How does linear SVC generate the predictions after learning the feature weights

Question

I'm fitting an SVC model with linear kernel, after that i'm checking the dot product of the fitted weights by the input to understand the prediction better. From my understanding of the linear SVC model, $x_1$ will be assigned to the positive class if

$$ w \cdot x_1 + b > 1 $$

and the negative class if

$$ w \cdot x_1 + b < -1 $$

However calculating the dot product by hand is not giving any clear cut as you can see below.

Is my understanding of how SVC generates the prediction after learning the weights of the features correct or am i missing something?

It's worth mentioning that the values in the histogram are identical to what i get from decision_function(X) so to rephrase my question: how did the estimator in scikit-learn decide what values of the decision function are assigned to what class? Since as you can see from the histogram decision_function > 0 are not assigned to the positive class and decisioN_function < 0 are not assigned to the negative class.

Answer 1

The conditions you quote are for training: SVM attempts to find $w$ and $b$ such that

$$ y_i (w \cdot x_i + b) \ge 1 $$

where $y_i \in \{-1, 1\}$.

However, this can only succeed if the classes are linearly separable. If not, as your case suggests, you need a soft margin SVM, by introducing slack variables:

$$ y_i (w \cdot x_i + b) \ge 1 - \xi_i $$

For classification, the result is simply compared to 0: If $w \cdot x_i + b > 0$, classify $x_i$ to class "$+1$", and otherwise to "$-1$".