Kurdyka-\L ojasiewicz exponent via square transformation

We consider one of the most common reparameterization techniques, the square transformation. Assuming the original objective function is the sum of a smooth function and a polyhedral function, we study the variational properties of the objective function after reparameterization. In particular, we first study the minimal norm of the subdifferential of the reparameterized objective function. Second, we compute the second subderivative of the reparameterized objective function on a linear subspace, which allows for fully characterizing the subclass of stationary points of the reparameterized objective function that correspond to stationary points of the original objective function. Finally, utilizing the representation of the minimal norm of the subdifferential, we show that the Kurdyka-\L ojasiewicz (KL) exponent of the reparameterized function can be deduced from that of the original function.