Advances in Linear Sketching over Finite Fields Grigory Yaroslavtsev (Indiana University, Bloomington) http://grigory.us CCC’18, with Sampath Kannan (U. Pennsylvania), Elchanan Mossel (MIT) and Swagato Sanyal (NUS)

𝔽 2 -Sketching Input 𝒙∈ 0,1 𝑛 Parity = Linear function over 𝔾 𝐹 2 : ⊕ 𝑖∈𝑆 𝑥 𝑖 Deterministic linear sketch: set of 𝒌 parities: ℓ 𝒙 = ⊕ 𝑖 1 ∈ 𝑆 1 𝑥 𝑖 1 ; ⊕ 𝑖 2 ∈ 𝑆 2 𝑥 𝑖 2 ;…; ⊕ 𝑖 𝒌 ∈ 𝑆 𝒌 𝑥 𝑖 𝒌 E.g. 𝑥 4 ⊕ 𝑥 2 ⊕ 𝑥 42 ; 𝑥 239 ⊕ 𝑥 30 ; 𝑥 566 ;… Randomized linear sketch: distribution over 𝒌 parities (random 𝑆 1 , 𝑆 2 , …, 𝑆 𝒌 ):

Linear sketching over 𝔽 2 Given 𝒇 𝒙 : 0,1 𝑛 →{0,1} Question: Can one recover 𝒇(𝒙) from a small (𝒌≪𝑛) linear sketch over 𝔽 2 ? Allow randomized computation (99% success) Probability over choice of random sets Sets are known at recovery time Recovery is deterministic (w.l.o.g)

Puzzle: Open Problem 78 on Sublinear.info Shared randomness Alice: 𝒙 ∈ 0,1 𝑛 Bob: 𝒚 ∈ 0,1 𝑛 𝑀(𝒙) 𝒇 + =𝒇(𝒙⊕𝒚) Conjecture: (Almost) shortest message is a randomized 𝔽 2 -sketch https://sublinear.info/index.php?title=Open_Problems:78

Multi-player version … Thm [Hosseini, Lovett, Y.’18; ECCC TR18-169] Shared randomness 𝒙 𝟏 ∈ 0,1 𝑛 𝒙 𝟐 ∈ 0,1 𝑛 𝒙 𝟑 ∈ 0,1 𝑛 𝒙 𝑵 ∈ 0,1 𝑛 𝒙 𝒊 ∈ 0,1 𝑛 … 𝒇 + =𝒇( 𝒙 𝟏 ⊕…⊕ 𝒙 𝑵 ) 𝑀 1 ( 𝒙 𝟏 ) 𝑀 2 ( 𝒙 𝟐 , 𝑀 1 ) 𝑀 𝑖 ( 𝒙 𝒊 , 𝑀 𝑖−1 ) Thm [Hosseini, Lovett, Y.’18; ECCC TR18-169] For 𝑁≥10𝑛 a protocol where each 𝑀 𝑖 is at most c bits ⇒ ∃ a randomized 𝔽 2 -sketch of size 𝑂(𝑐)

Multi-player version over 𝑍 𝑝 Shared randomness 𝒙 𝟏 ∈𝑍 𝑝 𝑛 𝒙 𝟐 ∈𝑍 𝑝 𝑛 𝒙 𝟑 ∈𝑍 𝑝 𝑛 𝒙 𝑵 ∈𝑍 𝑝 𝑛 𝒙 𝒊 ∈𝑍 𝑝 𝑛 … 𝒇 + =𝒇( 𝒙 𝟏 +…+ 𝒙 𝑵 ) 𝑀 1 ( 𝒙 𝟏 ) 𝑀 2 ( 𝒙 𝟐 , 𝑀 1 ) 𝑀 𝑖 ( 𝒙 𝒊 , 𝑀 𝑖−1 ) Thm [Hosseini, Lovett, Y.’18; ECCC TR18-169] For 𝑁≥10𝑛 log 𝑝 a protocol where each 𝑀 𝑖 is at most c bits ⇒ ∃ a randomized 𝑍 𝑝 -sketch of dimension 𝑂(𝑐) *Holds even for 𝑀 𝑖 ( 𝒙 𝒊 , 𝑀 1 , 𝑀 2 , …, 𝑀 𝑖−1 ) instead of 𝑀 𝑖 ( 𝒙 𝒊 , 𝑀 𝑖−1 )

