Euler characteristic in topological persistence

In topological data analysis, persistence barcodes record the persistence of homological generators in a one-parameter filtration built on the data at hand. In contrast, computing the pointwise Euler characteristic (EC) of the filtration merely records the alternating sum of the dimensions of each homology vector space.

In this talk, we will show that despite losing the classical "signal/noise" dichotomy, EC tools are powerful descriptors, especially when combined with new integral transforms mixing EC techniques with Lebesgue integration. Our motivation is fourfold: their applicability to multi-parameter filtrations and time-varying data, their remarkable performance in supervised and unsupervised tasks at a low computational cost, their satisfactoryproperties as integral transforms (e.g., regularity and invertibility properties) and the expectation results on the EC in random settings. Along the way, we will give an insight into the information these descriptors record.

This talk is based on the work [https://arxiv.org/abs/2111.07829] and the joint work with Olympio Hacquard [https://arxiv.org/abs/2303.14040].