Numerical example on Support Vector Machines

Question

I was watching this numerical problem on SVM link here. At 4.42 he wrote this equation:

$$ \begin{align} \alpha_1 \tilde{s_1} \cdot \tilde{s_1} + \alpha_2 \tilde{s_2} \cdot \tilde{s_1} + \alpha_3 \tilde{s_3} \cdot \tilde{s_1} &= -1 \\ \alpha_1 \tilde{s_1} \cdot \tilde{s_2} + \alpha_2 \tilde{s_2} \cdot \tilde{s_2} + \alpha_3 \tilde{s_3} \cdot \tilde{s_2} &= +1 \\ \alpha_1 \tilde{s_1} \cdot \tilde{s_3} + \alpha_2 \tilde{s_2} \cdot \tilde{s_3} + \alpha_3 \tilde{s_3} \cdot \tilde{s_3} &= +1 \end{align} $$

where $\tilde{s_1}, \tilde{s_2}, \tilde{s_3}$ are support vectors. But from the support vector we see that: $$ w = \sum \alpha_i y_i s_i$$ where $y_i$ is class label and $s_i$ is support vector. Since $\alpha$'s are zero for all other points except support vector so I am considering only the support vector here. My question is if we look at the above equation he just wrote: $$ w = \sum \alpha_i s_i$$

I can't understand how he wrote $ w = \sum \alpha_i s_i$ and get the correct answer yet?

Answer 1

he just wrote: w=∑αisi

I dont think so. He wrote $ \sum_i \alpha_ix_i^Tx$ where $x$ is the test sample. He took $x$ as each support vector and hence that summation will result to 1 or -1 (depending on class).