PosterSession

Poster session titles and abstracts

The virtual poster session is on Wednesday, May 11, at 9am Eastern Daylight Time (UTC-4).

• Nikola Milicevic, Homotopy, Homology and Persistent Homology Using Čech's Closure Spaces

  • We use Čech closure spaces, also known as pretopological spaces, to develop a uniform framework that encompasses the discrete homology of metric spaces, the singular homology of topological spaces, and the homology of (directed) clique complexes, along with their respective homotopy theories. We obtain nine homology and six homotopy theories of closure spaces. We show how metric spaces and more general structures such as weighted directed graphs produce filtered closure spaces. For filtered closure spaces, our homology theories produce persistence modules. We extend the definition of Gromov-Hausdorff distance to filtered closure spaces and use it to prove that our persistence modules and their persistence diagrams are stable. We also extend the definitions Vietoris-Rips and Čech complexes to closure spaces and prove that their persistent homology is stable.

• Mehmet Aktas, Liars are more influential: Effect of Deception in Influence Maximization on Social Networks

  • Detecting influential users, called the influence maximization problem on social networks, is an important graph mining problem with many diverse applications such as information propagation, market advertising, and rumor controlling. There are many studies in the literature for influential users detection problem in social networks. Although the current methods are successfully used in many different applications, they assume that users are honest with each other and ignore the role of deception on social networks. On the other hand, deception appears to be surprisingly common among humans within social networks. In this paper, we study the effect of deception in influence maximization on social networks. We first model deception in social networks. Then, we model the opinion dynamics on these networks taking the deception into consideration thanks to a recent opinion dynamics model via sheaf Laplacian. We then extend two influential node detection methods, namely Laplacian centrality and DFF centrality, for the sheaf Laplacian to measure the effect of deception in influence maximization. Our experimental results on synthetic and real-world networks suggest that liars are more influential than honest users in social networks.

• Francesco Conti, A topological pipeline for machine learning methods

  • The development of a topological pipeline for machine learning involves two crucial steps that strongly influence the performance of the pipeline. The first step is the choice of the filtration that associates a persistence diagram with digital data. The second step is the choice of the representation method for the persistence diagrams, which often relies on several parameters. In this work we develop a pipeline that associates persistence diagrams to digital data, via the most appropriate filtration for the type of data considered. Using a grid search approach, this pipeline determines optimal representation methods and parameters. We assess the performance of our pipeline, and in parallel we compare the different representation methods, on popular benchmark datasets. This work is a first step towards both an easy, ready to use, pipeline for data classification using persistent homology and machine learning, and to understand the theoretical reasons why, given a dataset and a task to be performed, a pair (filtration, topological representation) is better than another.

• Fabian Lenzen, Efficient 2-D persistent cohomology computation

  • During the recent years, persistent homology computation has undergone several important performance optimisations, one of which is the so-called clearing scheme, which extends a basis of the (co)boundaries to a basis of the (co)cycles, rather than recomputing the latter from scratch. In 1-D persistent homology of Vietoris-Rips complexes, this has proven particularly effective when applied to the computation of (relative) cohomology, rather than (absolute) homology – although both computations yield equivalent output. In 2-D, the computation of (absolute or relative) cohomology is more involved, due to the fact that cochain modules are not free anymore. We show how to deal with this non-freeness in a way that allows to apply a clearing scheme in 2-D nevertheless. We also show that in 2-D, there exists a certain equivalence between free resolutions of homology and cohomology.

• Fabian Roll, A Unified View on the Functorial Nerve Theorem and its Variations

  • The nerve theorem is a basic result of algebraic topology that plays a central role in computational and applied aspects of the subject. In applied topology, one often needs a nerve theorem that is functorial in an appropriate sense, and furthermore one often needs a nerve theorem for closed covers, as well as for open covers. While the techniques for proving such functorial nerve theorems have long been available, there is unfortunately no general-purpose, explicit treatment of this topic in the literature. We address this by proving a variety of functorial nerve theorems. First, we show how one can use relatively elementary techniques to prove nerve theorems for covers by closed convex sets in Euclidean space, and for covers of a simplicial complex by subcomplexes. Then, we prove a more general, ``unified'' nerve theorem that recovers both of these, using standard techniques from abstract homotopy theory.

