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Suppose we have a linear classifier for the classes ω1 and ω2 with characteristics vectors Xa=[a a]^T and Xb=[-a -a]^T correspondingly. Also suppose that the decision boundary that is defined by Perceptron at the repetition t is given by the relation g(X1,X2; γ_t)= X1+X2+γ_t*a=0 Also suppose P_t=1 for each t, and the correct classification is done for γ_t in the (-2,2) range, where Xa is misclassified for γ_t<-2 and Xb for γ_t>2

How would I prove that if during the (t)th application of the perceptron algorithm the value of γ_t is outside the range of correct classification (-2,2), after the (t+1)th application, the updated form of the decision boundary will be gt(X1,X2;γ_t)=X1+X2+γ_(t+1)*a=0

and that γ_(t+1) will be

(γ_t*a+1)/(a+1), if γ_t<-2 and

(γ_t*a-1)/(a+1), if γ_t>2

I thought about defining original weights Wt=[1,1] and updating the γ_t=γ_t+1 but that didn't get me anywhere

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  • $\begingroup$ Please, edit your post to format the formulas with MathJax. $\endgroup$
    – nbro
    Sep 11, 2023 at 9:59

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