  Machine Learning (ML) applications are proliferating in the enterprise.
Relational data which are prevalent in enterprise applications are typically
normalized; as a result, data has to be denormalized via primary/foreign-key
joins to be provided as input to ML algorithms. In this paper, we study the
implementation of popular nonlinear ML models, Gaussian Mixture Models (GMM)
and Neural Networks (NN), over normalized data addressing both cases of binary
and multi-way joins over normalized relations.
  For the case of GMM, we show how it is possible to decompose computation in a
systematic way both for binary joins and for multi-way joins to construct
mixture models. We demonstrate that by factoring the computation, one can
conduct the training of the models much faster compared to other applicable
approaches, without any loss in accuracy.
  For the case of NN, we propose algorithms to train the network taking
normalized data as the input. Similarly, we present algorithms that can conduct
the training of the network in a factorized way and offer performance
advantages. The redundancy introduced by denormalization can be exploited for
certain types of activation functions. However, we demonstrate that attempting
to explore this redundancy is helpful up to a certain point; exploring
redundancy at higher layers of the network will always result in increased
costs and is not recommended.
  We present the results of a thorough experimental evaluation, varying several
parameters of the input relations involved and demonstrate that our proposals
for the training of GMM and NN yield drastic performance improvements typically
starting at 100%, which become increasingly higher as parameters of the
underlying data vary, without any loss in accuracy.
