Fourier Transforms in Computer Science

Transcription

1 Fourier Transforms in Computer Science

2 Can a function [0,2π] R be expressed as a linear combination of sin nx, cos nx? If yes, how do we find the coefficients?

3 Σ If f(x)= a exp(2πi n x) then n Z n 1 a = n f(x)exp(-2πi n x) dx The reason that this works is that 0 the exp(-2πi nx) are orthonormal with respect to the inner product 1 <f,g> = f(x)g(x) dx 0 Fourier s recipe

4 Given good f:[0,1] C we define its Fourier transform as f:z C 1 f(n) = f(x)exp(-2πi n x) dx 0 space of functions space of functions Fourier Transform

5 space of functions space of functions Fourier Transform 2 2 L [0,1] isometry L (Z) Plancherel formula <f,g> = <f,g> Parseval s f = f identity 2 2

6 space of functions space of functions Fourier Transform convolution f*g pointwise multiplication f g 1 (f*g)(x)= f(y)g(x-y) dy 0

7 Can be studied in a more general setting: interval [0,1] LCA group G Lebesgue measure and integral Haar measure and integral exp(2πi n x) characters of G form a topological group G, dual of G continuous homomorphisms G C

8 Fourier coefficients 0 of AC functions Linial, Mansour, Nisan 93

9 circuit output AND OR OR OR depth=3 AND AND AND AND size=8 Input: x x x x x x

10 AC 0 circuits output constant depth, polynomial size AND OR OR OR depth=3 AND AND AND AND size=8 Input: x x x x x x

11 Random restriction of a function n f : {0,1} {0,1} x 1 x 2... x n 0 1 x 2 (1-p)/2 (1-p)/2 p

12 Fourier transform over Z 2 n characters χ A (x)= Π (-1) i A xi for each subset of {1,...,n} Fourier coefficients f(a)= P(f(x)= χ (x))-p(f(x)= χ (x)) A A

13 Hastad o switching lemma f AC high Fourier coefficients of a random restriction are zero with high probability 0 All coefficients of size >s are 0 with probability at least 1-M(5p 1/d s 1-1/d ) s size of the circuit depth

14 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 Σ 2 y y x E[r(x) ]=p f(x+y) (1-p)

15 Sum of the squares of the high Fourier coefficients of an AC Function is small Σ f(x) < 2M exp(- (t/2) ) x >s 2 1 5e 1/d 0 Learning of AC 0 functions

16 Influence of variables on Boolean functions Kahn, Kalai, Linial 88

17 The Influence of variables I (x ) f i The influence of x on f(x,x,,x ) i 1 2 n set the other variables randomly the probability that change of xiwill change the value of the function Examples: for the AND function of n variables each variable has influence 1/2n for the XOR function of n variables each variable has influence 1

18 The Influence of variables I (x ) f i The influence of x in f(x,x,,x ) i 1 2 n set the other variables randomly the probability that change of xiwill change the value of the function For balanced f there is a variable with influence > (c log n)/n

19 We have a function f such that i i p p I(x )= f, and the Fourier coefficients of f can be expressed i using the Fourier coefficients of f i f (x)=f(x)-f(x+i) i f (x) =2f(x) if i is in x i = 0 otherwise

20 We can express the sum of the influences using the Fourier coefficients of f ΣI(x i ) Σ = 4 x f(x) 2 if f has large high Fourier coefficients then we are happy How to inspect small coefficients?

21 Beckner s linear operators f(x) x a f(x) a<1 Norm 1 linear operator 1+a 2 n 2 2 from L (Z ) to L (Z ) n 2 Can get bound ignoring high FC 4/3 ΣI(x i ) > 4 Σ 2 x x f(x) (1/2)

22 Explicit Expanders Gaber, Galil 79 (using Margulis 73)

23 Expander Any (not too big) set of vertices W has many neighbors (at least (1+a) W ) positive constant W

24 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

25 Why do we want explicit expanders of small degree? extracting randomness sorting networks

26 Example of explicit bipartite expander of constant degree: Z mx Zm Z x Z m m (x,y) (x+y,y) (x,y) (x+y+1,y) (x,x+y) (x,x+y+1)

27 Transform to a continuous problem T 2 M(s(A)-A)+M(t(A)-A)>2cM(A) measure For any measurable A one of the transformations s:(x,y) (x+y,y) and t:(x,y) (x,x+y) displaces it

28 Estimating the Rayleigh quotient of an operator on X L(T ) 2 2 Functions with f(0)=0 (T f) (x,y)=f(x-y,y)+f(x,y-x) r(t)=sup { <Tx,x> ; x =1}

29 It is easier to analyze the corresponding linear operator in Z 2 (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) (x+y,y) and (x,y) (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) C.

30 Lattice Duality: Banaszczyk s Transference Theorem Banaszczyk 93

31 Lattice: given n x n regular matrix B, n a lattice is { Bx; x Z }

32 Successive minima: λk smallest r such that a ball centered in 0 of diameter r contains k linearly independent lattice points

33 Dual lattice: Lattice L * with matrix B -T Transference theorem: λ λ k * n-k+1 n 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)

34 Poisson summation formula: For nice f:r C Σ x Z Σ f(x) = f(x) x Z

35 Define Gaussian-like measure on the subsets of R n r(a)= Σ 2 exp(-π x ) x A Prove using Poisson summation formula: r((l+u)\b)<0.285 r(l) 1/2 a ball of diameter (3/4)n centered around 0

36 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)

37 If λ λ k * n-k+1 > n then we have a vector u perpendicular to all lattice points in * L B and L B outside ball small + small r(l * +u)/r(l *) inside ball, moved by u p(u)= Σ T r(x)exp(-2πiu x) x L large

38 Weight of a function sum of columns (mod m), Therien 94

39 m n Is there a set of columns which sum to the zero column (mod t)?

40 11/2 If m>c n t 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) w(f)w(g) w(f+g) w(f)+w(g)

41 t+1 is a prime f (x)=g x 1 a i, x m a i i,m If no set of columns sums to the zero column mod t then g= Π (1-(1-f ) ) m restricted to B={0,1} is 10 i t m t = wt(g 1 ) B t n (t-1) m