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I think that your missing the fact that $r$ is not a random constant, but precisely the minimum distance we want to ensure between each point and the separation hyperplane (called margin). And indeed in the book they use $r$ to define the distance from a generic vector $a$ to the hyperplane as well: $a = a' + r\frac{w}{||w||}$ Since $\frac{w}{||w||}$ is a ...


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