Topology of Learning in Feedforward Neural Networks

Understanding how neural networks learn remains one of the central challenges in machine learning research. From random at the start of training, the weights of a neural network evolve in such a way as to be able to perform a variety of tasks, such as classifying images. Here, we study the emergence of structure in the weights by applying methods from topological data analysis. We train simple feedforward neural networks on the MNIST data set and monitor the evolution of the weights. When initialized to zero, the weights follow trajectories that branch off recurrently, thus generating trees that describe the growth of the effective capacity of each layer. When initialized to tiny random values, the weights evolve smoothly along 2-D surfaces. We show that natural coordinates on these learning surfaces correspond to important factors of variation.