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15-451 Avrim Blum 11/25/03

FFT Recap (or, what am I expected to know?) - Learning Finite State Environments. 15-451 Avrim Blum 11/25/03. FFT Recap. The basic result: Given vectors A = (a 0 , a 1 , a 2 , ..., a n-1 ), and B = (b 0 , b 1 , ..., b n-1 ),

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15-451 Avrim Blum 11/25/03

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  1. FFT Recap (or, what am I expected to know?)- Learning Finite State Environments 15-451 Avrim Blum 11/25/03

  2. FFT Recap The basic result: Given vectors • A = (a0, a1, a2, ..., an-1), and • B = (b0, b1, ..., bn-1), the FFT allows us in O(n log n) time to compute the convolution • C = (c0, c1, ..., c2n-2) where cj = a0bj + a1bj-1 + ... + ajb0. I.e., this is polynomial multiplication, where A,B,C are vectors of coefficients.

  3. How does it work? Compute F-1(F(A)¢F(B)), where “F” is FFT. • F(A) is evaluation of A at 1,w,w2,...,wm-1 • w is principal mth root of unity. m = 2n-1. E.g., w = e2pi/m. Or use modular arithmetic. • Able to do this quickly with divide-and-conquer. • F(A)¢F(B) gives C(x) at these points. • We then saw that F-1 = (1/m)F’, where F’ is same as F but using w-1.

  4. Applications (not on test) • signal analysis, lots more • Pattern matching with don’t cares: • Given text string X = x1,x2,...,xn. xi 2 {0..25} • Given pattern Y = y1,y2,...,yk. yi2 {0..25} [ {*}. • Want to find instances of Y inside X. • Idea [Adam Kalai based on Karp-Rabin]: • Pick random R: r1,r2,...,rk, ri2 1..N. E.g, N=n2 • Set ri = 0 if yk-i+1 = *. • Let T = r1yk + ... + rky1. (can do mod p > N) • Now do convolution of R and X. See if any entries match T. Each entry at most 1/N chance of false positive.

  5. OK, on to machine learning...

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