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Appropriately representing elements in a database so that queries may be accurately matched is a central task in information retrieval; recently, this has been achieved by embedding the graphical structure of the database into a manifold in…

Topological abstractions offer a method to summarize the behavior of vector fields but computing them robustly can be challenging due to numerical precision issues. One alternative is to represent the vector field using a discrete approach,…

In recent years, many large directed networks such as online social networks are collected with the help of powerful data engineering and data storage techniques. Analyses of such networks attract significant attention from both the…

Energy-based models for discrete domains, such as graphs, explicitly capture relative likelihoods, naturally enabling composable probabilistic inference tasks like conditional generation or enforcing constraints at test-time. However,…

We propose a novel perspective on varied-density clustering for high-dimensional data by framing it as a label propagation process in neighborhood graphs that adapt to local density variations. Our method formally connects density-based…

Persistent homology is a method for computing the topological features present in a given data. Recently, there has been much interest in the integration of persistent homology as a computational step in neural networks or deep learning. In…

Given a large graph, a graph sample determines a subgraph with similar characteristics for certain metrics of the original graph. The samples are much smaller thereby accelerating and simplifying the analysis and visualization of large…

In this paper, we propose a path following replicator dynamic, and investigate its potentials in uncovering the underlying cluster structure of a graph. The proposed dynamic is a generalization of the discrete replicator dynamic. The…

While modern representation learning relies heavily on global error signals, decentralized algorithms driven by local interactions offer a fundamental distributed alternative. However, the macroscopic convergence properties of these…

Data-stream clustering is an ever-expanding subdomain of knowledge extraction. Most of the past and present research effort aims at efficient scaling up for the huge data repositories. Our approach focuses on qualitative improvement, mainly…

Persistent homology is a topological feature used in a variety of applications such as generating features for data analysis and penalizing optimization problems. We develop an approach to accelerate persistent homology computations…

Bayesian inference plays an important role in advancing machine learning, but faces computational challenges when applied to complex models such as deep neural networks. Variational inference circumvents these challenges by formulating…

In this paper, we construct and study derived character maps of finite-dimensional representations of $\infty$-groups. As models for $\infty$-groups we take homotopy simplicial groups, i.e. homotopy simplicial algebras over the algebraic…

The rapid growth of graph data creates significant scalability challenges as most graph algorithms scale quadratically with size. To mitigate these issues, Graph Condensation (GC) methods have been proposed to learn a small graph from a…

Persistent homology is a popular tool in Topological Data Analysis. It provides numerical characteristics of data sets which reflect global geometric properties. In order to be useful in practice, for example for feature generation in…

This paper builds on the connection between graph neural networks and traditional dynamical systems. We propose continuous graph neural networks (CGNN), which generalise existing graph neural networks with discrete dynamics in that they can…

Cartograms are maps that rescale geographic regions (e.g., countries, districts) such that their areas are proportional to quantitative demographic data (e.g., population size, gross domestic product). Unlike conventional bar or pie charts,…

Graph-structured data arise in many scenarios. A fundamental problem is to quantify the similarities of graphs for tasks such as classification. R-convolution graph kernels are positive-semidefinite functions that decompose graphs into…

Topological data analysis is an emerging area in exploratory data analysis and data mining. Its main tool, persistent homology, has become a popular technique to study the structure of complex, high-dimensional data. In this paper, we…

Modern continuous-time generative models typically induce \emph{V-shaped} flows: each sample travels independently along a nearly straight trajectory from the prior to the data. Although effective, this independent movement overlooks the…