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The aim of this paper is to rigorously study dynamics of Heterogeneously Coupled Maps (HCM). Such systems are determined by a network with heterogeneous degrees. Some nodes, called hubs, are very well connected while most nodes interact…
Real-world optimization problems are generally not just black-box problems, but also involve mixed types of inputs in which discrete and continuous variables coexist. Such mixed-space optimization possesses the primary challenge of modeling…
This paper presents a new clustering algorithm for space-time data based on the concepts of topological data analysis and in particular, persistent homology. Employing persistent homology - a flexible mathematical tool from algebraic…
A topological shape analysis is proposed and utilized to learn concepts that reflect shape commonalities. Our approach is two-fold: i) a spatial topology analysis of point cloud segment constellations within objects. Therein constellations…
We present an approach to learning features that represent the local geometry around a point in an unstructured point cloud. Such features play a central role in geometric registration, which supports diverse applications in robotics and 3D…
Deep learning methods have achieved impressive performance for multi-class medical image segmentation. However, they are limited in their ability to encode topological interactions among different classes (e.g., containment and exclusion).…
In persistent homology analysis, interval modules play a central role in describing the birth and death of topological features across a filtration. In this work, we extend this setting, and propose the use of bipath persistent homology,…
Extracting high-level structural information from 3D point clouds is challenging but essential for tasks like urban planning or autonomous driving requiring an advanced understanding of the scene at hand. Existing approaches are still not…
Topological data analysis is a powerful tool for describing topological signatures in real world data. An important challenge in topological data analysis is matching significant topological signals across distinct systems. In geometry and…
Persistence diagrams have been widely recognized as a compact descriptor for characterizing multiscale topological features in data. When many datasets are available, statistical features embedded in those persistence diagrams can be…
A wide variety of complex systems are characterized by interactions of different types involving varying numbers of units. Multiplex hypergraphs serve as a tool to describe such structures, capturing distinct types of higher-order…
This paper aims to discuss a method of quantifying the 'shape' of data, via a methodology called topological data analysis. The main tool within topological data analysis is persistent homology; this is a means of measuring the shape of…
Clustering of mixed-type datasets can be a particularly challenging task as it requires taking into account the associations between variables with different level of measurement, i.e., nominal, ordinal and/or interval. In some cases,…
Network theory provides various tools for investigating the structural or functional topology of many complex systems found in nature, technology and society. Nevertheless, it has recently been realised that a considerable number of systems…
A suitable feature representation that can both preserve the data intrinsic information and reduce data complexity and dimensionality is key to the performance of machine learning models. Deeply rooted in algebraic topology, persistent…
In standard persistent homology, a persistent cycle born and dying with a persistence interval (bar) associates the bar with a concrete topological representative, which provides means to effectively navigate back from the barcode to the…
In real-world systems, the relationships and connections between components are highly complex. Real systems are often described as networks, where nodes represent objects in the system and edges represent relationships or connections…
The statistical mechanical approach to complex networks is the dominant paradigm in describing natural and societal complex systems. The study of network properties, and their implications on dynamical processes, mostly focus on locally…
Graph-based collaborative filtering is capable of capturing the essential and abundant collaborative signals from the high-order interactions, and thus received increasingly research interests. Conventionally, the embeddings of users and…
Persistent homology, a central tool of topological data analysis, provides invariants of data called barcodes (also known as persistence diagrams). A barcode is simply a multiset of real intervals. Recent work of Edelsbrunner, Jablonski,…