i don't understand why a specific function satiesfies the Kurdika - Lojasewicz inequality. I have the function:
$\sum_{i=1}^{K}\rho^2_{\text{hinge}}(y_i(w^Tx_i+b))+f(w)$
where $w\in\mathbb{R}^n,b,x,y\in\mathbb{R}$ and $ \rho^2_{\text{hinge}}$ is just the squared hinge loss and the function f is defined as:
$(\forall w=(w_i)_{1\leq i\leq N}\in\mathbb{R}^N) \quad f(w)=\sum^N_{i=1} \varphi(w_i) +||w||^2$
for the function $\varphi$ we have the 2 choices
$(\forall w\in\mathbb{R})\quad \varphi(w)=\lambda\sqrt{w^2+\delta^2}$
and
$(\forall w\in\mathbb{R})\quad \varphi(w)=\lambda(1-\exp{(-\frac{w^2}{2\delta^2})})$
where $\lambda,\delta>0$.
In the first choice of $\varphi$ we only have algebraic functions, so its pretty clear that it satasifies the Kurdika - Lojasewicz inequality.
But I am not sure how to proof that in the second case. Would really appreciate your help!