Motivation: Distributed Computing Distributed computation among 𝑴 machines: 𝒙=( 𝒙 𝟏 , 𝒙 𝟐 , …, 𝒙 𝑴 ) (more generally 𝒙= ⊕ 𝑖=1 𝑴 𝒙 𝒊 ) 𝑴 machines can compute sketches locally: ℓ 𝒙 𝟏 , …, ℓ( 𝒙 𝑴 ) Send them to the coordinator who computes: ℓ 𝑖 𝒙 = ℓ 𝑖 𝒙 𝟏 ⊕⋯⊕ ℓ 𝑖 ( 𝒙 𝑴 ) (coordinate-wise XORs) Coordinator computes 𝒇 𝒙 with 𝒌𝑴 communication 1 𝒙 𝒙 𝟏 𝒙 𝟐

Motivation: Streaming 𝒙 generated through a sequence of updates Updates 𝑖 1 ,…, 𝑖 𝑚 : update 𝑖 𝑡 flips bit at position 𝑖 𝑡 𝒙 𝟎 Updates: (1, 3, 8, 3) 𝒙 𝟏 1 𝒙 𝟐 1 𝒙 𝟑 1 𝒙 1 ℓ 𝒙 allows to recover 𝒇(𝒙) with 𝒌 bits of space

Frequently Asked Questions Q: Why 𝔽 2 updates instead of ±1? Often doesn’t help if you know the sign Q: How to store random sets? Derandomize using Nisan’s PRG – extra O(log n) factor in space Q: Some applications? Essentially all dynamic graph streaming algorithms can be based on 𝐿 0 -sampling 𝐿 0 -sampling can be done optimally using 𝔽 2 -sketching [Kapralov et al. FOCS’17] Q: Why not allow to compute 𝒇 approximately? Stay tuned

Deterministic vs. Randomized Fact: 𝒇 has a deterministic sketch if and only if 𝒇=𝒈( ⊕ 𝑖 1 ∈ 𝑆 1 𝑥 𝑖 1 ; ⊕ 𝑖 2 ∈ 𝑆 2 𝑥 𝑖 2 ;…; ⊕ 𝑖 𝑘 ∈ 𝑆 𝑘 𝑥 𝑖 𝑘 ) Equivalent to “𝒇 has Fourier dimension 𝒌" Randomization can help: OR: 𝒇 𝒙 = 𝑥 1 ∨⋯∨ 𝑥 𝑛 Has “Fourier dimension” =𝑛 Pick 𝒕= log 1/𝜹 random sets 𝑆 1 ,…, 𝑆 𝒕 If there is 𝑗 such that ⊕ 𝑖∈ 𝑆 𝑗 𝑥 𝑖 =1 output 1, otherwise output 0 Error probability 𝜹

Fourier Analysis 𝒇 𝑥 1 , …, 𝑥 𝑛 : 0,1 𝑛 →{0,1} Notation switch: 𝒇 𝑥 1 , …, 𝑥 𝑛 : 0,1 𝑛 →{0,1} Notation switch: 0→1 1→−1 𝒇 ′ : −1,1 𝑛 →{−1,1} Functions as vectors form a vector space: 𝒇: −1,1 𝑛 →{−1,1}⇔𝒇∈ {−1,1} 2 𝑛 Inner product on functions = “correlation”: 𝒇,𝒈 = 2 −𝑛 𝑥∈ −1,1 𝑛 𝒇 𝑥 𝒈(𝑥) = 𝔼 𝑥∼ −1,1 𝑛 𝒇 𝑥 𝒈 𝑥 𝒇 2 = 𝑓,𝑓 = 𝔼 𝑥∼ −1,1 𝑛 𝒇 2 𝑥 =1 (for Boolean only)

