Decision Boundary Line (Linear/Non-Linear)

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SUMMARY

The discussion centers on the transformation of a non-linear decision boundary, specifically the equation (1 + X1)^2 + (2 − X2)^2 = 4, into a linear form by introducing quadratic terms. The participants confirm that by extending the feature space to include X1, X1^2, X2, and X2^2, the non-linear boundary can be expressed linearly. The algebraic manipulation shows that the original equation simplifies to a linear equation in the new variables, demonstrating the effectiveness of feature engineering in machine learning.

PREREQUISITES
  • Understanding of decision boundaries in machine learning
  • Familiarity with quadratic equations and algebraic manipulation
  • Knowledge of feature engineering techniques
  • Basic concepts of linear versus non-linear models
NEXT STEPS
  • Explore feature engineering techniques in machine learning
  • Learn about polynomial regression and its applications
  • Study the implications of non-linear decision boundaries in classification tasks
  • Investigate the use of kernel methods in support vector machines
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Data scientists, machine learning practitioners, and students studying classification algorithms who want to deepen their understanding of decision boundaries and feature transformations.

brojesus111
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Homework Statement



Given a non-linear decision boundary line: (1 + X1)^2 + (2 − X2)^2 = 4

Argue that while the decision boundary is not linear in terms of X1 and X2, it is linear in terms of X1,X1^2 , X2, and X2^2 .

The Attempt at a Solution



I'm honestly not sure. I realize the curve is a circle, but I don't understand how it could be turned linear by having it terms of X1,X1^2 , X2, and X2^2
 
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Is it because we are extending the feature space by including quadratic terms that can address this non-linearity?
 
It's pretty basic algebra that (1+ X1)^2+ (2- X2)^2= X1^2- 2X1+ 1+ X2^2- 4X2+ 4= 4
so X1^2- 2X1+ X2^2- 4X2+ 1= 0.

If you let Y1= X1^2 and Y2= X2^2, then you have Y1- 2X1+ Y2- 4Y1+ 1= 0 which is 'linear in X1, X2, Y1, and Y2".
 

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