Towards Topological Machine Learning

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I am incredibly grateful about how my academic year started so far: four preprints were at least conditionally accepted for publication in a forthcoming book on topological methods in data visualization, while another publication of my new lab was accepted as a poster for ICLR 2019.

The underlying theme of all these publications is to shift the focus of machine learning towards topological methods, i.e. methods that focus on connectivity properties of input data. I am convinced that thinking about these types of properties is worthwhile, as the resulting shift in perspective often leads to novel insights.

This spring of papers follows two themes: in the first, topology is used directly to drive algorithms, for example to classify data, or to elucidate its properties. In the second theme, topology is used indirectly to learn something about the behaviour of other algorithms.

Topology as a driver for algorithms

In Persistent Intersection Homology for the Analysis of Discrete Data, Markus Banagl, Filip Sadlo, Heike Leitte, and I describe how to use persistent intersection homology, an extension of persistent homology in order to describe spaces that do not consist of a single manifold, but of multiple ones. It turns out that as long as these multiple manifolds intersect in somewhat controlled ways, we are able to extract more information than “ordinary” persistent homology. This has the potential to result in a new view on traditional manifold learning algorithms because it seems highly unlikely that input data only come from a single manifold—despite our best efforts to pretend that this is the case.

Moreover, in Topological Machine Learning with Persistence Indicator Functions, Filip Sadlo, Heike Leitte, and I describe a functional summary of persistence diagrams—topological descriptors that typically arise during the calculation of persistent homology— that is easy to calculate, can be used for statistical hypothesis testing, and gives rise to a kernel function. Without going into the details, the last property is the most exciting one, at least for me, because it permits using several powerful machine learning algorithms, such as kernel support vector machines, to use these summaries for classification. In the paper, we demonstrate that we are capable of outperforming a state-of-the art method for non-attributed graph analysis, but this is just the beginning. I hope that the availability of our method will encourage more research in this direction.

Topology to understand deep learning

Following the second theme, in Neural Persistence: A Complexity Measure for Deep Neural Networks Using Algebraic Topology, Matteo Togninalli, Christian Bock, Michael Moor, Max Horn, Thomas Gumbsch, Karsten Borgwardt, and I developed a novel topology-based method for analysing (deep) fully-connected networks. Our method determines the amount of topological activity of the network during the training process and relates it to a theoretical maximum that depends on the corresponding architecture. We demonstrate that our measure—dubbed neural persistence— is capable of assessing the complexity of different (simple) networks. Moreover, we show how it can be used as a valid criterion for early stopping that does not rely on additional validation data. Thanks to a thorough review process, we added many additional experiments in the supplementary materials, including an extremely detailed analysis of early stopping scenarios on different data sets, compared with a standard criterion that employs validation loss. Of course, the code is publicly available.

All in all, this was an extremely productive spring of papers. I am grateful for my collaborators, team mates, and friends, in particular Matteo and Christian, who were instrumental in the conception, planning, and execution of this project.

Happy reading, until next time!