“Main Characters” are Parities For 𝑺⊆[𝑛] let character 𝝌 𝑺 (𝑥)= 𝑖∈𝑺 𝑥 𝑖 Fact: Every function 𝒇: −1,1 𝑛 →{−1,1} is uniquely represented as a multilinear polynomial 𝒇 𝑥 1 , …, 𝑥 𝑛 = 𝑺⊆[𝑛] 𝒇 𝑺 𝝌 𝑺 (𝑥) 𝒇 𝑺 a.k.a. Fourier coefficient of 𝒇 on 𝑺 𝒇 𝑺 ≡ 𝒇, 𝝌 𝑺 = 𝔼 𝑥∼ −1,1 𝑛 𝒇 𝑥 𝝌 𝑺 𝑥 𝑺 𝒇 𝑺 2 =1 (Parseval)

Fourier Dimension Fourier sets 𝑺 ≡ vectors in 𝔽 2 𝑛 “𝒇 has Fourier dimension 𝒌“ = a 𝒌-dimensional subspace 𝐴 𝒌 in Fourier domain has all weight 𝑺⊆ 𝐴 𝒌 𝒇 𝑺 2 =1 𝒇 𝑥 1 , …, 𝑥 𝑛 = 𝑺⊆[𝑛] 𝒇 𝑺 𝝌 𝑺 (𝒙) = 𝑺⊆ 𝐴 𝒌 𝒇 𝑺 𝝌 𝑺 (𝒙) Pick a basis 𝑺 𝟏 , …, 𝑺 𝒌 in 𝐴 𝒌 : Sketch: 𝝌 𝑺 𝟏 (𝒙), …, 𝝌 𝑺 𝒌 (𝒙) For every 𝑺∈ 𝐴 𝒌 there exists 𝒁⊆ 𝒌 : 𝑺= ⊕ 𝒊∈𝒁 𝑺 𝒊 𝝌 𝑺 𝒙 = ⊕ 𝒊∈𝒁 𝝌 𝑺 𝒊 (𝒙)

Deterministic Sketching and Noise Suppose “noise” has bounded norm 𝒇 =𝒈⊕𝒉= 𝒌-dimensional ⊕ “ noise” Sparse Fourier noise (via [Sanyal’15]) 𝒇 = 𝒌-dimensional ⊕ “Fourier 𝐿 0 -noise” 𝑛𝑜𝑖𝑠𝑒 0 = # non-zero Fourier coefficients of noise (aka “Fourier sparsity”) Linear sketch size: 𝒌+𝑂( 𝑛𝑜𝑖𝑠𝑒 0 1/2 ) Our work: can’t be improved even with randomness and even for uniform 𝑥, e.g for ``addressing function’’.

How Randomization Handles Noise 𝐿 0 -noise in original domain (via hashing a la OR) 𝒇= 𝒌-dim. ⊕ “ 𝐿 0 -noise” 𝔽 2 -sketch size: 𝒌 + O(log 𝑛𝑜𝑖𝑠𝑒 0 ) Optimal (but only existentially, i.e. ∃𝒇:…) 𝐿 1 -noise in the Fourier domain (via [Bruck, Smolensky ‘92; Grolmusz’97]) 𝒇 = 𝒌-dim. ⊕ “Fourier 𝐿 1 -noise” 𝔽 2 -sketch size: 𝒌+𝑂( 𝑛𝑜𝑖𝑠𝑒 1 2 ) Example = 𝒌-dim. ⊕ small decision tree / DNF / etc.

