  Scientists and engineers rely on accurate mathematical models to quantify the
objects of their studies, which are often high-dimensional. Unfortunately,
high-dimensional models are inherently difficult, i.e. when observations are
sparse or expensive to determine. One way to address this problem is to
approximate the original model with fewer input dimensions. Our project goal
was to recover a function f that takes n inputs and returns one output, where n
is potentially large. For any given n-tuple, we assume that we can observe a
sample of the gradient and output of the function but it is computationally
expensive to do so. This project was inspired by an approach known as Active
Subspaces, which works by linearly projecting to a linear subspace where the
function changes most on average. Our research gives mathematical developments
informing a novel algorithm for this problem. Our approach, Active Manifolds,
increases accuracy by seeking nonlinear analogues that approximate the
function. The benefits of our approach are eliminated unprincipled parameter,
choices, guaranteed accessible visualization, and improved estimation accuracy.
