Lossless Dimension Expanders via Linearized Polynomials and - - PowerPoint PPT Presentation

Lossless Dimension Expanders via Linearized Polynomials and Subspace Designs

Venkat Guruswami, Nicolas Resch and Chaoping Xing

Algebraic Pseudorandomness

´ Traditional pseudorandom objects (e.g., expander graphs, randomness extractors, pseudorandom generators, list- decodable codes etc.) are largely combinatorial objects. ´ Algebraic pseudorandom objects have recently been studied. Here, dimension of subspaces corresponds to subset size. ´ Examples include dimension expanders, subspace designs, subspace-evasive sets, rank-preserving condensers, list- decodable rank-metric codes. ´ Applications include constructions of Ramsey graphs [PR’04], list- decodable codes [GX’12, GX’13, GW’14], affine extractors [Gab’11], polynomial identity testing [KS’11, FS’12].

Dimension Expander

´ An !, # -dimension expander is a collection of linear maps Γ%, … , Γ': )* → )* such that, for any , ⊆ )* of dimension at most !., we have

dim 2

34% '

Γ

3 ,

≥ # dim, .

´ The degree is 7. ´ For !# < 1 and 7 = ;(1), a random collection of maps will be a dimension expander with good probability; goal is to obtain an explicit construction.

!" !" # Γ%(#) Γ( # Γ)(#) Γ*(#) + ,

• Γ
• (+)

History

´ Wigderson defined problem in 2004. Authors Parameters Field Restriction

Lubotzky–Zelmanov, Harrow, Ben-Aroya–Ta- Shma ‘08 1 2 , 1 + Ω 1 &, ' Bourgain–Yehudayoff ‘13 1 2 , 1 + Ω 1 None Forbes–Guruswami ‘15 1 Ω ( , Ω ( ) ≥ Ω(,-)

Our results

• Theorem. For any ! > 0, there exists $ = $(!) such that there

is an explicit

()* + , 1 − ! $ -dimension expander of degree

$ over /0

1 when 2 ≥ Ω(5). Since dim(∑:;(

+

Γ

:(=)) ≤ $ dim =, ? =

1 − ! $ is optimal. We call the expander lossless. Also, @ =

()* +

is optimal. $ = A

( *B , whereas random gets $ = A ( *C .

Our results

• Theorem. For any ! > 0, there exists $ = $(!) such that there

is an explicit

()* + , 1 − ! $ -dimension expander of degree

$ over /0

1 when 2 ≥ Ω(5).

• Theorem. For any 6 > 0, there exists $ = $(6) such that

there is an explicit

( 7(8+) ,Ω(6$) -dimension expander of

degree $ over /0

1 when 2 ≥ Ω 58 .

Linearized Polynomials

´ A linearized polynomial is a polynomial in !"#[%] of the form

' % = )

*+,

• ./

'

*%"0

´ Max 1 such that '

* ≠ 0 is the 4-degree of '(%).

´ Denote set of all linearized polynomials of 4-degree < 8 by !"# %; ⋅ " ;-. ´ If <, > ∈ !"@ and A, B ∈ !", then ' A< + B> = A' < + B'(>). That is, as a map !"# → !"#, ' is !"-linear.

• Fact. If ' ∈ !"# %; ⋅ " ;- ∖ {0}, then dim!K(ker ') ≤ 8 − 1.

The Construction

´ Fix !", … , !%, a basis for &'/&), where ℎ = ,% and - is degree. ´ Fix . ⊆ &)0 1; ⋅ ) 45 of &)-dimension 6. ´ For 7 = 1, … , -, define

Γ

::. → &)0

by = ↦ =(!

:) . &)0 &' &) A

• Intuitively, we’d like to show that

subspaces of dimension B are expanded to subspaces of dimension

• − D + 1 B.

Contrapositive Characterization

´ Thus, for ! ⊆ #$%, we need to understand & ∈ (: ∀+ ∈ , , & ./ ∈ ! = & ∈ (: & #1 ⊆ ! . ´ First, let us study & ∈ #$% 2; ⋅ $ 56: & #1 ⊆ ! .

• Proposition. Let Γ8, … , Γ:: #; → #;. Suppose that ∀ ! ⊆ #; s.t.

dim ! ≤ ABC, we have dim D ∈ #;: ∀ + ∈ , , Γ

/ D ∈ ! ≤ 8 E dim ! .

Then {Γ

/: + ∈ [,]} forms a (A, B)-dimension expander.