Randomized Sketching: Hardness Not 𝜸-concentrated on 𝒌 -dim. Fourier subspaces For ∀ 𝒌 -dim. Fourier subspace 𝐴 : 𝑆∉𝐴 𝒇 𝑺 2 ≥ 1−𝜸 Thm. Any 𝒌 -dim. 𝔽 2 -sketch makes error 1− 𝜸 2 Converse doesn’t hold, i.e. concentration is not enough

Randomized Sketching: Hardness Not 𝜸-concentrated on 𝑜(𝑛)-dim. Fourier subspaces: Almost all symmetric functions, i.e. 𝒇 𝒙 =𝒉( 𝑖 𝑥 𝑖 ) If not Fourier-close to constant or 𝑖=1 𝑛 𝑥 𝑖 E.g. Majority (not an extractor even for O( 𝑛 )) Tribes (balanced DNF) Recursive majority: 𝑀𝑎 𝑗 ∘𝑘 =𝑀𝑎 𝑗 3 ∘𝑀𝑎 𝑗 3 …∘𝑀𝑎 𝑗 3

Approximate Fourier Dimension Not 𝜸-concentrated on 𝒌 -dim. Fourier subspaces ∀ 𝒌 -dim. Fourier subspace 𝐴: 𝑆∉𝐴 𝒇 𝑺 2 ≥ 1−𝜸 Any 𝒌 -dim. linear sketch makes error ½(1− 𝜸 ) Definition (Approximate Fourier Dimension) dim 𝜸 𝒇 = smallest 𝒅 such that 𝒇 is 𝜸-concentrated on some Fourier subspace of dimension 𝒅 𝒇 ( 𝑆 1 ) 𝒇 ( 𝑆 2 ) 𝒇 ( 𝑆 3 ) 𝒇 ( 𝑆 2 + 𝑆 3 ) 𝒇 ( 𝑆 1 + 𝑆 3 ) 𝒇 ( 𝑆 1 +𝑆 2 + 𝑆 3 ) 𝑆∈𝐴 𝒇 𝑺 2 ≥ 𝜸

Sketching over Uniform Distribution + Approximate Fourier Dimension Sketching error over uniform distribution of 𝒙. dim 𝝐 𝒇 -dimensional sketch gives error 𝟏−𝝐: Fix dim 𝝐 𝒇 -dimensional 𝐴: 𝑆∈𝐴 𝒇 𝑺 2 ≥ 𝝐 Output: 𝒈 𝑥 =sign 𝑺∈𝐴 𝒇 𝑺 𝝌 𝑺 𝑥 : Pr 𝑥∼𝑈 −1,1 𝑛 𝒈 𝑥 =𝒇 𝑥 ≥𝝐⇒ error 𝟏−𝝐 We show a basic refinement ⇒ error 𝟏−𝝐 𝟐 Pick 𝜽 from a carefully chosen distribution Output: 𝒈 𝜽 𝑥 =sign 𝑺∈𝐴 𝒇 𝑺 𝝌 𝑺 𝑥 −𝜽 -1 +1 1

1-way Communication Complexity of XOR-functions Shared randomness Alice: 𝒙 ∈ 0,1 𝑛 Bob: 𝒚 ∈ 0,1 𝑛 𝑀(𝒙) 𝒇 + =𝒇(𝒙⊕𝒚) Examples: 𝒇(z) = 𝑂 𝑅 𝑖=1 𝑛 ( 𝑧 𝑖 ) ⇒ 𝒇 + : (not) Equality 𝒇(z) = ( 𝑧 0 > d) ⇒ 𝒇 + : Hamming Dist > d 𝑅 𝜖 1 𝒇 + = min.|M| so that Bob’s error prob. 𝜖

Communication Complexity of XOR-functions Well-studied (often for 2-way communication): [Montanaro,Osborne], ArXiv’09 [Shi, Zhang], QIC’09, [Tsang, Wong, Xie, Zhang], FOCS’13 [O’Donnell, Wright, Zhao,Sun,Tan], CCC’14 [Hatami, Hosseini, Lovett], FOCS’16 Connections to log-rank conjecture [Lovett’14]: Even special case for XOR-functions still open

