Fourier%20Transforms

1
Fourier Transforms in Computer Science
2
Can a function 0,2pzR be expressed as a linear
combination of sin nx, cos nx ?
If yes, how do we find the coefficients?
3
S
If f(x) a exp(2pi n x)
n
n Z
Fouriers recipe
then
a f(x)exp(-2pi n x) dx
n
The reason that this works is that
the exp(-2pi nx) are orthonormal with respect to
the inner product
ltf,ggt f(x)g(x) dx
4
Given good f0,1zC we define its Fourier
transform as fZzC
-
-
f(n) f(x)exp(-2pi n x) dx
space of functions
space of functions
Fourier Transform
5
space of functions
space of functions
Fourier Transform
2
2
L 0,1
L (Z)
isometry
-
-
Plancherel formula
ltf,ggt
ltf,ggt

-
Parsevals identity
f
f

2
2
6
space of functions
space of functions
Fourier Transform
pointwise multiplication
convolution
-
-
f g
fg
(fg)(x) f(y)g(x-y) dy
7
Can be studied in a more general setting
LCA group G
interval 0,1
Lebesgue measure and integral
Haar measure and integral
characters of G
exp(2pi n x)
form a topological group G, dual of G
continuous homomorphisms GzC
-
8
Fourier coefficients of AC functions
0
Linial, Mansour, Nisan 93
9
circuit
output
AND
depth3
OR
OR
OR
AND
AND
AND
AND
size8
x
x
x
x
x
x
Input
1
2
3
3
2
1
10
AC
circuits
0
constant depth, polynomial size
output
AND
depth3
OR
OR
OR
AND
AND
AND
AND
size8
x
x
x
x
x
x
Input
1
2
3
3
2
1
11
Random restriction of a function
n
f 0,1 z 0,1
x
x
x
. . .
1
2
n
x
1
0
2
p
(1-p)/2
(1-p)/2
12
n
Fourier transform over Z
2
characters
x
for each subset of 1,...,n
P
i
c
(-1)
(x)
A
i A
Fourier coefficients
-
c
c
f(A) P(f(x) (x))-P(f(x) (x))
A
A
13
Hastad switching lemma
o
0
f AC z high Fourier coefficients of a random
restriction are zero with high probability
All coefficients of size gts are 0 with
probability at least
s
1/d
1-1/d
1-M(5p s )
depth
size of the circuit
14
We can express the Fourier coef- ficients of the
random restriction of f using the Fourier
coefficients of f
-
-
x
Er(x)p f(x)
-
-
S
2
2
y
x
Er(x) p f(xy) (1-p)
ymx
15
Sum of the squares of the high Fourier
coefficients of an AC Function is small
0
-
1/d
1
S
2
f(x) lt 2M exp(- (t/2) )
5e
xgts
0
Learning of AC functions
16
Influence of variables on Boolean functions
Kahn, Kalai, Linial 88
17
The Influence of variables
I (x )
i
f
The influence of x on f(x ,x , ,x )
i
n
1
2
set the other variables randomly
the probability that change of x will change the
value of the function
i
Examples for the AND function of n variables
each variable has influence 1/2
for the XOR
function of n variables each
variable has influence 1
n
18
The Influence of variables
I (x )
i
f
The influence of x in f(x ,x , ,x )
i
n
1
2
set the other variables randomly
the probability that change of x will change the
value of the function
i
For balanced f there is a variable with influence
gt (c log n)/n
19
We have a function f such that I(x )f ,
and the Fourier coef- ficients of f can be
expressed using the Fourier coefficients of f
i
p
i
i
p
i
f (x)f(x)-f(xi)
i
-
-
f (x) 2f(x) if i is in x 0
otherwise
i
20
We can express the sum of the influences using
the Fourier coefficients of f
-
S
S
2
I(x )
4
x f(x)

i
if f has large high Fourier coefficients then we
are happy
How to inspect small coefficients?
21
Beckners linear operators
-
-
x
f(x)
a f(x)
alt1
Norm 1 linear operator from L (Z ) to L (Z
)
2
2
n
n
1a
2
2
Can get bound ignoring high FC
-
S
4/3
S
x
2
I(x )
gt
4
x f(x) (1/2)
i
22
Explicit Expanders
Gaber, Galil 79 (using Margulis 73)
23
Expander
Any (not too big) set of vertices W has many
neighbors (at least (1a)W)
positive constant
W
24
Expander
Any (not too big) set of vertices W has many
neighbors (at least (1a)W)
positive constant
N(W)
W
N(W)gt(1a)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 x Z
Z x Z
m
m
m
m
(x,y)
(xy,y) (x,y) (xy1,y)
(x,xy) (x,xy1)
27
Transform to a continuous problem
M(s(A)-A)M(t(A)-A)gt2cM(A)
2
T
measure
For any measurable A one of the transformations
s(x,y)z(xy,y) and t(x,y)z(x,xy) displaces
it
28
Estimating the Rayleigh quotient of an operator
on X L(T )
2
2
m
-
Functions with f(0)0
(T f) (x,y)f(x-y,y)f(x,y-x)
r(T)sup ltTx,xgt x1
29
It is easier to analyze the corres- ponding
linear operator in Z
2
(S f)(x,y)f(xy,y)f(x,xy)
Let L be a labeling of the arcs of the graph with
vertex set Z x Z and edges (x,y)z(xy,y) and
(x,y)z(x,xy) 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.
30
Lattice Duality Banaszczyks
Transference Theorem
Banaszczyk 93
31
Lattice
given n x n regular matrix B, a lattice is Bx
x Z
n
32
Successive minima

• smallest r such that a ball
• centered in 0 of diameter r
• contains k linearly
• independent lattice points

k
33
Dual lattice
-T
Lattice L with matrix B

Transference theorem
l l c n

k
n-k1
can be used to show that O(n) approximation of
the shortest lattice vector in 2-norm is
not NP-hard unless NPco-NP
(Lagarias, H.W. Lenstra, Schnorr 90)
34
Poisson summation formula
For nice
fRzC
-
S
S
f(x) f(x)
x Z
x Z
35
Define Gaussian-like measure on the subsets of R
n
S
2
r(A) exp(-p x )
x A
Prove using Poisson summation formula
r((Lu)\B)lt0.285 r(L)
a ball of diameter (3/4)n centered around 0
1/2
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(Lu)/r(L)


37
l l gt n

If

then we have
k
n-k1
a vector u perpendicular to all lattice points in
L B and L B
outside ball

small small
inside ball, moved by u
r(L u)/r(L )


-
S
T
p(u) r(x)exp(-2piu x)
x L
large
38
Weight of a function sum of columns (mod m)
,
Therien 94
39
m
2
1
2
4
1
1
4
4
1
3
1
4
4
3
4
1
n
3
1
3
6
1
1
6
2
2
3
2
5
2
3
4
5
Is there a set of columns which sum to the zero
column (mod t) ?
40
If mgtc n t then there always exists a set of
columns which sum to zero column (mod t)
11/2
wt(f) number of non-zero Fourier
coefficients
w(fg)cw(f)w(g)
w(fg)cw(f)w(g)
41
t1 is a prime
x a
x a
...
1
i,1
m
i,m
f (x)g
i
If no set of columns sums to the zero column mod
t then
P
t
g
(1-(1-f ) )
i
m
restricted to B0,1 is 1
0
m
m
n
wt(g 1 )c t (t-1)
t
B