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 Γ 3 , ≥ # dim, . 34% ´ 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, 1 &, ' 2 , 1 + Ω 1 Harrow, Ben-Aroya–Ta- Shma ‘08 Bourgain–Yehudayoff ‘13 1 None 2 , 1 + Ω 1 ) ≥ Ω(, - ) Forbes–Guruswami ‘15 1 , Ω ( Ω (

Our results ( ( * B , whereas random gets $ = A * C . $ = A Theorem. For any ! > 0 , there exists $ = $(!) such that there ()* is an explicit + , 1 − ! $ -dimension expander of degree 1 when 2 ≥ Ω(5) . $ over / 0 + Since dim(∑ :;( : (=)) ≤ $ dim = , ? = Γ 1 − ! $ ()* Also, @ = is optimal. is optimal. We call the expander lossless. +

Our results Theorem. For any ! > 0 , there exists $ = $(!) such that there ()* is an explicit + , 1 − ! $ -dimension expander of degree 1 when 2 ≥ Ω(5) . $ over / 0 Theorem. For any 6 > 0 , there exists $ = $(6) such that ( there is an explicit 7(8+) ,Ω(6$) -dimension expander of 1 when 2 ≥ Ω 5 8 . degree $ over / 0

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 by = ↦ =(! : ) . Γ : :. → & ) 0 & ) 0 A Intuitively , we’d like to show that & ' subspaces of dimension B are - expanded to subspaces of dimension - − D + 1 B . & )

Contrapositive Characterization Proposition. Let Γ 8 , … , Γ : : # ; → # ; . Suppose that ∀ ! ⊆ # ; s.t. dim ! ≤ ABC , we have 8 dim D ∈ # ; : ∀ + ∈ , , Γ E dim ! . / D ∈ ! ≤ Then {Γ / : + ∈ [,]} forms a (A, B) -dimension expander. ´ Thus, for ! ⊆ # $ % , we need to understand ∈ ! = & ∈ (: & # 1 ⊆ ! . & ∈ (: ∀+ ∈ , , & . / ´ First, let us study & ∈ # $ % 2; ⋅ $ 56 : & # 1 ⊆ ! .

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 896 ) , 3 4 5 , 4 6 , … , 4 896 = ; 5 4 5 + ; 6 4 6 + ⋯ + ; 896 (4 @ ; ⋅ $ (B . @ ∈ # $ A 4 each ; @ 4 ´ Let C ≔ dim # H - and suppose C < .0 ≤ J − L + 1 . Can find 3 ≠ 0 as above such that, if ! # + ⊆ - , 3 !, ! P , … , ! P QRS (%) = 0 , where ! P T ≔ ∑ @ ! + T % $ V . @

Periodic Subspace Structure % " & ' ∈ ) & * ["; ⋅ & ] such that ´ The space of ! " = ∑ % ! / !, ! 1 , …, ! 1 345 " = 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 . AB@ ∈ 8 A . ´ Morally : can pretend ! > , ! @ , …, ! … ? @ + 8 ? AB@ + 8 8 ∈ ∈ ∈ ! ! ! AB@ @ >

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 dim 7 C> ? = : , *+, D dim 7 C 1 ' ∩ ? ≤ ;: . '()

! &'$ ! " … G $ + I ! $ G &'$ + I I ∈ ∈ ∈ E E E &'$ $ " Theorem. Let ! " , ! $ , … , ! &'$ give a (), *, +) -subspace design for all ) ≤ ./ , where 0 ≤ . < 1/+ , and assume each dim 7 8 ! 9 = //; . Then Γ $ , … , Γ = : ? → 7 A B gives a .*, ='&C$ -dimension D expander.

Constructions of Subspace Designs Theorem. [GK’16] Suppose ! ≤ # ≤ $ < & . There exists an explicit & + , subspaces - 5 , each of collection of ' ≥ Ω ⁄ . , …, - 1 ⊆ 3 4 56. codimension ,# , which form an !, + 7689. , 1 -subspace design. Construction 1. Let ; > 0 . If & ≥ $ > there exists a !, ? > , @ -subspace design for all ! ≤ .6A> BC $ . Moreover each subspace has 3 4 -dimension $/E . Construction 2. Let F > 0 . If & ≥ $/@ there exists a !, 1 + F, @ - subspace design for all ! ≤ .6H C $ . Moreover each subspace has 3 4 - . . dimension $/E and E = J H L . H K , @ = J

“High-degree” Folded Reed-Solomon Construction " , … , ! % ⊆ ' ( ) *+, , subspace design from [GK ’16] with - ≈ 1/√2 . ´ Take ! ´ Let 3: ' ( 5 the field automorphism mapping ) ↦ 8) , where 8 is a → ' ( 5 ∗ . generator for ' ( < by ´ Define the map :: ' ( ) *+, → ' ( ; = > , = > ? , … , = > ? @ 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 I J = : ! J for K = 1,…,2 .

Summary and Open Problems ´ Explicit construction of a !"# $ , 1 − ( ) -dimension expander ! when * ≥ Ω(.) , or -dimension expander when 0 1$ , Ω 2) * ≥ . 1 . ´ Main ingredients: linearized polynomials, subspace designs. ´ Decrease the field size? Or obtain same result over 3, 4 ? ´ Applications of dimension expanders? Thank You!