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Perceptron Algorithm

Perceptron Algorithm. x 1. x 2. x n. Perceptron. Linear threshold unit (LTU). x 0 = 1. w 0. w 1. w 2. .  i =0 n w i x i. g. w n. 1 if  i =0 n w i x i >0 o ( x i ) = -1 otherwise. {. Possibilities for function g. Sign function.

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Perceptron Algorithm

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  1. Perceptron Algorithm

  2. x1 x2 xn Perceptron • Linear threshold unit (LTU) x0=1 w0 w1 w2  . . . i=0nwi xi g wn 1 if i=0nwi xi>0 o(xi) = -1 otherwise {

  3. Possibilities for function g Sign function Sigmoid (logistic) function Step function sign(x) = +1, if x > 0 -1, if x 0 step(x) = 1, if x> threshold 0, if x threshold (in picture above, threshold = 0) sigmoid(x) = 1/(1+e-x) Adding an extra input with activation x0 = 1 and weight wi, 0 = -T (called the bias weight) is equivalent to having a threshold at T. This way we can always assume a 0 threshold.

  4. Using a Bias Weight to Standardize the Threshold 1 -T w1 x1 w2 x2 w1x1+ w2x2 < T w1x1+ w2x2 - T< 0

  5. t = -1 t = 1 o=1 w = [0.25 –0.1 0.5] x2 = 0.25 x1 – 0.5 o=-1 (x, t)=([2,1], -1) o =sgn(0.45-0.6+0.3) =1 (x, t)=([-1,-1], 1) o = sgn(0.25+0.1-0.5) =-1 w = [0.2 –0.2 –0.2] w = [-0.2 –0.4 –0.2] (x, t)=([1,1], 1) o = sgn(0.25-0.7+0.1) = -1 w = [0.2 0.2 0.2] -0.5x1+0.3x2+0.45>0  o = 1 Perceptron Learning Rule x2 x2 x1 x1 x2 x2 x1 x1

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