Understanding Kernel Ridge Regression: Common behaviors from simple functions to density functionals

Machine learning (ML) is a powerful data-driven method for learning patterns in high-dimensional spaces via induction, and has had widespread success in many fields including more recent applications in quantum chemistry and materials science [1-9]. Here we are interested in applications of ML to construction of density functionals [10-14], which have focused so far on approximating the kinetic energy (KE) of non-interacting electrons. An accurate, general approximation to this could make orbital-free DFT a practical reality. However, ML methods have been developed within the areas of statistics and computer science, and have been applied to a huge variety of data, including neuroscience, image and text processing, and robotics [15]. Thus, they are quite general and have not been tailored to account for specific details of the quantum problem.