Deterministic 1-way Communication Complexity of XOR-functions Alice: 𝒙 ∈ 0,1 𝑛 Bob: 𝒚 ∈ 0,1 𝑛 𝑀(𝒙) 𝒇 + =𝒇(𝒙⊕𝒚) 𝐷 1 𝒇 + = min.|M| so that Bob is always correct [Montanaro-Osborne’09]: 𝐷 1 𝒇 = 𝐷 𝑙𝑖𝑛 𝒇 𝐷 𝑙𝑖𝑛 𝒇 = deterministic 𝔽 2 -sketch complexity of 𝒇 + 𝐷 1 𝒇 + = 𝐷 𝑙𝑖𝑛 𝒇 = Fourier dimension of 𝒇

1-way Communication Complexity of XOR-functions Shared randomness Alice: 𝒙 ∈ 0,1 𝑛 Bob: 𝒚 ∈ 0,1 𝑛 𝑀(𝒙) 𝒇(𝒙⊕𝒚) 𝑅 𝜖 1 𝒇 = min.|M| so that Bob’s error prob. 𝜖 𝑅 𝜖 𝑙𝑖𝑛 𝒇 + = rand. 𝔽 2 -sketch complexity (error 𝜖 ) 𝑅 𝜖 1 𝒇 + ≤ 𝑅 𝜖 𝑙𝑖𝑛 𝒇 Conjecture: 𝑅 𝜖 1 𝒇 + ≈ 𝑅 𝜖 𝑙𝑖𝑛 𝒇 ?

𝑅 𝜖 1 𝒇 + ≈ 𝑅 𝜖 𝑙𝑖𝑛 𝒇 ? As we show holds for: 𝑅 𝜖 1 𝒇 + ≈ 𝑅 𝜖 𝑙𝑖𝑛 𝒇 ? As we show holds for: Majority, Tribes, recursive majority, addressing function (Almost all) symmetric functions Degree-𝒅 𝔽 2 -polynomials: 𝑅 5𝜖 𝑙𝑖𝑛 𝒇 =𝑂(𝒅 𝑅 𝜖 1 𝒇 + ) Analogous question for 2-way is wide open: [HHL’16] 𝑄 𝜖 ⊕−𝑑𝑡 𝒇 =𝑝𝑜𝑙𝑦( 𝑅 𝜖 𝒇 + )?

Distributional 1-way Communication under Uniform Distribution Alice: 𝒙 ∼𝑈( 0,1 𝑛 ) Bob: 𝒚 ∼𝑈( 0,1 𝑛 ) 𝑀(𝒙) 𝒇(𝒙⊕𝒚) 𝑅 𝜖 1 𝒇 = sup 𝐷 𝕯 𝜖 1,𝐷 𝒇 𝕯 𝜖 1,𝑈 𝒇 = min.|M| so that Bob’s error prob. 𝜖 is over the uniform distribution over (𝒙,𝒚) Enough to consider deterministic messages only Motivation: streaming/distributed with random input

Communication for Uniform Distribution Thm: If dim 𝝐 𝒇 = 𝒅−1 then 𝕯 1− 𝝐 𝟔 1,𝑈 𝒇 + ≥ 𝒅 6 . Optimal up to constant factors in dimension and error 𝒅-dim. linear sketch has error 1− 𝝐 𝟐

Application: Random Streams 𝒙∈ 0,1 𝑛 generated via a stream of updates Each update flips a random coordinate Goal: maintain 𝒇 𝒙 during the stream (error prob. 𝝐) Question: how much space necessary? Answer: 𝕯 𝜖 1,𝑈 and best algorithm is linear sketch After first O(𝑛 log⁡𝑛) updates input 𝒙 is uniform Big open question: Is the same true if 𝒙 is not uniform? True for VERY LONG ( 2 2 2 Ω 𝑛 ) streams (via [LNW’14]) True for streams of length (𝑂( 𝑛 2 )) (via [HLY’18])

