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1

Your solution changes because you scale all features but you neglect to rescale the misclassification penalty $C$. The SVM cost function is as follows: $$\min \|\mathbf{w}\|^2 + C \sum_i \xi_i$$ where $\mathbf{w}$ is the separating hyperplane, $\xi$ is a vector of slack variables associated to misclassification of training instances and $C$ is a ...

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The standard "distance" metric that one typically uses for this type of problem is the Mahalanobis distance. Assuming that you choose to adopt the Mahalanobis distance as your preferred distance metric, then step 1 that you outlined is indeed correct: normalize all vectors in the training set $V$, plus the new test vector $\overline{v}$, using the L2-norm. ...

2

Feature normalization is to make different features in the same scale. The scaling speeds up gradient descent by avoiding many extra iterations that are required when one or more features take on much larger values than the rest(Without scaling, the cost function that is visualized will show a great asymmetry). I think it makes sense that use the mean and ...

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Your approach is entirely correct. Although data transformations are often undervalued as "preprocessing", one cannot emphasize enough that transformations in order to optimize model performance can and should be treated as part of the model building process. Reasoning: A model shall be applied on unseen data which is in general not available at the time ...

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Look up Box-Cox transformation, I think that is what you mean. The Wikipedia page looks pretty good and has a bunch of references.

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