Topological Approaches to Deep Learning

Deep neural networks [10] are a powerful and fascinating methodology for solving problems with large and complex data sets. They use directed graphs as a template for very large computations, and have demonstrated a great deal of success in the study of various kinds of data, including images, text, time series, and many others. One issue that restricts their applicability, however, is the fact that it is not understood in any kind of detail how they work. A related problem is that there is often a certain kind of overfitting to particular data sets, which results in the possibility of so-called adversarial behavior, where they can be made to fail by making very small changes to image data that is almost imperceptible to a human. For these reasons, it is very desirable to develop methods for gaining understanding of the internal states of the neural networks.