Approximate 𝔽 2 -Sketching [Y.’17] 𝒇 𝑥 1 , …, 𝑥 𝑛 : 0,1 𝑛 →ℝ Normalize: 𝒇 𝟐 =1 Question: Can one compute 𝒇′: 𝔼[ 𝒇 𝒙 − 𝒇 ′ 𝒙 𝟐 ≤𝝐] from a small (𝒌≪𝑛) linear sketch over 𝔽 2 ?

Approximate 𝔽 2 -Sketching [Y.’17] Interesting facts: All results under the uniform distribution generalize directly to approximate sketching 𝐿 1 -sampling has optimal dependence on parameters: Optimal dependence: O 𝑓 1 2 𝜖 Open: Is 𝐿 1 -sampling optimal for Boolean functions?

Approximate 𝔽 2 -Sketching of Valuation Functions [Y.,Zhou’18] Additive ( 𝑖=1 𝑛 𝑤 𝑖 𝑥 𝑖 ): Θ min |𝑤 ​ 1 2 𝜖 ,𝑛 (optimal via weighted Gap Hamming) Budget-additive (min(b, 𝑖=1 𝑛 𝑤 𝑖 𝑥 𝑖 )): Θ min |𝑤 ​ 1 2 𝜖 ,𝑛 Coverage: Optimal Θ 1 𝜖 (via 𝐿 1 -Sampling) Matroid rank (various results depending on rank 𝑟) 𝛼-Lipschitz submodular functions: Ω(𝑛) communication lower bound for 𝛼=Ω(1/𝑛) Uses a large family of matroids from [Balcan, Harvey’10]

Thanks! Questions? Other stuff [Y., Zhou 18]: Linear Threshold Functions: Θ 𝜃 𝑚 log 𝜃 𝑚 Resolves a communication conjecture of [MO’09] Simple neural nets: LTF(ORs), LTF(LTFs) Blog post: http://grigory.us/blog/the-binary-sketchman

Sketching over Uniform Distribution Thm: If dim 𝝐 𝒇 = 𝒅−1 then 𝕯 1− 𝝐 𝟔 1,𝑈 𝒇 + ≥ 𝒅 6 . Optimal up to error as 𝒅-dim. linear sketch has error 1− 𝝐 𝟐 Weaker: If 𝝐 𝟐 > 𝝐 𝟏 , dim 𝝐 𝟏 (𝒇) = dim 𝝐 𝟐 (𝒇) = 𝒅−1 then: 𝕯 𝜹 1,𝑈 (𝒇)≥𝒅, where 𝜹=( 𝝐 𝟐 − 𝝐 𝟏 )/4. Corollary: If 𝒇 (∅)<𝐶 for 𝐶<1 then there exists 𝒅: 𝕯 Θ 1 𝑛 1,𝑈 𝒇 ≥𝒅. Tight for the Majority function, etc.

𝕯 𝜖 1,𝑈 and Approximate Fourier Dimension Thm: If 𝝐 𝟐 > 𝝐 𝟏 >0, dim 𝝐 𝟏 (𝒇) = dim 𝝐 𝟐 (𝒇) = 𝒅−1 then: 𝕯 𝜹 1,𝑈 (𝒇)≥𝒅, where 𝜹=( 𝝐 𝟐 − 𝝐 𝟏 )/4. 𝒚∈ 𝟎,𝟏 𝒏 𝒇(𝒙⊕𝒚) = 𝒇 𝒙 (𝒚) 𝒇 𝒙 𝟏 𝒇 𝒙 𝟐 00 01 10 11 𝒇 𝒙 𝟑 𝑀(𝒙)= 𝒙∈ 𝟎,𝟏 𝒏

