Huge data sets containing millions of training examples with a large number of attributes are relatively easy to gather. However one of the bottlenecks for successful inference is the computational complexity of machine learning algorithms. Most state-of-the-art nonparametric machine learning algorithms have a computational complexity of either O(N^2) or O(N^3), where N is the number of training examples. This has seriously restricted the use of massive data sets. The bottleneck computational primitive at the heart of various algorithms is the multiplication of a structured matrix with a vector, which we refer to as matrix-vector product (MVP) primitive. The goal of my thesis is to speedup up some of these MVP primitives by fast approximate algorithms that scale as O(N) and also provide high accuracy guarantees. I use ideas from computational physics, scientific computing, and computational geometry to design these algorithms. The proposed algorithms have been applied to speedup kernel density estimation, optimal bandwidth estimation, projection pursuit, Gaussian process regression, implicit surface fitting, and ranking.