6.896 Sublinear Time Algorithms
May 10, 2007
Lecture 26
Lecturer: Ronitt Rubinfeld
Scribe: Chihyu Chao
1R
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c
a
p
f
r
o
m
L
a
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T
i
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Defnition 1 (Fourier coﬃcient)
ˆ
f
(
s
)=
h
f, χ
s
i
=
1
2
n
∑
x
∈{±
1
}
n
f
(
x
)
χ
s
(
x
)
Theorem 1
∀
f,x
f
(
x
∑
s
⊂
[
n
]
ˆ
f
(
s
)
χ
s
(
x
)
Last time, we learned that the magnitute of the Fourier coeﬃcient is in agreement with the linear
function. Based on this, we can use Fourier analysis to test linearity.
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s
e
s
o
f
F
o
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r
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r
A
n
a
l
y
s
i
s
Now, we are going to use Fourier analysis to:
1. estimate sums of powers of degree 1 Fourier coeﬃcient of
f
2. estimate the maximum of degree 1 Fourier coeﬃcient
2.1
Estimate sums of powers
We would like to estimate sums of powers of degree 1 Fourier coeﬃcients of
f
, given oracle access to
f
.
(We are only inerested in sets of size 1, and thus “degree 1”.)
X

s

=1
ˆ
f
(
s
)
p
(for
p
≥
2)
The algorithm is a random process:
•
Pick
x
(1)
,x
(2)
,...,x
(
p
−
1)
∈
R
{±
1
}
n
uniformly
•
Pick noise vector
μ
such that entry
+1
−
1
with probability
1
2
+
η
2
1
2
−
η
2
•
y
←
f
(
x
(1)
)
f
(
x
(2)
)
···
f
(
x
(
p
−
1)
)
f
(
x
(1)
²
x
(2)
x
(
p
−
1)
²
μ
)
(note that for
p
=3,italmost
looks like a linearity test with noise)
We will show: E [
y
]=
∑
s
⊆
[
n
]
η

s

ˆ
f
(
s
)
p
, which will look a lot like the analysis of the linearity test last
time.
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 Fall '04
 RonittRubinfeld
 Algorithms, Mathematical analysis, Fourier Coefficient

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