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# l26 - 6.896 Sublinear Time Algorithms Lecture 26 Lecturer...

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6.896 Sublinear Time Algorithms May 10, 2007 Lecture 26 Lecturer: Ronitt Rubinfeld Scribe: Chih-yu Chao 1R e c a p f r o m L a s t T i m e 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. 2U s e s o f F o u r i e 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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l26 - 6.896 Sublinear Time Algorithms Lecture 26 Lecturer...

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