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I am new to Machine Learning. Suppose a training set of positive (square) and negative (circle) points is given like:

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Obviously there would be no nice linear separator of positive and negative points. So assume there exist a transformation for (x1,x2) points as: y1 = (x1-5)^2, y2 = (x2-5)^2 in (y1,y2) space. We will then map all positive points to (25,25) and all negative points to (1,1).

Now, the question is how can I find the maximum-margin separator in new space? That is, what curve in the original space transforms to the straight-line boundary in the transformed space? Using that I need to determine that should a new point be considered positive or negative.

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Map all the points to the new space. Then find the maximum-margin linear separator of those mapped points. Algorithms for finding a maximum-largin linear separator are described in the literature on SVMs (that's exactly what you need to do to train a linear SVM). In other words, you can map the points to the new space, then can take those mapped points and train a linear SVM on them. That's conceptually how to do it.

In practice, nonlinear SVMs use clever tricks to do the same thing, but do it faster (using the kernel trick).

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  • $\begingroup$ Thank you. All the points are mapped into 2 points in new space (1,1) and (25,25). The question would then be how to find the line equation (without using tools like sklearn)? $\endgroup$ – fhm Apr 5 '18 at 12:45
  • $\begingroup$ @fhm, as I said in my question, the same way you find a separating line for a linear SVM. You do know how to do that, right? If not, you should study linear SVMs -- that algorithm is explained in many standard places. $\endgroup$ – D.W. Apr 5 '18 at 16:17

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