Snippets from the topological data analysis workshop

 I had the pleasure of visiting KIAS last week to attend their Workshop on Topological Data Analysis: Mathematics, Physics, and Beyond.

The programme featuring so many pure mathematics-focused talks was daunting at first, but ultimately I learnt a lot more than I would have by presenting the same work at a physics conference. This serves as a nice reminder: When we become experts at a highly specialised topic in our PhD studies (and beyond), it is easy to lose sight of the bigger picture. Changing fields or even attending a wide range of seminars outside our own expertise sparks new ideas.

Some useful tidbits from the talks:

• The Vietoris-Rips complex is not stable with respect to outliers. For example, adding a few points to a middle of a circular point cloud will completely change the form of the 1D persistence diagram (replacing the single high persistence cycle with low-persistence cycles). Stability theorems for persistence diagrams refer to small perturbations of a fixed set of point.
• Similar persistence diagrams do not imply similar data, in fact datasets with an arbitrarily large Hausdorff distance can have the same persistence diagrams.
• Mathematicians like TDA because it involves interesting problems with elegant solutions. But not all these elegant methods end up being useful in practice.
• The chemical and biological sciences benefit from having large datasets publicly available for benchmarking and standardised scoring of new machine learning / data analysis methods including TDA. This allows demonstrations of new heuristic methods to be convincing. 
• TDA can be used to estimate geometric features of data, including their (fractal) dimension, which is useful for understanding the performance of neural networks.
• The generators of 0-dimensional persistence diagrams give the minimal spanning tree for the data. This widely-used "folklore" was only rigorously proved in 2020!