomness reduction? Better upper bounds?

▶ Other applications of dimension reduction for polynomials?▶ Derandomization of the dimension reduction lemma?

▶ (NP-)hardness of deciding Non-Interactive Simulation?

▶ Other Tensor Power problems?

OpenQuestions

Thanks!Questions?

9 / 9









OpenQuestions

Thanks!Questions?

9 / 9




Proof via dimension reduction for low-degree multilinear polynomials






OpenQuestions

Thanks!Questions?

9 / 9

non-interactivesimulationand …computing sdp integrality gaps for csps [raghavendra-steurer ’09]...

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