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Fourier Transforms in Computer Science Can a function [0,2p]zR be expressed as a linear combination of sin nx, cos nx ? If yes, how do we find the coefficients? If f(x)= anexp(2pi n x) S nZ then Fourier’s recipe 1 an = f(x)exp(-2pi n x) dx 0 The reason that this works is that the exp(-2pi nx) are orthonormal with respect to the inner product 1 <f,g> = f(x)g(x) dx 0 Given “good” f:[0,1]zC we define its Fourier transform as f:ZzC f(n) = 1 f(x)exp(-2pi n x) dx 0 space of functions space of functions Fourier Transform space of functions space of functions Fourier Transform 2 L [0,1] Plancherel formula Parseval’s identity isometry <f,g> ||f|| 2 = -<f,g> = ||f||2 2 L (Z) space of functions space of functions Fourier Transform convolution f*g 1 (f*g)(x)= 0 f(y)g(x-y) dy pointwise multiplication -fg Can be studied in a more general setting: interval [0,1] Lebesgue measure and integral exp(2pi n x) form a topological group G, dual of G LCA group G Haar measure and integral characters of G continuous homomorphisms GzC Fourier coefficients 0 of AC functions Linial, Mansour, Nisan ‘93 circuit output AND OR AND OR AND OR AND depth=3 AND size=8 Input: x1 x2 x3 x1 x2 x3 AC circuits 0 constant depth, polynomial size output AND OR AND OR AND OR AND depth=3 AND size=8 Input: x1 x2 x3 x1 x2 x3 Random restriction of a function n f : {0,1} z {0,1} x 1 x2 ... 0 1 x2 (1-p)/2 (1-p)/2 p xn Fourier transform over n Z2 characters c A(x)= P (-1) iA xi for each subset of {1,...,n} Fourier coefficients f(A)= P(f(x)=cA(x))-P(f(x)=c A(x)) Hastad switching lemma o f AC0z high Fourier coefficients of a random restriction are zero with high probability All coefficients of size >s are 0 with probability at least 1/d 1-1/d s 1-M(5p size of the circuit s depth ) We can express the Fourier coefficients of the random restriction of f using the Fourier coefficients of f |x| E[r(x)]=p f(x) - 2 |x| 2 |y| E[r(x) ]=p S f(x+y) (1-p) ymx Sum of the squares of the high Fourier coefficients of an AC0 Function is small - 2 1 1/d S f(x) < 2M exp(- 5e(t/2) ) |x|>s 0 Learning of AC functions Influence of variables on Boolean functions Kahn, Kalai, Linial ‘88 The Influence of variables If (x i) The influence of xi on f(x1 ,x2 , …,xn ) set the other variables randomly the probability that change of x i will change the value of the function Examples: for the AND function of n variables each variable has influence 1/2 n for the XOR function of n variables each variable has influence 1 The Influence of variables If (x i) The influence of xi in f(x1 ,x2 , …,xn ) set the other variables randomly the probability that change of x i will change the value of the function For balanced f there is a variable with influence > (c log n)/n We have a function f i such that p I(x i)=||f i || p , and the Fourier coefficients of f i can be expressed using the Fourier coefficients of f fi (x)=f(x)-f(x+i) f i(x) =2f(x) =0 if i is in x otherwise We can express the sum of the influences using the Fourier coefficients of f - 2 S I(x i ) = 4S |x| f(x) if f has large high Fourier coefficients then we are happy How to inspect small coefficients? Beckner’s linear operators f(x) |x|- a f(x) a<1 Norm 1 linear operator 1+a2 from L n (Z 2 ) 2 to L n (Z 2 ) Can get bound ignoring high FC - 2 4/3 |x| S I(x i ) > 4S |x| f(x) (1/2) Explicit Expanders Gaber, Galil ’79 (using Margulis ’73) Expander Any (not too big) set of vertices W has many neighbors (at least (1+a)|W|) positive constant W Expander Any (not too big) set of vertices W has many neighbors (at least (1+a)|W|) positive constant N(W) W |N(W)|>(1+a)|W| Why do we want explicit expanders of small degree? extracting randomness sorting networks Example of explicit bipartite expander of constant degree: Zm x Z m (x,y) Zm x Z m (x,y) (x+y,y) (x+y+1,y) (x,x+y) (x,x+y+1) Transform to a continuous problem M(s(A)-A)+M(t(A)-A)>2cM(A) 2 T measure For any measurable A one of the transformations s:(x,y)z(x+y,y) and t:(x,y)z(x,x+y) “displaces” it Estimating the Rayleigh quotient 2 2 of an operator on XmL(T ) Functions with f(0)=0 (T f) (x,y)=f(x-y,y)+f(x,y-x) r(T)=sup { |<Tx,x>| ; ||x||=1} It is easier to analyze the corres2 ponding linear operator in Z (S f)(x,y)=f(x+y,y)+f(x,x+y) Let L be a labeling of the arcs of the graph with vertex set Z x Z and edges (x,y)z(x+y,y) and (x,y)z(x,x+y) such that L(u,v)=1/L(v,u). Let C be maximum over all vertices of sum of the labels of the outgoing edges. Then r(S)cC. Lattice Duality: Banaszczyk’s Transference Theorem Banaszczyk ‘93 Lattice: given n x n regular matrix B, n a lattice is { Bx; x Z } Successive minima: k smallest r such that a ball centered in 0 of diameter r contains k linearly independent lattice points Dual lattice: Lattice L* with matrix B -T Transference theorem: * k n-k+1 n c can be used to show that O(n) approximation of the shortest lattice vector in 2-norm is not NP-hard unless NP=co-NP (Lagarias, H.W. Lenstra, Schnorr’ 90) Poisson summation formula: For “nice” f:RzC S f(x) xZ = S f(x) xZ Define Gaussian-like measure n on the subsets of R r(A)= exp(-p ||x|| S x A 2 ) Prove using Poisson summation formula: r((L+u)\B)<0.285 r(L) a ball of diameter (3/4)n1/2 centered around 0 Define Gaussian-like measure on the subsets of R p(A)= r(A L)/r(L) Prove using Poisson summation formula: * * p(u) = r(L+u)/r(L) > then we have n a vector u perpendicular to If * k n-k+1 all lattice points in L B and L* B outside ball small + small r(L* +u)/r(L* ) inside ball, “moved by u” T p(u)= S r(x)exp(-2piu x) xL large Weight of a function – sum of columns (mod m) , Therien ‘94 m 4 1 2 4 1 4 1 2 3 4 1 3 4 1 4 1 6 1 3 6 1 2 1 3 3 2 5 3 2 4 2 5 n Is there a set of columns which sum to the zero column (mod t) ? If m>c n t 11/2 then there always exists a set of columns which sum to zero column (mod t) wt(f)= number of non-zero Fourier coefficients w(fg)cw(f)w(g) w(f+g)cw(f)+w(g) t+1 is a prime x1ai,1 + fi (x)=g ... + xma i,m If no set of columns sums to the zero column mod t then t g= P (1-(1-f i ) ) m restricted to B={0,1} is 10 m n t = wt(g 1B )c t m (t-1)