CS 6815: Lecture 4. September 4, 2018

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1 XS = X s... X st CS 685: Lecture 4 Instructor: Eshan Chattopadhyay Scribe: Makis Arsenis and Ayush Sekhari September 4, 208 In this lecture, we will first see an algorithm to construct ɛ-biased spaces. Then we will take a detour to define characters for finite Abelian groups, and, using them define Fourier transformation for functions over F n p. We will later give an introduction on how having small fourier coefficeints is related to ɛ biasedness, which will be continued over to the next lecture. Construction of ε-biased spaces (continued from the last lecture) We present an algorithm to construct n random variables in F 2 which are ε-biased. Procedure: Let r = log 2 ( n ε ), q = 2 r n ε. (i) Pick random y, z F q. (ii) For i {0,..., n } set X i = y i, z F 2. Randomness used: 2 log ( n ε ). Claim.. Consider n random variables X = {X 0,..., X n } constructed using the procedure above. Then X is an ε-biased space. Proof. For a random variable Z taking values in {0, }, let us define the function Bias(Z) as the bias of the random variable Z, i.e., Bias(Z) = Pr[Z = ] Pr[Z = 0] Additionally, for any set S = {s,..., s t } [0, n ], S, define: Recalling that a set X = {X 0,..., X n } is said to be ɛ-biased, if for all sets S [n], ( ) Bias XS ɛ We need to show that the set X = {X 0,..., X n } constructed using the procedure above forms an ɛ-biased space. Consider a set S = {s,..., s t } [0, n ], S The notation a, b for a, b F q means inner product with a, b viewed as vectors of the space F r 2, i.e. a, b = (mod 2) where ai, bi are the i-th bits of a and, b respectively. r i= aibi

2 XS = X s... X st = y s, z y st, z = y s y st, z 2 Let P (y) be be the univariate polynomial over F q defined by P (y) = t i= ys i. We can now rewrite the above as: XS = P (y), z Notice the following facts: (i) If y is sampled from a distribution Y for which we are guaranteed that P (y) 0 then: (ii) Otherwise, Pr [ P (y), z = 0] = y Y,z F2 r 2 0 Pr [P (y) = 0] deg(p ) n < ε () y F q q q The above follows using the Schwartz-Zippel lemma and the fact that P is a univariate polynomial over F q of degree at most n. Thus, we have: [ ] Pr XS = 0 = Pr[ P (y), z = 0 P (y) 0] Pr[P (y) 0] + Pr[P (y) = 0] = ( Pr[P (y) = 0]) + Pr[P (y) = 0] 2 = 2 + Pr[P (y) = 0] 2 Using () with the above expression, we get: and consequently, [ ] 2 Pr XS = ε 2 2 ε [ ] 2 Pr XS = 2 And thus, Bias( X S ) = ε [ ] [ Pr XS = Pr XS = 0] We now take a detour and discuss Fourier Analysis on Finite Abelian groups which will be useful in constructing ɛ-biased spaces. 2 Notice that in the last equality, the symbol + on the left-hand side denotes addition over F 2 whereas in the right-hand side denotes addition over F q. 2

