We present Topological Point Cloud Clustering (TPCC), a new method to cluster points in an arbitrary point cloud based on their contribution to global topological features. TPCC synthesizes desirable features from spectral clustering and topological data analysis and is based on considering the spectral properties of a simplicial complex associated to the considered point cloud. As it is based on considering sparse eigenvector computations, TPCC is similarly easy to interpret and implement as spectral clustering. However, by focusing not just on a single matrix associated to a graph created from the point cloud data, but on a whole set of Hodge-Laplacians associated to an appropriately constructed simplicial complex, we can leverage a far richer set of topological features to characterise the data points within the point cloud and benefit from the relative robustness of topological techniques against noise. We test the performance of TPCC on both synthetic and real-world data and compare it with classical spectral clustering.

Knowledge networks can organize complex data by constructing graphs where nodes are concepts or ideas and edges represent connections of significance. Understanding the structure of these knowledge networks to uncover how science progresses over time is of interest to researchers studying the “Science of Science.” In this project, we are interested in understanding cycles or holes within a network, which can be thought of as gaps in knowledge. We use topological data analysis, and in particular, persistent homology filtered through time where the nodes represent scientific concepts and edges between two nodes are added at the time when they appear together in an abstract of a scientific paper. We study properties of these knowledge gaps in multiple dimensions such as when they form, when they no longer remain, and the concepts and papers that make up the cycles. We observe that papers involved in the knowledge gaps are cited more frequently than papers that are not.

We consider persistent homology obtained by applying homology to the open Rips filtration of a compact metric space $(X,d)$. We show that each decrease in zero-dimensional persistence and each increase in one-dimensional persistence is induced by local minima of the distance function $d$. When $d$ attains local minimum at only finitely many pairs of points, we prove that each above mentioned change in persistence is induced by a specific critical edge in Rips complexes, which represents a local minimum of $d$.

We use this fact to develop a theory (including interpretation) of critical edges of persistence. The obtained results include upper bounds for the rank of one-dimensional persistence and a corresponding reconstruction result. Of potential computational interest is a simple geometric criterion recognizing local minima of $d$ that induce a change in persistence.

The discrete homotopy theory is a homotopy theory defined on graphs, simplicial comlexes and metric spaces.

The discrete homotopy mesures the ``combinatorial holes'', but it can also be seen as the classical homotopy of realization of the associated cubical sets, which has been recently proved by Carranza and Kapulkin.

The discrete homotopy theory is interesting not only itself but also for its application to the other area, such as matroid theory, hyperplane/subspace arrangements, topological data analysis etc. It is interesting and necessary to develop the discrete homotopy theory itself to expand such applications. In this poster, we consider the relative version of the discrete homotopy theory. In particular, we define a discrete relative homotopy groups and provide long exact sequences for a pair of graphs $(G, A)$. The work is jointly with Ye Liu.

Since 2010, our group has developed and refined a method, Regressor Interpolation at Progressive Time Delays (RIPTiDe) to extract hemodynamic parameters (blood arrival time delay and relative blood volume) from resting state fMRI data (rs-fMRI) [Tong 2010, Erdogan 2016, Aslan 2019, 2020]. By using the RIPTiDe algorithm, we infer the relative blood arrival time and volume from the strength and delay of the peak of the crosscorrelation between each voxels’ timecourse and a “probe” timecourse representing the moving blood. RIPTiDe iteratively extracts the travelling systemic low frequency global mean signal, which exists in some form in every voxel from fMRI data, and uses the crosscorrelation function to track its progress through the brain. In our former works, we showed that an important component of the rs-fMRI data contains brain hemodynamic and cardiac related signals.

In this study, we used the publicly available Open Access Series of Imaging Studies (OASIS-3) dataset [LaMontagne 2019], which contains rs-fMRI as well as Arterial Spin Labeling (ASL) data in subjects with Alzheimer’s Disease (AD), and matched healthy subjects. Recent studies show that vascular alterations are present in more than 50% of clinically diagnosed AD cases [Cortes-Canteli 2020], Detecting incipient vascular pathology could allow early intervention, mitigating the damage caused by these changes. In this study we hypothesized that RIPTiDe derived blood correlation strength maps would offer a promising tool to retrospectively extract hemodynamic information which can be used to study this newly emerging focus in Alzheimer’s studies.

We hypothesize that vascular deformation in Alzheimer people can be detected by measuring and comparing topological invariants by use of persistent homology.

We examined the maps derived from rs-fMRI to find potential differences between the 2 groups. We compared rs-fMRI data from 372 images from Alzheimer subjects and 2161 images from Control subjects in the OASIS-3 dataset. We applied persistent homology calculations to these images and for downstream analysis we calculated silhouette graphs using giotta software. The resulting 1000-dimensional vectors averaged over the Alzheimer and Control Groups separately. We calculate the difference of the silhouette graphs as a comparison we also subsample the control subjects for 372 subjects averaged the plot and take the difference as well. We observed that there is difference in the average of the silhouette graphs which indicates difference in topological invariant of Alzheimer and control subjects.

