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I am looking at the wikipedia article for hard-margin SVMs and it looks like the optimization problem they use is

"minimize ||w|| such that the classes are linearly separable"

However isn't the point of SVMs to find the hyperplane with the largest margin that separates the datapoints? But since it is trying to minimize ||w|| wouldn't this result in the smallest separating hyperplane? So shouldn't this be a maximization problem rather than a minimization problem?

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Because a distance from the origin to a plane is inversely proportional to $\lVert w \rVert$, where $w$ is a vector normal to the plane. For an appropriate parametrisation, the margin to be maximized is $2/\lVert w \rVert$.

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  • $\begingroup$ I see. Is there an intuitive explanation for what ||w|| represents? I assumed it meant the length of the normal vector but I don't think that is correct. $\endgroup$ – user1893354 May 11 at 22:47
  • $\begingroup$ I don't see much of an intuition in this or equivalent formulae. Onee needs to got through necessary geometric and algebraic steps to get to the final form. Perhaps you can start at this wiki page and then explore some university lecture notes: en.wikipedia.org/wiki/Distance_from_a_point_to_a_plane $\endgroup$ – dnqxt May 11 at 23:18

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