Explore a 23-minute conference talk from the Workshop on "One World Optimization Seminar in Vienna" held at the Erwin Schrödinger International Institute for Mathematics and Physics (ESI). Delve into L1-regularized optimization problems and their associated smooth "over-parameterized" optimization problems built on the Hadamard difference parametrization (HDP). Discover how second-order stationary points of the HDP-based model correspond to stationary points of the L1-regularized model. Learn about the Kurdyka-Łojasiewicz (KL) exponent of the HDP-based model and how it relates to the L1-regularized model under specific assumptions. Examine the applicability of these concepts to various loss functions commonly used in L1-regularizations, such as least squares and logistic loss functions. Gain insights into how KL exponents can be used to determine the local convergence rate of standard gradient methods for minimizing HDP-based models.