Higher order Fourier analysis and algebraic property testing Hamed - PowerPoint PPT Presentation

Higher order Fourier analysis and algebraic property testing Hamed Hatami School of Computer Science McGill University July 22, 2019 Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 1 / 26

Based on joint works with Arnab Bhattacharyya, Eldar Fischer, Pooya Hatami, Shachar Lovett. Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 2 / 26

Property Testing Given a function: Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 3 / 26

Property Testing Given a function: Evaluate it on a small number of points: Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 3 / 26

Property Testing Given a function: Evaluate it on a small number of points: Decide whether ◮ it satisfies a property P ◮ or is “far” from satisfying P . far from P P Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 3 / 26

Property Testing Given a function: Evaluate it on a small number of points: Decide whether ◮ it satisfies a property P ◮ or is “far” from satisfying P . far from P P Definition dist ( f , g ) = Pr [ f ( x ) � = g ( x )] . dist ( f , P ) = min g ∈ P dist ( f , g ) . Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 3 / 26

The field of property testing has emerged from [Blum, Luby, Rubinfeld 93], [Babai, Fortnow, Lund 91], etc. Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 4 / 26

The field of property testing has emerged from [Blum, Luby, Rubinfeld 93], [Babai, Fortnow, Lund 91], etc. Closely related to the concepts of regularity and uniformity [Ruzsa-Szemerédi 76], [Rödl-Duke 85]. Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 4 / 26

The field of property testing has emerged from [Blum, Luby, Rubinfeld 93], [Babai, Fortnow, Lund 91], etc. Closely related to the concepts of regularity and uniformity [Ruzsa-Szemerédi 76], [Rödl-Duke 85]. Formally defined by [Rubinfeld, Sudan 96], [Goldreich, Goldwasser, Rubinfeld 98]. Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 4 / 26

The field of property testing has emerged from [Blum, Luby, Rubinfeld 93], [Babai, Fortnow, Lund 91], etc. Closely related to the concepts of regularity and uniformity [Ruzsa-Szemerédi 76], [Rödl-Duke 85]. Formally defined by [Rubinfeld, Sudan 96], [Goldreich, Goldwasser, Rubinfeld 98]. Closely related to limit theories of combinatorial objects [Lovász-Szegedy 2010]. Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 4 / 26

Example Let P = { functions f : F n p → { 0 , 1 } where f ≡ 0 } . Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 5 / 26

Example Let P = { functions f : F n p → { 0 , 1 } where f ≡ 0 } . Test Pick x ∈ F n p at random. If f ( x ) = 0 accept otherwise reject. Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 5 / 26

Example Let P = { functions f : F n p → { 0 , 1 } where f ≡ 0 } . Test Pick x ∈ F n p at random. If f ( x ) = 0 accept otherwise reject. Analysis If f ∈ P , then Pr [ accept ] = 1. If dist ( f , P ) > ǫ , then Pr [ reject ] ≥ ǫ . Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 5 / 26

Property Testing In this talk we focus on the following version of property testing: One-sided error: If f ∈ P , then Pr [ accept ] = 1. Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 6 / 26

Property Testing In this talk we focus on the following version of property testing: One-sided error: If f ∈ P , then Pr [ accept ] = 1. If dist ( f , P ) ≥ ǫ > 0, then Pr [ reject ] ≥ δ ( ǫ ) > 0. Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 6 / 26

Property Testing In this talk we focus on the following version of property testing: One-sided error: If f ∈ P , then Pr [ accept ] = 1. If dist ( f , P ) ≥ ǫ > 0, then Pr [ reject ] ≥ δ ( ǫ ) > 0. Proximity Oblivious: Number of queries is a fixed constant that does not depend on ǫ . Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 6 / 26

BLR linearity testing Linearity: f : F n 2 → F 2 , f ( x ) = a 1 x 1 + . . . + a n x n . Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 7 / 26

BLR linearity testing Linearity: f : F n 2 → F 2 , f ( x ) = a 1 x 1 + . . . + a n x n . Local characterization: f ( x + y ) = f ( x ) + f ( y ) for all x , y . Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 7 / 26

