Kurdyka-{\L}ojasiewicz exponent via Hadamard parametrization

Understand

We consider a class of $\ell_1$-regularized optimization problems and the associated smooth "over-parameterized" optimization problems built upon the Hadamard parametrization, or equivalently, the Hadamard difference parametrization (HDP).

• We characterize the set of second-order stationary points of the HDP-based model and show that they correspond to some stationary points of the corresponding $\ell_1$-regularized model.
• More importantly, we show that the Kurdyka-Lojasiewicz (KL) exponent of the HDP-based model at a second-order stationary point can be inferred from that of the corresponding $\ell_1$-regularized model under suitable assumptions.
• Our assumptions are general enough to cover a wide variety of loss functions commonly used in $\ell_1$-regularized models, such as the least squares loss function and the logistic loss function.