Topological data analysis leverages topological features to analyze datasets, with applications in diverse fields like medical sciences and biology. A key tool of this theory is the persistence diagram, which encodes topological information but poses challenges for integration into standard machine learning pipelines. We introduce Qupid (QUantized Persistence and Integral transforms of Diagrams), a novel and simple method for vectorizing persistence diagrams. First, Qupid uses a binning procedure to turn persistence diagrams into finite measures on a grid and then applies discrete transforms to these measures. Key features are the choice of log-scaled grids that emphasize information contained near the diagonal in persistence diagrams, combined with the use of discrete transforms to enhance and efficiently encode the obtained topological information. We conduct an in-depth experimental analysis of Qupid, showing that the simplicity of our method results in very low computational costs while preserving highly competitive performances compared to state-of-the-art methods across numerous classification tasks on both synthetic and real-world datasets. Finally, we provide experimental evidence that our method is robust to a decrease in the grid resolution used.

In this article, we study Euler characteristic techniques in topological data analysis. Pointwise computing the Euler characteristic of a family of simplicial complexes built from data gives rise to the so-called Euler characteristic profile. We show that this simple descriptor achieves state-of-the-art performance in supervised tasks at a meagre computational cost. Inspired by signal analysis, we compute hybrid transforms of Euler characteristic profiles. These integral transforms mix Euler characteristic techniques with Lebesgue integration to provide highly efficient compressors of topological signals. As a consequence, they show remarkable performances in unsupervised settings. On the qualitative side, we provide numerous heuristics on the topological and geometric information captured by Euler profiles and their hybrid transforms. Finally, we prove stability results for these descriptors as well as asymptotic guarantees in random settings.

We study a regression problem on a compact manifold M. In order to take advantage of the underlying geometry and topology of the data, the regression task is performed on the basis of the first several eigenfunctions of the Laplace-Beltrami operator of the manifold, that are regularized with topological penalties. The proposed penalties are based on the topology of the sub-level sets of either the eigenfunctions or the estimated function. The overall approach is shown to yield promising and competitive performance on various applications to both synthetic and real data sets. We also provide theoretical guarantees on the regression function estimates, on both its prediction error and its smoothness (in a topological sense). Taken together, these results support the relevance of our approach in the case where the targeted function is “topologically smooth”.