• Chunyin Siu, Antman Persistence: Detection of Small Holes with the Robust Density-Aware Distance (RDAD) Filtration

  • A novel topological-data-analytical (TDA) method is proposed to distinguish, from noise, small holes surrounded by high-density regions of a probability density function whose mass is concentrated near a manifold (or more generally, a CW complex) embedded in a high-dimensional Euclidean space. The proposed method is robust against additive noise and outliers. In particular, sample points are allowed to be perturbed away from the manifold. Traditional TDA tools, like those based on the distance filtration, often struggle to distinguish small features from noise, because of their short persistence. An alternative filtration, called Robust Density-Aware Distance (RDAD) filtration, is proposed to prolong the persistence of small holes surrounded by high-density regions. This is achieved by weighting the distance function by the density in the sense of Bell et al. Distance-to-measure is incorporated to enhance stability and mitigate noise due to the density estimation. The utility of the proposed filtration in identifying small holes, as well as its robustness against noise, are illustrated through an analytical example and extensive numerical experiments.

• Boštjan Lemež, Reconstruction properties of selective Rips complexes

  • Selective Rips complexes are generalization of Vietoris-Rips complexes. The main ideas of new custom made simplicial complexes are that they are specifically designed to detect local features and they are »thin« in each dimension. We will present reconstruction results with selective Rips complexes up to homotopy type. In particular, we show that if a metric space Y is close (in Gromov-Hausdorff distance) to a closed Riemannian manifold X, then selective Rips complexes of Y for certain parameters attain the homotopy type of X. This result is a generalization of Latschev’s reconstruction result from Vietoris-Rips complexes to selective Rips complexes. In particular, we present a novel proof for the Latschev’s theorem as a special case using the Nerve thoerem. This is a joint work with Žiga Virk.

• Rachael Schwartz, Applied Topology for Visual Perceptual Computing: A Case Study from Hand-drawn Animation Technology

  • Because Topological Data Analysis (TDA) is sensitive to shape and structure, it naturally captures important aspects of human visual perception. However, applications of TDA to visual perceptual computing have not been substantially explored. We briefly discuss parallels between human visual processing and persistent homology, and outline potential applications of sheaf theory and geometric group theory to visual perceptual computing. We then present our case study, in which we investigate the utility of persistent homology for perception-based hand-drawn animation technology. In our experiment, professional animators at the animation studio Cartoon Saloon were shown 54 pairs of hand-drawn frames from the Oscar-nominated film Wolfwalkers, and were asked to numerically rate how helpful one frame in each pair would be as drawing reference for the other. Persistent homology and Wasserstein distances were respectively computed for all frames and frame pairs used in the experiment. These TDA results were compared to our perceptual experiment results, and correlations between persistent homology and animators' visual perception were assessed. Our work supports our aim of using TDA-based computer vision to find high-quality reference drawings for professional animators, and invites further exploration of TDA for visual perceptual computing.

• Faraz Ahmad, Compactification of Spaces of Equivariant Operators in TDA

  • There is an increasing interest in data analysis and deep learning on equivariant operators, giving impetus to the search for metric and topological properties of the spaces of group equivariant non-expansive operators (GENEOs). These operators help us better understand the role of observers in the analysis of data. In our model, we are concerned with the aspects of data analysis that pertain to the representations of data observers as structured collections of equivariant operators. One of the most desirable properties for these spaces to have is compactness, which provides us with fundamental guarantees in deep learning. Under some mild conditions, we show that it is always possible to construct a compactification of a given space of GENEOs.

• Mario Gomez, Curvature Sets Over Persistence Diagrams

  • We study an invariant of compact metric spaces inspired by the Curvature Sets defined by Gromov. The (n,k)-Persistence Set of X is the collection of k-dimensional VR persistence diagrams of any subset of X with n or less points. These invariants are generally computationally cheaper than the persistence diagram of X, and the (4,1)-Persistence Set can detect the homotopy type of certain family of graphs and the curvature of surfaces.

• Sushovan Majhi, Vietoris–Rips Complexes of Metric Spaces near a Metric Graph

  • For a sufficiently small scale, the Vietoris–Rips complex of a metric space with a small Gromov–Hausdorff distance to a closed Riemannian manifold has been already known to recover the manifold up to homotopy type. While the qualitative result is remarkable and generalizes naturally to the recovery of spaces beyond Riemannian manifolds—such as geodesic metric spaces with a positive convexity radius—the generality comes at a cost. Although the scale parameter is known to depend only on the geometric properties of the geodesic space, how to quantitatively choose it for a given geodesic space is still elusive. In this work, we focus on the topological recovery of a special type of geodesic space, called a metric graph. For an abstract metric graph and a sample with a small Gromov–Hausdorff distance to it, we provide a description of the scale parameter based on the convexity radius of the underlying graph in order for the Vietoris–Rips complex of the sample to be homotopy equivalent. Our investigation also extends to the study of the Vietoris–Rips complexes of a Euclidean point-cloud with a small Hausdorff distance to a (hidden) embedded metric graph. From the pairwise Euclidean distances of points of the data, we introduce a family of path-based Vietoris–Rips complexes. Based on the convexity radius and distortion of the embedding of the underlying graph, we show how to choose a suitable scale in order to recover the graph up to homotopy type.