3 2 Fourier Analysis on Finite Abelian Groups Suggested Reading: Chapter of the book Analysis of Boolean Functions by Ryan O Donnel. 2. Characters of Finite Abelian Groups Definition 2. (Group Homomorphism). A group homomorphism χ : G G 2 is a map between two groups (G, ) and (G 2, ) such that the group operation is preserved, i.e. x, y G, χ(x y) = χ(x) χ(y) A consequence of the above definition is that: χ() = and χ(g ) = (χ(g)) for all g G. Example 2.2. Let (G, ) = (Z, +), (G 2, ) = (Z m, +). homomorphism from Z to Z m. Then χ(a) = (a mod m) is a group From now on, we will be restricting ourselves to Finite Abelian Groups, and denote them by G. Aditionally, we will define S to be the set of unit norm complex numbers, i.e. S := {x C : x = } where C denotes complex numbers. We are ready to define characters of a group: Definition 2.3. (Character) A character of G is a homomorphism χ : G S. Definition 2.4. (Trivial Character) A character χ : G S is called trivial if χ(g) =, for all g G. The following gives examples of characters for G = (Z m, +). Claim 2.5. Let G = (Z m, +) be a finite Abelian group. Also, let ω denote the m th primitive root of unity (over C), i.e., ω = e 2πi/m where i 2 =. Then, the mapping χ j : G S defined by χ j (x) := ω jx for all j [m], is a group homomorphism from G to S. Proof. Let x, y G. Then: Thus, X j is a group homomorphism. χ j (x + y) = ω j(x+y) = ω jx ω jy = χ j (x) χ j (y) 3 We will now show that χ j distinct for j [m] and exhaustive. Claim 2.6. G = (Z m, +) has exactly m characters. Additionally, the set χ = {χ j j [m]} of characters as defined above has cardinality m, i.e., χ = m, and includes all the characters of G. Proof. Let us consider χ to be a character of G. characterized by setting the value of χ(), as: Then the mapping χ : G S is completely a [m], χ(a) = χ( ) = ( χ()) }{{} a a Thus, the number of characters is equal to the number of ways to set χ(). And, there are only m possible values to set χ() as χ(m) = (χ()) m = implies χ() is an m th root of unity. Additionally, the set χ contains all of the characters χ characterized by setting χ to ω j for some j [m]. All of them are distinct and χ = m. Thus, χ is the complete set of characters of Z m. 3 Note that ( ) in the RHS is on S. 3

4 Before proceeding, we also recall the fundamental theorem of finite Abelian groups which allows us to define characters for product groups, Theorem 2.7 (Fundamendal Theorem of finite Abelian groups). A finite Abelian group is isomorphic to a direct product of cyclic groups of prime-power order, where the decomposition is unique up to the order in which the factors are written. The fundamental theorem allows one to look at Z n as Z q Z q... Z qr, where q,..., q r are powers of prime numbers and r i= q i = n. Claim 2.8. Consider a finite Abelian group G = G G 2. If χ : G S is a character for G and χ 2 : G 2 S is a character for G 2 then χ : G C defined as χ(g) = χ (g ) χ 2 (g 2 ), where g (g, g 2 ), is a character for the direct product G = G G 2. 4 We also define some more properties of characters- Definition 2.9. Let f, g : G C be characters of the finite Abelian group G 5, (i) Inner Product: f, g = E [f(x) g(x)] x G where z is defined as the complex conjugate of z C. (ii) l p norm: ( ) /p f l p = f(x) p x G where z is defined as the absolute value of z C. (iii) L p norm: ( f L p = E x G [ f(x) p ] ) /p Claim 2.0. Let χ, χ 2 be two distinct characters of Z m. Then, (i) χ, χ 2 = 0 (ii) χ, χ = where χ, χ 2 is defined as in definition 2.9. Proof. Without the loss of generality, let us assume that χ (x) = ω jx and χ 2 (x) = ω kx, for some j, k < m, then, χ, χ 2 = E x Zm [ω jx ω kx ] = E x Zm [ω (j k)x ] (2) (i) If i k then: χ, χ 2 = (2) = m = m = 0 x= α= ω (j k)x 4 The operation ( ) in the RHS is defined over S 5 These definitions are more genereral and do not need f, g to be characters ω α (since Z m is a cyclic group) 4