BLR linearity testing Linearity: f : F n 2 → F 2 , f ( x ) = a 1 x 1 + . . . + a n x n . Local characterization: f ( x + y ) = f ( x ) + f ( y ) for all x , y . Test Pick x , y ∈ F n 2 at random. If f ( x + y ) = f ( x ) + f ( y ) accept otherwise reject. Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 7 / 26

BLR linearity testing Linearity: f : F n 2 → F 2 , f ( x ) = a 1 x 1 + . . . + a n x n . Local characterization: f ( x + y ) = f ( x ) + f ( y ) for all x , y . Test Pick x , y ∈ F n 2 at random. If f ( x + y ) = f ( x ) + f ( y ) accept otherwise reject. Analysis If f linear, then Pr [ accept ] = 1. Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 7 / 26

BLR linearity testing Linearity: f : F n 2 → F 2 , f ( x ) = a 1 x 1 + . . . + a n x n . Local characterization: f ( x + y ) = f ( x ) + f ( y ) for all x , y . Test Pick x , y ∈ F n 2 at random. If f ( x + y ) = f ( x ) + f ( y ) accept otherwise reject. Analysis If f linear, then Pr [ accept ] = 1. ⇒ max � dist ( f , P ) > ǫ = f ( S ) < 1 − 2 ǫ = ⇒ Pr [ reject ] ≥ ǫ. Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 7 / 26

Classical Fourier Analysis over F n 2 For a ∈ F n 2 , define χ a : x �→ ( − 1 ) a 1 x 1 + ... + a n x n . Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 8 / 26

Classical Fourier Analysis over F n 2 For a ∈ F n 2 , define χ a : x �→ ( − 1 ) a 1 x 1 + ... + a n x n . These characters are orthonormal: � 1 a = b � χ a , χ b � = E x χ a ( x ) χ b ( x ) = a � = b . 0 Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 8 / 26

Classical Fourier Analysis over F n 2 For a ∈ F n 2 , define χ a : x �→ ( − 1 ) a 1 x 1 + ... + a n x n . These characters are orthonormal: � 1 a = b � χ a , χ b � = E x χ a ( x ) χ b ( x ) = a � = b . 0 Every f : F n 2 → R is uniquely expanded as � � f = f ( a ) χ a . a Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 8 / 26

Linearity and Fourier Analysis f : F n 2 → F 2 . Change the range to f : F n 2 → {− 1 , 1 } (by considering ( − 1 ) f ). Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 9 / 26

Linearity and Fourier Analysis f : F n 2 → F 2 . Change the range to f : F n 2 → {− 1 , 1 } (by considering ( − 1 ) f ). f : F n 2 → {− 1 , 1 } linear ⇔ it is a character f = χ a . Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 9 / 26

Linearity and Fourier Analysis f : F n 2 → F 2 . Change the range to f : F n 2 → {− 1 , 1 } (by considering ( − 1 ) f ). f : F n 2 → {− 1 , 1 } linear ⇔ it is a character f = χ a . If dist ( f , Linear ) > ǫ then f is far from all characters: max � f ( S ) < 1 − 2 ǫ. Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 9 / 26

Linearity and Fourier Analysis f : F n 2 → F 2 . Change the range to f : F n 2 → {− 1 , 1 } (by considering ( − 1 ) f ). f : F n 2 → {− 1 , 1 } linear ⇔ it is a character f = χ a . If dist ( f , Linear ) > ǫ then f is far from all characters: max � f ( S ) < 1 − 2 ǫ. BLR: Then Pr [ reject ] = 1 2 ( 1 − E f ( x ) f ( y ) f ( x + y )) ≥ ǫ. Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 9 / 26

Algebraic Property Testing Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 10 / 26

Our setting Functions of the form f : F n p → { 0 , . . . , R } where p is a fixed prime. R is a fixed integer. Hamed Hatami (McGill Universities) Higher order Fourier analysis and algebraic property testing July 22, 2019 11 / 26