  Artificial intelligence is revolutionizing our lives at an ever increasing
pace. At the heart of this revolution is the recent advancements in deep neural
networks (DNN), learning to perform sophisticated, high-level tasks. However,
training DNNs requires massive amounts of data and is very computationally
intensive. Gaining analytical understanding of the solutions found by DNNs can
help us devise more efficient training algorithms, replacing the commonly used
mthod of stochastic gradient descent (SGD). We analyze the dynamics of SGD and
show that, indeed, direct computation of the solutions is possible in many
cases. We show that a high performing setup used in DNNs introduces a
separation of time-scales in the training dynamics, allowing SGD to train
layers from the lowest (closest to input) to the highest. We then show that for
each layer, the distribution of solutions found by SGD can be estimated using a
class-based principal component analysis (PCA) of the layer's input. This
finding allows us to forgo SGD entirely and directly derive the DNN parameters
using this class-based PCA, which can be well estimated using significantly
less data than SGD. We implement these results on image datasets MNIST, CIFAR10
and CIFAR100 and find that, in fact, layers derived using our class-based PCA
perform comparable or superior to neural networks of the same size and
architecture trained using SGD. We also confirm that the class-based PCA often
converges using a fraction of the data required for SGD. Thus, using our method
training time can be reduced both by requiring less training data than SGD, and
by eliminating layers in the costly backpropagation step of the training.