𝕯 𝜖 1,𝑈 and Approximate Fourier Dimension If 𝑴 𝒙 =𝒅−1 average “rectangle” size = 2 𝒏−𝒅+𝟏 A subspace 𝐴 distinguishes 𝒙 𝟏 and 𝒙 𝟐 if: ∃𝑺∈𝐴 : 𝜒 𝑺 𝒙 𝟏 ≠ 𝜒 𝑺 𝒙 𝟐 Lem 1: Fix a 𝒅-dim. subspace 𝐴 𝒅 : typical 𝒙 𝟏 and 𝒙 𝟐 in a typical “rectangle” are distinguished by 𝐴 𝒅 Lem 2: If a 𝒅-dim. subspace 𝐴 𝒅 distinguishes 𝒙 𝟏 and 𝒙 𝟐 + 1) 𝒇 is 𝝐 𝟐 -concentrated on 𝐴 𝒅 2) 𝒇 is not 𝝐 𝟏 -concentrated on any (𝒅−1)-dim. subspace ⇒ Pr 𝑧∼𝑈( −1,1 𝑛 ) 𝒇 𝒙 𝟏 𝑧 ≠ 𝒇 𝒙 𝟐 𝑧 ≥ 𝝐 𝟐 − 𝝐 𝟏

𝕯 𝜖 1,𝑈 and Approximate Fourier Dimension Thm: If 𝝐 𝟐 > 𝝐 𝟏 >0, dim 𝝐 𝟏 (𝒇) = dim 𝝐 𝟐 (𝒇) = 𝒅−1 then: 𝕯 𝜹 1,𝑈 (𝒇)≥𝒅, Where 𝜹=( 𝝐 𝟐 − 𝝐 𝟏 )/4. Pr 𝑧∼𝑈( −1,1 𝑛 ) 𝒇 𝒙 𝟏 𝑧 ≠ 𝒇 𝒙 𝟐 𝑧 ≥ 𝝐 𝟐 − 𝝐 𝟏 Error for fixed 𝒚 = min( Pr 𝑥∈𝑅 [ 𝒇 𝒙 𝒚 =0], Pr 𝑥∈𝑅 [ 𝒇 𝒙 𝒚 =1]) Average error for (𝒙,𝒚)∈𝑅 = Ω( 𝝐 𝟐 − 𝝐 𝟏 ) 𝒚 𝒈 𝒙 𝟏 1 𝑅=“typical rectangle” 𝒈 𝒙 𝟐

Example: Majority 𝕯 𝑂(1/ 𝑛 ) 1,𝑈 (𝑴𝒂 𝒋 𝒏 )≥𝒏 Majority function: 𝑴𝒂 𝒋 𝒏 𝑧 1 ,…, 𝑧 𝑛 ≡ 𝑖=1 𝑛 𝑧 𝑖 ≥𝑛/2 𝑴𝒂 𝒋 𝒏 𝑺 only depends on 𝑺 𝑴𝒂 𝒋 𝒏 𝑺 =0 if |𝑺| is odd 𝑊 𝒌 𝑴𝒂 𝒋 𝒏 = 𝑺: 𝑺 =𝒌 𝑴𝒂 𝒋 𝒏 𝑺 =𝛼 𝒌 − 3 2 1±𝑂 1 𝒌 (𝑛−1)-dimensional subspace with most weight: 𝑨 𝒏−𝟏 =𝑠𝑝𝑎𝑛( 1 , 2 ,…,{𝑛−1}) 𝑺∈ 𝑨 𝒏−𝟏 𝑴𝒂 𝒋 𝒏 𝑺 =1− 𝛾 𝑛 ±𝑂 𝑛 −3/2 Set 𝝐 𝟐 =1−𝑂( 𝑛 −3/2 ), 𝝐 𝟏 =1− 𝛾 𝑛 +𝑂 𝑛 −3/2 𝕯 𝑂(1/ 𝑛 ) 1,𝑈 (𝑴𝒂 𝒋 𝒏 )≥𝒏