In this study, we conducted hierarchical clustering-based spatial imaging analysis using the largest dataset from the UK Biobank. We extracted the amplitude of low-frequency fluctuations (ALFF) from resting-state functional magnetic resonance imaging (rs-fMRI) data for 35,000 subjects. ALFF represents the average spectral content within the low-frequency regime for every voxel in the brain image. While the majority of rs-fMRI literature focuses on neuronal activation studies, the low-frequency regime in rs-fMRI contains important structural information. We averaged the data over 400 regions of interest (ROIs) using the Schaefer parcellation scheme. For downstream tasks, this number of ROIs might be too high, and the aim is to identify major clusters within these 400 ROIs. To this end, we calculated a spatial correlation matrix condensing the group information into a 400 by 400 matrix. We then performed hierarchical clustering. To determine the optimal number of clusters, we plotted the elbow plot from the clustering algorithm. Based on the dendrogram produced by the clustering, we plotted the reordered correlation matrix. The resulting matrix makes it easier to see the coherent blocks as well. By choosing 6 as the number of clusters, we assigned a cluster index to every voxel. Remarkably, although we worked with a 2D correlation matrix, new clusters were observed in spatially coherent regions.

The information in bi-parameter persistent homology is usually compared by the so-called matching distance. However, the algorithms for calculating such a metric have a rather high complexity. In this poster we illustrate how some geometric techniques can make easier the study of this distance, possibly opening the way to more efficient methods for its approximation. In particular, we present a theorem that restricts the set of parameter values involved in the algorithms for the computation of such a distance, therefore reducing their complexity.

The category of semi-coarse spaces has similar categorical properties to topological spaces and it has been studied to establish foundations of Topological Data Analysis and develop discrete homotopy; other examples of these categories are Čech closure spaces and Choquet Spaces. In this exposition we introduce these spaces and develop the beginnings of homotopy theory for semi-coarse spaces. Finally, we contrast different graphs whose fundamental group is the integers, but they are not necessarily homotopy equivalent. This is joint work with Antonio Rieser.

Topological Data Analysis (TDA) is an emerging field that aims to discover a dataset’s underlying topological information. TDA tools have been commonly used to create filters and topological descriptors to improve Machine Learning (ML) methods. This paper proposes a different TDA pipeline to classify balanced and imbalanced multi-class datasets without additional ML methods. Our proposed method was designed to solve multi-class and imbalanced classification problems with no data resampling preprocessing stage. The proposed TDA-based classifier (TDABC) builds a filtered simplicial complex on the dataset representing high-order data relationships. Following the assumption that a meaningful sub-complex exists in the filtration that approximates the data topology, we apply Persistent Homology (PH) to guide the selection of that sub-complex by considering detected topological features. We use each unlabeled point’s link and star operators to provide different-sized and multi-dimensional neighborhoods to propagate labels from labeled to unlabeled points. The labeling function depends on the filtration’s entire history of the filtered simplicial complex and it is encoded within the persistence diagrams at various dimensions. We select eight datasets with different dimensions, degrees of class overlap, and imbalanced samples per class to validate our method. The TDABC outperforms all baseline methods classifying multi-class imbalanced data with high imbalanced ratios and data with overlapped classes. Also, on average, the proposed method was better than K Nearest Neighbors (KNN) and weighted KNN and behaved competitively with Support Vector Machine and Random Forest baseline classifiers in balanced datasets.

We introduce an intrinsic estimator for the scalar curvature of a data set presented as a finite metric space (e.g., a distance matrix, a point cloud, or a network with the shortest-path metric). Our estimator depends only on the metric structure of the data, and not on an embedding in Euclidean space. Our estimator is consistent in the sense that for points sampled randomly from a compact Riemannian manifold, the estimator converges to the scalar curvature as the number of points increases. Additionally, our estimator is stable with respect to perturbations of the metric (e.g., noise in the sample or error estimating the intrinsic metric), which justifies its use in applications. We validate our estimator experimentally on synthetic data that is sampled from manifolds with specified curvature.

 This is joint work with Andrew J. Blumberg.

"The Most Important Missing Infrastructure Project in Knot Theory" -Dr. Dror Bar-Natan

There are a number of great knot and link tables available to researchers; be that mathematicians, biologists, physicists, and many other domains. However, with only knot and link tables we are in the position of a chemist with a table of fatty acids but no periodic table. Our group at University of Iowa are striving to build that periodic table of knot "elements", the two string tangles.