This course gives an introduction to computational data analytics, with emphasis on its mathematical foundations. The goal is to carefully develop and explore mathematical theories and methods that make up the backbone of modern mathematical data sciences, such as knowledge discovery in databases, machine learning, and mathematical artificial intelligence. Topics include mathematical foundations of probability, linear approximation and its polynomial and high dimensional extensions, proper orthogonal decomposition methods, optimization, theories of nonlinear neural network and approximations. Students taking this course are expected to have knowledge of basic linear algebra.