# A Blast from the Past

## Tags: research

« Previous post: Metric Spaces and How To Compare Them — Next post: Taming the E-Mail Avalanche »In what I can only describe as a small wonder, four publications of mine
have been finally published as book chapters! They were initially
submissions of the workshop Topology-Based Methods in
Visualization, thus predating my
current career in machine learning quite a bit. It is *instructive*, to
say the least, to see how much my research interests shifted and
developed over the years, so seeing these papers finally being published
fills me with a weird form of nostalgia. Let’s briefly discuss them!

# Hierarchies and Ranks for Persistence Pairs

Hierarchies and Ranks for Persistence Pairs is the most theoretical of the four papers. We discuss a novel way of assessing the stability of persistence pairs—tuples describing topological features—based on their relation to other persistence pairs in the data. Our method makes it possible to capture more fine-grained relations than what one could normally get using persistence diagrams. In particular, our resulting hierarchies can be easily compared with one another! If you are interested in learning more about the expressivity of persistent homology, this might be an interesting paper for you.

# Persistent Intersection Homology for the Analysis of Discrete Data

In Persistent Intersection Homology for the Analysis of Discrete
Data, we
discuss a novel perspective for analysing data sets. Specifically, we
think about what happens when the manifold hypothesis does not hold any
more, but instead, we are dealing with data sets that are *almost* like
a manifold, except for some singularities. One good way to think about
such spaces is in terms of stratified spaces,
and intersection homology enables us to measure how ‘far’ such spaces
are from being manifolds.

# Topological Machine Learning with Persistence Indicator Functions

Topological Machine Learning with Persistence Indicator Functions can be seen as the foundation for what I
am doing these days, viz. the development of novel algorithms that
employ a topology-based perspective to improve machine learning
algorithms. This paper presents what is now known as a *Betti curve*.
These curves are simple summaries of topological descriptors, making it
possible to calculate similarities or even distances quite efficiently.
The flip side is a loss of predictive power—but as my preliminary
results show, this is still *good enough* in many cases.

This paper describes my first foray into the wonderful world of machine learning, and I like to believe that I learned a lot in the past few years. Nevertheless, this paper might make an interesting read for someone looking for a brief introduction to topological data analysis.

# Persistence Concepts for 2D Skeleton Evolution Analysis

Persistence Concepts for 2D Skeleton Evolution Analysis is an interesting paper, in which we describe new persistence-inspired concepts for analysing the evolution of 2D skeletons. We use this to assess the Saffman–Taylor instability in fluids. Our method enables the comparison of different experiments and simulations with real-world observations. Looking into this was fascinating, and I hope that I will have the opportunity to continue working on this.

# Parting thoughts

I am happy that these papers are available now, even more so because I am still an early-career researcher and glad for any kind of exposure that I can get. At the same time, my research interests have changed and evolved over the past few years, but in retrospective, the vestigial or rudimentary traces of the shape of things to come are already present in these papers. If I had known earlier how welcoming and open-minded the machine learning community can be, I would have made the switch earlier.

I hope that you, dear reader, find these papers somewhat exciting as well. Until next time!