How to show $\rho > 0$ when $\rho$ be minimum attainable from $y_n(W^{*T}X_n)$, where $W^*$ the vector that separates the data?

Question

In the book Learning from Data written (by Abu Mostafa), we have the following exercise:

Let $\rho$ be minimum attainable from $y_n(W^{*T}X_n)$ where $W^*$ is the vector that separates the data. Show $\rho > 0$. Also assume the Perceptron Learning Algorithm is initialized with the 0 vector.

How to prove the above statement?

I thought that it could be negative since a Perceptron function returns either +/-1?

Even I wonder if I comprehend this proof question correctly.

Answer 1

We have a vector $w*$ such that it has separated all of the data points. This implies if the correct classification of a point is -1, then $w^{*T}x_n$ is also negative. If $y_n$ is positive, then $w^{*T}x_n$ is positive. Thus is because we've stated every point is correctly classed. Thus $\rho>0$