Related papers: Learning on Persistence Diagrams as Radon Measures

Modern representation learning increasingly relies on unsupervised and self-supervised methods trained on large-scale unlabeled data. While these approaches achieve impressive generalization across tasks and domains, evaluating embedding…

Artificial neural networks can learn complex, salient data features to achieve a given task. On the opposite end of the spectrum, mathematically grounded methods such as topological data analysis allow users to design analysis pipelines…

We want to approximate general multivariate probability density functions by deterministic sample sets. For optimal sampling, the closeness to the given continuous density has to be assessed. This is a difficult challenge in multivariate…

We introduce a new model for planar point point processes, with the aim of capturing the structure of point interaction and spread in persistence diagrams. Persistence diagrams themselves are a key tool of TDA (topological data analysis),…

Low-rank matrices are pervasive throughout statistics, machine learning, signal processing, optimization, and applied mathematics. In this paper, we propose a novel and user-friendly Euclidean representation framework for low-rank matrices.…

We build on the dynamical systems approach to deep learning, where deep residual networks are idealized as continuous-time dynamical systems, from the approximation perspective. In particular, we establish general sufficient conditions for…

Persistence diagrams, the most common descriptors of Topological Data Analysis, encode topological properties of data and have already proved pivotal in many different applications of data science. However, since the (metric) space of…

Datasets containing geo-spatial resources are increasingly being represented according to the Linked Data principles. Several time-efficient approaches for discovering links between RDF resources have been developed over the last years.…

This paper presents a generalization of the Wasserstein distance for both persistence diagrams and merge trees [20], [66] that takes advantage of the regions of their topological features in the input domain. Specifically, we redefine the…

Random Projections have been widely used to generate embeddings for various graph learning tasks due to their computational efficiency. The majority of applications have been justified through the Johnson-Lindenstrauss Lemma. In this paper,…

We develop the theoretical foundations of a generalized Gromov-Hausdorff distance between functions on networks that has recently been applied to various subfields of topological data analysis and optimal transport. These functional…

Generative modeling is typically framed as learning mapping rules, but from an observer's perspective without access to these rules, the task becomes disentangling the geometric support from the probability distribution. We propose that…

This paper focuses on various decompositions of topological measures, deficient topological measures, signed topological measures, and signed deficient topological measures. These set functions generalize measures and correspond to certain…

Deep learning has been wildly successful in practice and most state-of-the-art machine learning methods are based on neural networks. Lacking, however, is a rigorous mathematical theory that adequately explains the amazing performance of…

Reeb graphs are a fundamental structure for analyzing the topological and geometric properties of scalar fields. Comparing Reeb graphs is crucial for advancing research in this domain, yet existing metrics are often computationally…

We improve and extend persistence spheres, introduced in~\cite{pegoraro2025persistence}. Persistence spheres map an integrable measure $\mu$ on the upper half-plane, including persistence diagrams (PDs) as counting measures, to a function…

To have a superior generalization, a deep learning neural network often involves a large size of training sample. With increase of hidden layers in order to increase learning ability, neural network has potential degradation in accuracy.…

Deep learning techniques have shown promise in many domain applications. This paper proposes a novel deep reservoir computing framework, termed deep recurrent stochastic configuration network (DeepRSCN) for modelling nonlinear dynamic…

Persistent homology is a popular computational tool for analyzing the topology of point clouds, such as the presence of loops or voids. However, many real-world datasets with low intrinsic dimensionality reside in an ambient space of much…

Throughout this paper, a persistence diagram ${\cal P}$ is composed of a set $P$ of planar points (each corresponding to a topological feature) above the line $Y=X$, as well as the line $Y=X$ itself, i.e., ${\cal P}=P\cup\{(x,y)|y=x\}$.…