# Confusion related to convexity and concavity of a problem

I was reading this paper http://www.ist.temple.edu/~vucetic/documents/wang11kdd.pdf related to adaptive multi-hyperplane machine for non linear classification

In that paper, they have mentioned about multiclass SVM, with multiple weights for each class.

The loss for any classification is

$l(x_n,y_n) = max_{i\epsilon y\\\y_n}(0,1 + max g(i,x_n) - g(y_n,x_n))$

where $y_n$ is the label for the nth example and $x_n$ is the features.

I have this confusion when they do the training of this algorithm. They call this SVM MM(Multiple Hyperplane).

They say the convex-approximated problem is defined as

$min_{W}P(W|z) = \frac{\lambda}{2}||W||^2 + \frac{1}{N}\sum_{n=1}^{N}l_{cvx}(W;(x_n,y_n);z_n)$

where they have the concave term $-g(y_n,z_n)$ replaced with the convex term $-w^T_{y_n,z_n}x_n$.

I am not sure if I have described it clearly. But I am going to attach the screenshot of the paper as well. The thing is I didn't get what's the difference between $-g(y_n,z_n)$ and $-w^T_{y_n,z_n}x_n$. They seem the same term to me.

I might be asking a lot. But can anyone provide some info?

I have marked by the red rectangle the part that I didn't understand. I might be asking a lot. But I didn't get that part. Why is it so?

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