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Unformatted text preview: 6.896 Sublinear Time Algorithms November 18, 2004 Lecture 19 Lecturer: Eli BenSasson Scribe: Ben Wagner 1 Linearity Testing Over Z 2 1.1 Introduction Today we will provide a tighter analysis of the BLR linearity test (defined in previous lecture) for the special case where G = Z n 2 and H = Z 2 . This analysis was originally shown by Bellare et al. [3]. We use the definitions from last lecture (specifically, the set of homomorphisms Hom( G, H ), the BLR tester T f and the definition of distance δ ( f, Hom)). Recall the BLR test T f picks x, y uniformly at random from G , and accepts iff f ( x ) + f ( y ) = f ( x + y ) and let Z 2 be the additive group with two elements 0 , 1 and addition modulo two. Theorem 1 [3] For every function f : Z n 2 → Z 2 , the BLR test T f gives δ ( f , Hom( Z n 2 , Z 2 )) ≤ Pr h T f rejects i where Hom( Z n 2 , Z 2 ) = n ` α : Z n 2 → Z 2 α ∈ Z n 2 , ` α ( x ) = X α i x i mod 2 o The BLR Theorem proved in last lecture is stronger than Theorem 1, as it applies to any G , H , (not even necessarily Abelian). On the down side, the connection between rejection probability and distance to Hom( G, H ) there is weaker, as the BLR theorem only holds when the rejection probability is small (less than 2 / 9) and even then only shows the distance is twice the rejection probability. 1.2 Adding Structure to the Problem To obtain the tighter Theorem 1, we need to add structure to the set of homomorphisms from Z n 2 to Z 2 . This structure will be provided by working in the realm of real numbers, and we start by casting our problem in this world. Definition 2 The real embedding of a Boolean function f : Z n 2 → Z 2 is the function f : Z n 2 → {− 1 , +1 } defined by f ( x ) = ( − 1) f ( x ) . For ` α ∈ Hom( Z n 2 , Z 2 ) let χ α : Z n 2 → {− 1 , +1 } be the real embedding of ` α . Using this change of representation, Theorem 1 is reduced to: Theorem 3 For every function f : Z n 2 → Z 2 and f its real embedding, the BLR test T f gives δ ( f , Hom( Z n 2 , Z 2 )) = min α (Pr x [ f ( x ) 6 = χ α ( x )]) ( ∗ ) ≤ Pr x,y h T f rejects i = Pr x,y [ f ( x ) · f ( y ) 6 = f ( x + y )] ( ∗∗ ) Proof (of Theorem 1 from Theorem 3): The first equality is our definition of distance, the second inequality is what we wish to prove, and the third equality is the definition of the BLR test. The labels (*) and (**) refer to the rhs of their respective equations. From here on, we focus on proving Theorem 3. 1 1.3 The Fourier Basis and its elementary properties It will be useful to think of all our functions as elements in R 2 n . This is done by indexing the coordinates of R 2 n by x ∈ Z n 2 and letting the x th coordinate of f have value f ( x ). We define two bases for working in R 2 n . The first is the ”standard” one, SB = standard basis for R 2 n = { e x = (0 , , . . . , 1 (index x) , . . . , 0) : x ∈ Z n 2 } In words, the x th basis element of SB has 1 in position x and is zero elsewhere. The second is known asand is zero elsewhere....
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 Spring '04
 RonittRubinfeld
 Linear Algebra, Algorithms, Fourier Series, inner product, Standard basis, PCP theorem, χα

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