From affine to two-source extractors via approximate duality Eli Ben-Sasson Noga Zewi Computer Science Department Technion — Israel Institute of Technology May 2011 Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 1 / 17
Outline Main points 1 Extractors, dispersers and bipartite Ramsey graphs 2 Description of results 3 Proof sketches 4 Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 2 / 17
Main points Approximate duality and discrepancy Given A , B ⊂ F n 2 let their duality constant (or discrepancy ) be n � � ( − 1) � a , b � �� � D ( A , B ) = � E a ∈ A , b ∈ B � , where � a , b � = a i b i . � � i =1 Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 3 / 17
Main points Approximate duality and discrepancy Given A , B ⊂ F n 2 let their duality constant (or discrepancy ) be n � � ( − 1) � a , b � �� � D ( A , B ) = � E a ∈ A , b ∈ B � , where � a , b � = a i b i . � � i =1 Clearly D ( A , B ) = 1 iff A ⊆ b + B ⊥ for some b ∈ F n 2 . Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 3 / 17
Main points Approximate duality and discrepancy Given A , B ⊂ F n 2 let their duality constant (or discrepancy ) be n � � ( − 1) � a , b � �� � D ( A , B ) = � E a ∈ A , b ∈ B � , where � a , b � = a i b i . � � i =1 Clearly D ( A , B ) = 1 iff A ⊆ b + B ⊥ for some b ∈ F n 2 . Question: What if D ( A , B ) > 0 . 99 ? Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 3 / 17
Main points Approximate duality and discrepancy Given A , B ⊂ F n 2 let their duality constant (or discrepancy ) be n � � ( − 1) � a , b � �� � D ( A , B ) = � E a ∈ A , b ∈ B � , where � a , b � = a i b i . � � i =1 Clearly D ( A , B ) = 1 iff A ⊆ b + B ⊥ for some b ∈ F n 2 . Question: What if D ( A , B ) > 0 . 99 ? Answers (this talk): Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 3 / 17
Main points Approximate duality and discrepancy Given A , B ⊂ F n 2 let their duality constant (or discrepancy ) be n � � ( − 1) � a , b � �� � D ( A , B ) = � E a ∈ A , b ∈ B � , where � a , b � = a i b i . � � i =1 Clearly D ( A , B ) = 1 iff A ⊆ b + B ⊥ for some b ∈ F n 2 . Question: What if D ( A , B ) > 0 . 99 ? Answers (this talk): There exist “large” subsets A ′ , B ′ of A , B such that D ( A ′ , B ′ ) = 1. 1 Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 3 / 17
Main points Approximate duality and discrepancy Given A , B ⊂ F n 2 let their duality constant (or discrepancy ) be n � � ( − 1) � a , b � �� � D ( A , B ) = � E a ∈ A , b ∈ B � , where � a , b � = a i b i . � � i =1 Clearly D ( A , B ) = 1 iff A ⊆ b + B ⊥ for some b ∈ F n 2 . Question: What if D ( A , B ) > 0 . 99 ? Answers (this talk): There exist “large” subsets A ′ , B ′ of A , B such that D ( A ′ , B ′ ) = 1. 1 Assuming the polynomial Freiman-Ruzsa conjecture (PFR), A ′ , B ′ as 2 above exist even when D ( A , B ) > 2 − Ω( n ) . Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 3 / 17
Main points One main point Theorem (Discrepancy in matrices of small rank) Assuming PFR, for every α, δ > 0 exists γ > 0 such that: If M ∈ F N × N has F 2 -rank at most log N and discrepancy greater than 2 − γ n 2 α then M contains a large monochromatic rectangle M [ S , T ] , | S | , | T | ≥ N 1 − δ α . Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 4 / 17
Main points One main point Theorem (Discrepancy in matrices of small rank) Assuming PFR, for every α, δ > 0 exists γ > 0 such that: If M ∈ F N × N has F 2 -rank at most log N and discrepancy greater than 2 − γ n 2 α then M contains a large monochromatic rectangle M [ S , T ] , | S | , | T | ≥ N 1 − δ α . Conjecture (Polynomial Freiman-Ruzsa (PFR)) Let σ 2 ( A ) = | A + A | / | A | where A + A = { a + a ′ | a , a ′ ∈ A } . There exists a constant c such that for every A ⊂ F n 2 we have: ∃ A ′ ⊂ A , | A ′ | ≥ | A | /σ 2 ( A ) c such that | A ′ | / | span ( A ′ ) | ≥ σ 2 ( A ) − c . Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 4 / 17