Aside: Connection to Coding Theory

´ The condition we study is like a “rank-metric list-recovery” problem. ´ [Guruswami-Wang-Xing ‘16] recently provided an explicit construction of a rank-metric code list-decodable up to the Singleton bound. ´ Our construction is similar, but the parameter regime is sufficiently different that we require novel constructions (particularly, the subspace design). ´ This is very much akin to [Guruswami-Umans-Vadhan ‘08], where an explicit construction of bipartite expander graphs were obtained from Parvaresh-Vardy codes.

Interpolation

´ Recall: we want to understand ! ∈ #$ %; ⋅ $ (): ! #+ ⊆ - . ´ For integers ., 0 ≤ 2, consider 3 4

5, 4 6, … , 4 896 = ;5 4 5 + ;6 4 6 + ⋯ + ;896(4 896) ,

each ;@ 4

@ ∈ #$A 4 @; ⋅ $ (B.

´ Let C ≔ dim#H - and suppose C < .0 ≤ J − L + 1. Can find 3 ≠ 0 as above such that, if ! #+ ⊆ -, 3 !, !P, … , !PQRS (%) = 0 , where !PT ≔ ∑@ !

@ +T%$V.

Periodic Subspace Structure

´ The space of ! " = ∑%!

%"&' ∈ )&*["; ⋅ &] such that

/ !, !1, …, !1345 " = 0 has the following structure:

´ There is an )7-subspace 8 ⊆ )&* of dimension ≤ ; − 1, and !

> ∈ 8.

´ There are ?@, … , ?AB@ ∈ )&* such that !

% ∈ ?% + 8.

´ We call such a subspace of )&* A (; − 1, E)-periodic subspace. ´ Morally: can pretend !

>, ! @, …, ! AB@ ∈ 8A. !

>

∈ !

@

∈ !

AB@

∈

…

?AB@ + 8 ?@ + 8 8

Choice of !

´ Thus, we would like to choose ! so as to have small intersection with any periodic subspace. ´ We will define

! ≔ # $ = ∑'()

*+, # '$-. : # ' ∈ 1' ∀3 , where 1), 1,, … , 1*+, ⊆ 7-8 form a subspace design.

• Definition. A collection 1), 1,, … , 1*+, ⊆ 7-8 of 7--subspaces is

called a (:, ;, <)-subspace design if, for every 7->-subspace ? ⊆ 7-8 with dim7C> ? = :, D

'() *+,

dim7C 1' ∩ ? ≤ ;: .

• Theorem. Let !", !$, … , !&'$ give a (), *, +)-subspace design

for all ) ≤ ./, where 0 ≤ . < 1/+, and assume each dim78 !9 = //;. Then Γ$, … , Γ=: ? → 7AB gives a .*, ='&C$

D

• dimension

expander.

E

"

∈ E

$

∈ E

&'$

∈

…

G&'$ + I G$ + I I !" !$ !&'$

Constructions of Subspace Designs

• Theorem. [GK’16] Suppose ! ≤ # ≤ $ < &. There exists an explicit

collection of ' ≥ Ω ⁄ &+ , subspaces -

., …, - 1 ⊆ 34 5, each of

codimension ,#, which form an !,

56. + 7689. , 1 -subspace design.

Construction 1. Let ; > 0. If & ≥ $> there exists a !, ?

>, @ -subspace

design for all ! ≤ .6A>

BC $. Moreover each subspace has 34-dimension

$/E. Construction 2. Let F > 0. If & ≥ $/@ there exists a !, 1 + F, @ - subspace design for all ! ≤ .6H

C $. Moreover each subspace has 34-

dimension $/E and E = J

. HK , @ = J . HL .

“High-degree” Folded Reed-Solomon Construction

´ Take !

", … , !% ⊆ '( ) *+,, subspace design from [GK ’16] with - ≈ 1/√2.

´ Let 3: '( 5 → '( 5 the field automorphism mapping ) ↦ 8), where 8 is a generator for '(

∗ .

´ Define the map :: '( ) *+, → '(;

< by

= ↦ = > , = >? , … , = >?@ A B . ´ > is an irreducible polynomial of degree C such that > ) ,>? = > 8) ,…,>?@AB = >(8<F")) are pairwise coprime, ´ = >?H is the residue of = when evaluated at the place >. ´ Define IJ = : !

J for K = 1,…,2.

Summary and Open Problems

´ Explicit construction of a !"#

$ , 1 − ( ) -dimension expander

when * ≥ Ω(.), or

! 0 1$ , Ω 2)

• dimension expander when

* ≥ .1. ´ Main ingredients: linearized polynomials, subspace designs. ´ Decrease the field size? Or obtain same result over 3, 4? ´ Applications of dimension expanders?

Thank You!