On the Expressivity of Persistent Homology in Graph Learning

Persistent homology, a technique from computational topology, has recently shown strong empirical performance in the context of graph classification. Being able to capture long range graph properties via higher-order topological features, such as cycles of arbitrary length, in combination with multi-scale topological descriptors, has improved predictive performance for data sets with prominent topological structures, such as molecules. At the same time, the theoretical properties of persistent homology have not been formally assessed in this context. This paper intends to bridge the gap between computational topology and graph machine learning by providing a brief introduction to persistent homology in the context of graphs, as well as a theoretical discussion and empirical analysis of its expressivity for graph learning tasks

'Shall I compare thee to a network?': Visualizing the Topological Structure of Shakespeare’s Plays

Many of the plays of William Shakespeare are almost universally known and continue to be played even 400 years after his death. Although the plots of the plays are in general very different, scholars are still discussing similarities in their language, their structure, and many other aspects. In this paper, we demonstrate that visualization approaches may support such an analysis. The presence of machine-readable annotations for each of the plays permits us to construct a set of weighted networks. Every network describes co-occurrence relations between individual characters of a play; its weights may be used to indicate the importance of a connection between two characters, for instance. We subject the networks to a topology-based analysis that permits us to assess their structural similarity. Moreover, we use the dissimilarity values to obtain a topology-based embedding of all the plays. We then proceed to show how features in the dramatic structure of the play manifest themselves in the embedding. This paper is thus a first step towards a more in-depth analysis of the plays, demonstrating the benefits of topology-based visualizations for the digital humanities

Filtration Surfaces for Dynamic Graph Classification

Existing approaches for classifying dynamic graphs either lift graph kernels to the temporal domain, or use graph neural networks (GNNs). However, current baselines have scalability issues, cannot handle a changing node set, or do not take edge weight information into account. We propose filtration surfaces, a novel method that is scalable and flexible, to alleviate said restrictions. We experimentally validate the efficacy of our model and show that filtration surfaces outperform previous state-of-the-art baselines on datasets that rely on edge weight information. Our method does so while being either completely parameter-free or having at most one parameter, and yielding the lowest overall standard deviation among similarly scalable methods

Evaluating the "Learning on Graphs" Conference Experience

With machine learning conferences growing ever larger, and reviewing processes becoming increasingly elaborate, more data-driven insights into their workings are required. In this report, we present the results of a survey accompanying the first "Learning on Graphs" (LoG) Conference. The survey was directed to evaluate the submission and review process from different perspectives, including authors, reviewers, and area chairs alike

Smoothness analysis of subdivision algorithms

In computer graphics, subdivision algorithms are common tools for smoothing down irregularly shaped meshes. Of special interest, due to their simple formulations, are algorithms that generalize B-spline subdivision. Their conceptual simplicity is in stark contrast to the complexity of analysing their results. A complete formal examination of smoothness properties for subdivision schemes was only recently performed by Jörg Peters and Ulrich Reif. This thesis presents a precise and detailed introduction to the analysis of subdivision algorithms. For this purpose, first of all, the necessary background in B-spline theory is established. Building on this, two of the most common subdivision algorithms, the Doo-Sabin and the Catmull-Clark scheme, are motivated. Their treatment is followed by an in-depth description of methods for analysing smoothness properties of subdivision schemes, as developed by Peters and Reif. Afterwards, these methods are applied to the two aforementioned algorithms, thereby establishing smoothness for both algorithms in their original form. Last, in order to demonstrate the effects of choosing unsuitable weights, a number of degenerate weights, which produce irregular shapes in almost all cases, are derived for both schemes—these have hitherto not been published

Unwrapping Highly-Detailed 3D Meshes of Rotationally Symmetric Man-Made Objects

Rotationally symmetric objects commonly occur at archæological finds. Instead of creating 2D images for documentation purposes by manual drawing or photographic methods, we propose a method based on digitally colored surface models that are acquired by 3D scanners, thereby including color information. We then transform these highly-detailed meshes using simple geometrical objects such as cones and spheres and unwrap the objects onto a plane. Our method can handle curved vessel profiles by dividing the surface into multiple segments and approximating each segment with a cone frustum that serves as an auxiliary surface. In order to minimize distortions, we introduce a simple quality measure based on distances of points to a fitted cone. We then extend our method to approximately spherical objects by fitting a sphere on the surface of the object and applying a map projection, namely the equirectangular projection known from cartography. Our implementation generates true-to-scale images from triangular meshes. Exemplary results demonstrate our methods on real objects, ranging from small and medium-sized objects such as clay cones from the Ancient Orient and figural friezes of Greek vessels to extremely large objects such as the remains of a cylindrical tower of Heidelberg Castle

Persistent Homology in Multivariate Data Visualization

Technological advances of recent years have changed the way research is done. When describing complex phenomena, it is now possible to measure and model a myriad of different aspects pertaining to them. This increasing number of variables, however, poses significant challenges for the visual analysis and interpretation of such multivariate data. Yet, the effective visualization of structures in multivariate data is of paramount importance for building models, forming hypotheses, and understanding intrinsic properties of the underlying phenomena. This thesis provides novel visualization techniques that advance the field of multivariate visual data analysis by helping represent and comprehend the structure of high-dimensional data. In contrast to approaches that focus on visualizing multivariate data directly or by means of their geometrical features, the methods developed in this thesis focus on their topological properties. More precisely, these methods provide structural descriptions that are driven by persistent homology, a technique from the emerging field of computational topology. Such descriptions are developed in two separate parts of this thesis. The first part deals with the qualitative visualization of topological features in multivariate data. It presents novel visualization methods that directly depict topological information, thus permitting the comparison of structural features in a qualitative manner. The techniques described in this part serve as low-dimensional representations that make the otherwise high-dimensional topological features accessible. We show how to integrate them into data analysis workflows based on clustering in order to obtain more information about the underlying data. The efficacy of such combined workflows is demonstrated by analysing complex multivariate data sets from cultural heritage and political science, for example, whose structures are hidden to common visualization techniques. The second part of this thesis is concerned with the quantitative visualization of topological features. It describes novel methods that measure different aspects of multivariate data in order to provide quantifiable information about them. Here, the topological characteristics serve as a feature descriptor. Using these descriptors, the visualization techniques in this part focus on augmenting and improving existing data analysis processes. Among others, they deal with the visualization of high-dimensional regression models, the visualization of errors in embeddings of multivariate data, as well as the assessment and visualization of the results of different clustering algorithms. All the methods presented in this thesis are evaluated and analysed on different data sets in order to show their robustness. This thesis demonstrates that the combination of geometrical and topological methods may support, complement, and surpass existing approaches for multivariate visual data analysis