Extractors, dispersers and bipartite Ramsey graphs Affine and two-source dispersers/extractors Definition (Affine Extractor) is a function f : F n 2 → F m 2 such that for all affine sources X ⊆ F n 2 of min-entropy rate dim( X ) at least ρ , n � f ( X ) − U m � ∞ ≤ ǫ. Definition (Two-source extractor) is a function f : F n 2 × F n 2 → F m 2 such that for all pairs of independent sources X , Y ⊆ F n 2 of min-entropy rate at least ρ , � f ( X , Y ) − U m � ∞ ≤ ǫ. Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 5 / 17
Extractors, dispersers and bipartite Ramsey graphs Affine and two-source dispersers/extractors Definition (Affine Disperser) is a function f : F n 2 → F 2 such that for all affine sources X ⊆ F n 2 of min-entropy rate dim( X ) at least ρ , n | f ( X ) | > 1 . Definition (Two-source disperser) is a function f : F n 2 × F n 2 → F 2 such that for all pairs of independent sources X , Y ⊆ F n 2 of min-entropy rate at least ρ , | f ( X , Y ) | > 1 . Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 5 / 17
Extractors, dispersers and bipartite Ramsey graphs Three main points “Good” affine extractors (small rate and error) can be converted in a black-box manner into “good” two-source dispersers (small rate). Two-source dispersers of low F 2 -rank are extractors with bounded error. Under the polynomial Freiman-Ruzsa conjecture (PFR), extractor error is exponentially small. Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 6 / 17
Extractors, dispersers and bipartite Ramsey graphs Previous constructions Two-source Reference rate error O ( log n Erd¨ os ‘47 (prob. method) n ) disperser Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 7 / 17
Extractors, dispersers and bipartite Ramsey graphs Previous constructions Two-source Reference rate error O ( log n Erd¨ os ‘47 (prob. method) n ) disperser 1 Chor, Goldreich ‘88 2 + ǫ exponentially small Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 7 / 17
Extractors, dispersers and bipartite Ramsey graphs Previous constructions Two-source Reference rate error O ( log n Erd¨ os ‘47 (prob. method) n ) disperser 1 Chor, Goldreich ‘88 2 + ǫ exponentially small 1 2 − O ( 1 Pudl´ ak, R¨ odl ‘04 √ n ) dispersers Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 7 / 17
Extractors, dispersers and bipartite Ramsey graphs Previous constructions Two-source Reference rate error O ( log n Erd¨ os ‘47 (prob. method) n ) disperser 1 Chor, Goldreich ‘88 2 + ǫ exponentially small 1 2 − O ( 1 Pudl´ ak, R¨ odl ‘04 √ n ) dispersers 1 Bourgain ‘05 2 − ǫ 0 exponentially small Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 7 / 17
Extractors, dispersers and bipartite Ramsey graphs Previous constructions Two-source Reference rate error O ( log n Erd¨ os ‘47 (prob. method) n ) disperser 1 Chor, Goldreich ‘88 2 + ǫ exponentially small 1 2 − O ( 1 Pudl´ ak, R¨ odl ‘04 √ n ) dispersers 1 Bourgain ‘05 2 − ǫ 0 exponentially small BKSSW ‘05 ǫ disperser Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 7 / 17
Extractors, dispersers and bipartite Ramsey graphs Previous constructions Two-source Reference rate error O ( log n Erd¨ os ‘47 (prob. method) n ) disperser 1 Chor, Goldreich ‘88 2 + ǫ exponentially small 2 − O ( 1 1 Pudl´ ak, R¨ odl ‘04 √ n ) dispersers 1 Bourgain ‘05 2 − ǫ 0 exponentially small BKSSW ‘05 ǫ disperser n 1 − ǫ BRSW ‘06 disperser Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 7 / 17
Extractors, dispersers and bipartite Ramsey graphs Previous constructions Two-source Reference rate error O ( log n Erd¨ os ‘47 (prob. method) n ) disperser 1 Chor, Goldreich ‘88 2 + ǫ exponentially small 1 2 − O ( 1 Pudl´ ak, R¨ odl ‘04 √ n ) dispersers 1 Bourgain ‘05 2 − ǫ 0 exponentially small BKSSW ‘05 ǫ disperser n 1 − ǫ BRSW ‘06 disperser Affine Reference rate error O ( log n Folklore (prob. method) n ) random function Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 7 / 17
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