Topological obstructions in neural networks learning

We apply methods of topological data analysis to loss functions to gain insights on learning of deep neural networks and their generalization properties. We study global properties of the loss function gradient flow. We use topological data analysis of the loss function and its Morse complex to relate local behavior along gradient trajectories with global properties of the loss surface. We define neural network Topological Obstructions score, TO-score, with help of robust topological invariants, barcodes of loss function, that quantify the badness of local minima for gradient-based optimization. We have made several experiments for computing these invariants, for small neural networks, and for fully connected, convolutional and ResNet-like neural networks on different datasets: MNIST, Fashion MNIST, CIFAR10, SVHN. Our two principal observations are as follows. Firstly, the neural network barcode and TO-score decrease with the increase of the neural network depth and width. Secondly, there is an intriguing connection between the length of minima segments in the barcode and the minima generalization error.