Regularization governs the theory of machine learning and inverse problems. Classical regularization schemes for data-fitting results in solutions to optimization problems that are splines. Exponential splines generalize the classical polynomial splines and have many desirable properties. This project would explore the proper formulation for regularization in learning/inverse problems that results in solutions which are exponential splines. This would entail defining the appropriate search space (or "native space") for the optimization problem. The project would then explore properties of these native spaces by studying the approximation rates of functions in these spaces with exponential splines.