  SNe Ia continue to play a key role in cosmological measurements. Their
interpretation over a range in redshift requires a rest-frame spectral energy
distribution model. For practicality, these models are parameterized with a
limited number of parameters and are trained using linear or nonlinear
dimensionality reduction. This work focuses on the related problem of
estimating the number of parameters underlying SN Ia spectral variation (the
dimensionality). I present a technique for using the properties of
high-dimensional space and the counting statistics of "twin" SNe Ia to estimate
this dimensionality. Applying this method to the supernova pairings from
Fakhouri et al. (2015) shows that a modest number of parameters (three to five,
not including extinction) explain those data well. The analysis also finds that
the intrinsic parameters are approximately Gaussian-distributed. The limited
number of parameters hints that improved SED models are possible that may
enable substantial reductions in SN cosmological uncertainties with current and
near-term datasets.
