  We propose to test, and when possible establish, an equivalence between two
different artificial neural networks by attempting to construct a data-driven
transformation between them, using manifold-learning techniques. In particular,
we employ diffusion maps with a Mahalanobis-like metric. If the construction
succeeds, the two networks can be thought of as belonging to the same
equivalence class.
  We first discuss transformation functions between only the outputs of the two
networks; we then also consider transformations that take into account outputs
(activations) of a number of internal neurons from each network. In general,
Whitney's theorem dictates the number of measurements from one of the networks
required to reconstruct each and every feature of the second network. The
construction of the transformation function relies on a consistent, intrinsic
representation of the network input space.
  We illustrate our algorithm by matching neural network pairs trained to learn
(a) observations of scalar functions; (b) observations of two-dimensional
vector fields; and (c) representations of images of a moving three-dimensional
object (a rotating horse). The construction of such equivalence classes across
different network instantiations clearly relates to transfer learning. We also
expect that it will be valuable in establishing equivalence between different
Machine Learning-based models of the same phenomenon observed through different
instruments and by different research groups.
