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Is it correct to say that the reason why ||w|| is set to 1 in the formula for the geometric margin is that it then is the same as the functional margin (i. e. gives the same information) while still being scale invariant?

I come to that from this site: Support Vector Machine (SVM) where it says "If ||w||=1, the functional margin is equal to geometric margin. "

If not, then why can ||w|| not be set to 0.7 oder 2.53 or 14?

I have read various blogs and pdfs on the topic. Some don't adress this at all or just say things along the lines of "this is a mathematical trick" and don't really explain it.

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The constraint is interesting in itself as it prevent divergences (or other strange behavior) for the optimisation problem that is the calibration process. The value it is set to not that much. $||w|| = 1$ is a practical convention. For $||w||$ to be constrained to 2.53 you would have need to :

  • Have some practical interest for it to being set to 2.53, which there is none. $||w|| = 1$ might help simplify some theoretical calculations.
  • Reach a consensus between users that it would be good to use this value. It's not hard to see why the consensus was reached for 1 instead of 2.53.
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