  A dimension reduction method based on the "Nonlinear Level set Learning"
(NLL) approach is presented for the pointwise prediction of functions which
have been sparsely sampled. Leveraging geometric information provided by the
Implicit Function Theorem, the proposed algorithm effectively reduces the input
dimension to the theoretical lower bound with minor accuracy loss, providing a
one-dimensional representation of the function which can be used for regression
and sensitivity analysis. Experiments and applications are presented which
compare this modified NLL with the original NLL and the Active Subspaces (AS)
method. While accommodating sparse input data, the proposed algorithm is shown
to train quickly and provide a much more accurate and informative reduction
than either AS or the original NLL on two example functions with
high-dimensional domains, as well as two state-dependent quantities depending
on the solutions to parametric differential equations.