• Dhananjay Bhaskar, Topological Analysis of Self-Organized Patterns in Heterogeneous Interacting Cell Populations

  • Heterogeneous cell populations exhibit coordinated motion, self-organization, and phase transitions during embryo formation, skin pigmentation, wound healing, and cancer metastasis. Interestingly, such complex behavior can be simulated with a minimal agent-based model (ABM) consisting of random polarization, differential adhesion, and cell division. Parameter sweeps of the ABM generate patterns of cell sorting, engulfment, and self-assembly into radially symmetric, spotted, and stripe patterns. Using simulation data, we demonstrate that a combination of topological data analysis (TDA) and machine learning can automatically identify distinct cell arrangements and delineate phase boundaries, thus uncovering the relationship between geometry at the tissue scale and cell-cell interactions at the local level. We test the robustness of our approach by performing ablation studies, finite-size scaling, and systematic perturbations to pattern size, frequency, population ratios, etc. Our technical contributions include distributed computation of persistent homology and overcoming the challenges associated with comparing non-constant population sizes. Additionally, our unsupervised and model-agnostic approach can be used to investigate a variety of phenomena in active and condensed matter.

• Mauricio Che, Metric geometry of spaces of persistence diagrams

  • In this poster we present some results obtained in collaboration with Fernando Galaz-García, Ingrid Membrillo-Solis and Luis Guijarro on the geometry of spaces of persistence diagrams. More precisely, we consider a family of functors $\mathcal{D}_p$ which, for $p \geq 1$, associate to each metric pair $(X,A)$ a pointed (pseudo) metric space $\mathcal{D}_p(X,A)$ consisting of persistence diagrams with points in $(X,A)$ and finite $p$-persistence, endowed with the $p$-Wasserstein distance. We prove that these functors preserve several useful properties, such as separability, completeness and certain curvature bounds. We also show that $\mathcal{D}_p$ is continuous with respect to the Gromov–Hausdorff convergence if and only if $p = \infty$ (that is, when we consider the “bottleneck” distance). Finally, we prove that the space of Euclidean persistence diagrams has infinite Hausdorff and asymptotic dimensions.

• Misha Tyomkin, Numbers on a barcode of a Morse function

  • Morse function f on a manifold M is called strong if all its critical points have different critical values. Given a strong Morse function f and a field F we construct a bunch of elements of F, which we call Bruhat numbers (they're defined up to sign). More concretely, Bruhat number is written on each bar in the barcode of f (a.k.a. Barannikov decomposition). It turns out that if homology of M over F is that of a sphere, then the product of all the numbers is independent of f. We then construct the barcode and Bruhat numbers with twisted (a.k.a. local) coefficients and prove that the mentioned product equals to the Reidemeister torsion of M. In particular, it's again independent of f. This way we link Morse theory to the Reidemeister torsion via barcodes. Based on a joint work with Petya Pushkar, https://arxiv.org/abs/2012.05307.

Guidelines for poster presenters

Here are some guidelines for poster presenters.

Guidelines for presenting your poster:

Your goal is to invite questions and interaction from those who attend your poster room. How can you do this?

• Have a short 3-5 minute verbal overview for your poster prepared. Don’t go into too many details, and don’t try to explain everything on the poster.

• After that 3-5 minute intro, take a break from speaking! Give folks time to read over your poster, to think of questions, and to ask questions. Do not be afraid of, say, half a minute of silence. Formulating questions takes time.

• Invite questions from folks! Once you get a first question, others will follow. Answering questions will allow you to get into some of the details behind your poster.

• If somebody new joins to see your poster, feel free to give another short intro as appropriate.

Guidelines for creating your poster:

• We recommend landscape posters, as they fit better on screens, but there is no restriction on poster size, background, or text color.

• Include figures! Posters are a visual medium.

• Select your text carefully. Some people like to make posters with section titles, figure captions, brief explanations, but few words overall (and instead say those words out loud). If you want to include more words on the poster that's great, but make sure the main messages are easily visible.

• Each presenter should make a PDF of their poster, which we will post on the webpage.

• Optionally, you can also create an HTML version of your poster, or a short video explaining your poster, for us to share next to your PDF — but these formats are optional.

Please contact us at aatrn.director@gmail.com if you have any questions!