  We present an approach to analyze $C^1(\mathbb{R}^m)$ functions that
addresses limitations present in the Active Subspaces (AS) method of
Constantine et al.(2015; 2014). Under appropriate hypotheses, our Active
Manifolds (AM) method identifies a 1-D curve in the domain (the active
manifold) on which nearly all values of the unknown function are attained, and
which can be exploited for approximation or analysis, especially when $m$ is
large (high-dimensional input space). We provide theorems justifying our AM
technique and an algorithm permitting functional approximation and sensitivity
analysis. Using accessible, low-dimensional functions as initial examples, we
show AM reduces approximation error by an order of magnitude compared to AS, at
the expense of more computation. Following this, we revisit the sensitivity
analysis by Glaws et al. (2017), who apply AS to analyze a magnetohydrodynamic
power generator model, and compare the performance of AM on the same data. Our
analysis provides detailed information not captured by AS, exhibiting the
influence of each parameter individually along an active manifold. Overall, AM
represents a novel technique for analyzing functional models with benefits
including: reducing $m$-dimensional analysis to a 1-D analogue, permitting more
accurate regression than AS (at more computational expense), enabling more
informative sensitivity analysis, and granting accessible visualizations(2-D
plots) of parameter sensitivity along the AM.
