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.

Topological integral transforms have found many applications in shape analysis, from prediction of clinical outcomes in brain cancer to analysis of barley seeds. Using Euler characteristic as a measure, these objects record rich geometric information on weighted polytopal complexes. While some implementations exist, they only enable discretized representations of the transforms, and they do not handle weighted complexes (such as for instance images). Moreover, recent hybrid transforms lack an implementation. In this paper, we introduce eucalc, a novel implementation of three topological integral transforms - the Euler characteristic transform, the Radon transform, and hybrid transforms - for weighted cubical complexes. Leveraging piecewise linear Morse theory and Euler calculus, the algorithms significantly reduce computational complexity by focusing on critical points. Our software provides exact representations of transforms, handles both binary and grayscale images, and supports multi-core processing. It is publicly available as a C++ library with a Python wrapper. We present mathematical foundations, implementation details, and experimental evaluations, demonstrating eucalc’s efficiency.

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.

This paper addresses two questions: (a) can we identify a sensible class of 2-parameter persistence modules on which the rank invariant is complete? (b) can we determine efficiently whether a given 2-parameter persistence module belongs to this class? We provide positive answers to both questions, and our class of interest is that of rectangle-decomposable modules. Our contributions include: on the one hand, a proof that the rank invariant is complete on rectangle-decomposable modules, together with an inclusion-exclusion formula for counting the multiplicities of the summands; on the other hand, algorithms to check whether a module induced in homology by a bifiltration is rectangle-decomposable, and to decompose it in the affirmative, with a better complexity than state-of-the-art decomposition methods for general 2-parameter persistence modules. Our algorithms are backed up by a new structure theorem, whereby a 2-parameter persistence module is rectangle-decomposable if, and only if, its restrictions to squares are. This local characterization is key to the efficiency of our algorithms, and it generalizes previous conditions derived for the smaller class of block-decomposable modules. It also admits an algebraic formulation that turns out to be a weaker version of the one for block-decomposability. By contrast, we show that general interval-decomposability does not admit such a local characterization, even when locality is understood in a broad sense. Our analysis focuses on the case of modules indexed over finite grids, the more general cases are left as future work.

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