Barcodes as summary of loss function's topology

We apply the canonical forms (barcodes) of Morse complexes to explore topology of loss surfaces. We present a novel algorithm for calculations of the objective function's barcodes of minima. We have conducted experiments for calculating barcodes of local minima forbenchmark functions and for loss surfaces of neural networks. Our experiments confirm two principal observations: (1) the barcodes of minima are located in a small lower part of the range of values of loss function of neural networks and (2) increase of the neural network's depth brings down the minima's barcodes. This has natural implications for the neural network learning and the ability to generalize.