5 (ii) If i = k then: χ, χ 2 = (2) = m = m = j= ω 0 j= Additive character of F n p (where p is a prime) Consider the group (F n p, +) where + means vector addition over F n p, and let χ be a non-trivial character of F p. Claim 2.. For v F n p, define χ v : F n p S as χ v (y) = χ( v, y ) 6. Then, for all v F n p, χ v is a character of (F n p, +). Proof. Let x, y F n p : χ v (x + y) = χ( v, x + y ) = χ( v, x + v, y ) = χ( v, x ) χ( v, y ) = χ v (x) χ v (y) Thus, for all v F n p, χ v is a character of (F n p, +). Claim 2.2. Consider the set X = {χ v v F n p } of the characters of (F n p, +) as defined above. Then X is a set of orthonormal functions., Proof. We will show this in two parts as follows:. For v v 2 F n p, χ v and χ v2 are orthogonal, i.e., χ v, χ v2 = 0. By definition, χ v (x) = χ( x, v ) and without loss of generality, let us assume that χ(z) = ω z for all z F p, where ω is a p th root of unity. Thus, Thus, χ v (x) = χ( v, x ) = ω v,x, and, χ v2 (x) = χ( v 2, x ) = ω v 2,x χ v, χ v2 [χ v (x) χ v2 (x)] [ω v,x ω v 2,x ] [ω v 2 v,x ] = 0 where the last equality follows from the fact that the sum of all p th roots of unity is 0, i.e., p i= ωi = 0. 6 x, y = n i= xi yi for vectors x, y Fn p. 5

6 2. For any v F n p, χ v, χ v =. Similar to the part-, without loss of generality, χ v (x) = χ( v, x ) = ω v,x, and, Thus, χ v, χ v [χ v (x) χ v (x)] [ω v,x ω v,x ] [ω v v,x ] [ω 0 ] = Thus, the set X = {χ v v F n p } is orthonormal and has cardinality p n, and correspondingly, forms an orthonormal basis to represent functions mapping F n p F p, as formalized in the following: Claim 2.3. Let V be the vector space of functions f : F n p C. Then the set X = {χ v v F n p } forms an orthonormal basis for V. In the next section, we will see exact decomposition of the given function in terms of its Fourier components. 2.2 Fourier Analysis over F n p Theorem 2.4. Given a function f : F n p C. Define ˆf : F n p C as follows: ˆf(v) = E s F n p [f(x) χ v (x)] = f, χ v Then, the function f can be written in an alternate form using ˆf as follows: f(x) = ˆf(v) χ v (x) Proof. Using the claim 2.3, {χ v } v F p n is an orthonormal basis of V. Thus, f can be expressed as: f(x) = C v χ v (x) for some constants C v, which can be calculated as follows: f, χ v = E x F n p [f(x) χ v (x)] = p n x F n p = p n x F n p u F n p C v = C v C u χ u (x) χ v (x) 6

7 where the last equality follows from the fact that χ u, χ v = 0 for u v and is otherwise. Thus, f(x) = ˆf(v) χ v (x) Note that the set X is fixed and known in advance. Thus, as shown above, f can be alternately represented using ˆf or the vector ( ˆf(v) v F n p ). This is called as the Fourier transform on the basis X, and the constant ˆf(v) is called as the Fourier coefficient for basis χ v. As we will see thoughout the course, looking at functions under the lens of Fourier transformation, provides many computational benefits and simplicity. In the following theorem, we provide an identity to compute inner product of functions in terms using their Fourier coefficients. The following theorem relates expectations in function space to dot product in the Fourier space. Theorem 2.5 (Parseval s Theorem). Let function f, g : F n p C. Then: Proof. E x F n p [f(x) g(x)] = ˆf(v) ĝ(v) E x F n p [f(x) g(x)] = f, g = ˆf(v) χ v, v w = v,w ˆf(v)χ v, ĝ(w)χ w ĝ(w) χ v = v,w = v ˆf(v) ĝ(w) χ v, χ w ˆf(v) ĝ(v) Corollary 2.6. For a given function f : F n p C, f L 2 = ˆf l 2 3 ε-biased definition under the lens of Fourier Analysis In this section, we will see how to construct ε biased distributions using Fourier coefficients. Claim 3.. Let D : F n p R be a distribution on F n p. D is an ε-biased distribution on F n p if: where ˆD(v) = p n E x D [χ v (x)]. continue in the next lecture... v F n p, v 0 : ˆD(v) ε p n 7