  Many modern data sets are sampled with error from complex high-dimensional
surfaces. Methods such as tensor product splines or Gaussian processes are
effective/well suited for characterizing a surface in two or three dimensions
but may suffer from difficulties when representing higher dimensional surfaces.
Motivated by high throughput toxicity testing where observed dose-response
curves are cross sections of a surface defined by a chemical's structural
properties, a model is developed to characterize this surface to predict
untested chemicals' dose-responses. This manuscript proposes a novel approach
that models the multidimensional surface as a sum of learned basis functions
formed as the tensor product of lower dimensional functions, which are
themselves representable by a basis expansion learned from the data. The model
is described, a Gibbs sampling algorithm proposed, and is investigated in a
simulation study as well as data taken from the US EPA's ToxCast high
throughput toxicity testing platform.
