Anton Ayzenberg (Higher School of Economics)
Title: Flag complexes of multi-graphs and related constructions
Abstract: Flag complex of a graph is the basic tool of topological data analysis, since it translates common data structures (such as graphs) into topological objects. In 2019 Govc, Levi, and Smith introduced several generalizations of flag complexes to the case of directed graphs, and tested them on the neural data from the Blue Brain Project. One of their constructions, the tournaplexes, made us think there should be a reasonable generalization of a flag complex for graphs with multiple edges.
Our basic construction is rather straightforward. A clique in a multigraph is sub(multi)graph isomorphic to a complete graph. The poset of all cliques is a simplicial poset, its geometrical realization is called the clique complex of a multigraph. We related this construction to inflated complexes studied by Wachs and introduced a natural common generalization of these two constructions. We proved a result on homotopy wedge decomposition which paves the way to parallel homology (and potentially persistent homology) computations of such topological spaces.
I want to present several open questions that still remain in this area, varying from purely theoretical to pure engineering.
Suyoung Choi (Ajou University)
Title: GPU-friendly enumeration on simplicial complexes
Abstract: In this talk, we discuss the use of parallel computing for mathematical problems. In particular, we provide a GPU-friendly algorithm for obtaining all weak pseudo-manifolds whose facets are all in an input set of facets satisfying given conditions.
We use this algorithm to completely list up toric colorable seed PL-spheres with a few vertices implying the complete characterization of PL-spheres of dimension n-1 with n+4 vertices having maximal Buchstaber numbers. This result is an essential intermediate step for the classification of non-singular complete toric varieties with Picard number 4.
Moo K. Chung (University of Wisconsin-Madison)
Title: Unified Topological Inference for Brain Networks in Temporal Lobe Epilepsy Using the Wasserstein Distance
Abstract: Persistent homology can extract hidden topological signals present in brain networks. Persistent homology summarizes the changes of topological structures through over multiple different scales called filtrations. Doing so detect hid- den topological signals that persist over multiple scales. However, a key obstacle of applying persistent homology to brain network studies has always been the lack of coherent statistical inference framework. To address this problem, we present a unified topological inference framework based on the Wasserstein distance. Our approach has no explicit models and distributional assumptions. The inference is performed in a completely data driven fashion. The method is applied to the resting-state functional magnetic resonance images (rs-fMRI) of the temporal lobe epilepsy patients. We made MATLAB package available at https://github.com/laplcebeltrami/dynamicTDA that was used to perform all the analysis in this study.
Bukyoung Jhun (Seoul National University)
Title: Topological estimation of the latent geometry of a complex network
Abstract: Most real-world networks are embedded in their latent geometries; if two nodes in a network are found close to each other in their latent geometry, they have a disproportionately high probability of being connected by a link. For instance, each node (airport) in the airline network is embedded in a location on Earth, and if two nodes are close, they are likely to be connected by a link (airway). The latent geometry of a complex network is a central topic of research in network science with extensive practical applications. However, unlike the airline network, the latent geometry of most networks is not given; hence, it must be discovered using an appropriate method. In this study, we developed a framework that can estimate the topological properties of the latent geometry of a complex network.
Jae-Hun Jung (POSTECH)
Title: Topological data analysis of time-series data
Abstract: Time-series data analysis is found in various applications that deal with sequential data over the given interval of, e.g. time. In this talk, we discuss time-series data analysis based on topological data analysis (TDA). The commonly used TDA method for time-series data analysis utilizes the embedding techniques such as sliding window embedding. With sliding window embedding the given data points are translated into the point cloud in the embedding space and the method of persistent homology is applied to the obtained point cloud. In this talk, we first show some examples of time-series data analysis with TDA. The first example is from music data for which the dynamic processes in time is summarized by low dimensional representation based on persistence homology. The second is the example of the gravitational wave detection problem and we will discuss how we concatenate the real signal and topological features. Then we will introduce our recent work of exact and fast multi-parameter persistent homology (EMPH) theory. The EMPH method is based on the Fourier transform of the data and the exact persistent barcodes. The EMPH is highly advantageous for time-series data analysis in that its computational complexity is as low as O(N log N) and it provides various topological inferences almost in no time. The presented works are in collaboration with Mai Lan Tran, Chris Bresten and Keunsu Kim.
Seungchan Ko (Sungkyunkwan University)
Title: A novel approach for wafer defect pattern classification based on topological data analysis
Abstract: In semiconductor manufacturing, wafer map defect pattern provides critical information for facility maintenance and yield management, so the classification of defect patterns is one of the most important tasks in the manufacturing process. In this talk, I propose a novel way to represent the shape of the defect pattern as a finite-dimensional vector, which will be used as an input for a neural network algorithm for classification. The main idea is to extract the topological features of each pattern by using the theory of persistent homology from topological data analysis. Through some experiments with a simulated dataset, I will show that the proposed method is faster and much more efficient in training with higher accuracy, compared with the method using convolutional neural networks, which is the most common approach for wafer map defect pattern classification. Moreover, this new method outperforms the CNN-based method when the number of training data is not enough and is imbalanced.
Kang-Ju Lee (Seoul National University)
Title: Learning covers in mapper graphs
Abstract: Mapper is a visualization technique of topological data analysis (TDA), which takes as input a point cloud data and produces as output a graph reflecting the structure of the underlying data. Tuning parameters for mapper is an essential step in generating nice mapper graphs, which has been a challenging problem in the TDA community. In this paper, we present an algorithm that searches covers for better mapper graphs based on statistical theory. A previous method splits covers for optimizing the Bayesian Information Criterion (BIC), which was employed in an algorithm, called x-means clustering, estimating the number of clusters in k-means clustering. Our method is based on g-means clustering, an improved algorithm of the x-means clustering. While splitting covers of mapper graphs, we conduct a statistical test for the hypothesis that a subset of data comes from a Gaussian distribution. Our experiments demonstrate that the proposed algorithm generates nice mapper graphs and is better than the previous one using the BIC criterion. This is joint work with Alvarado, Belton, Fischer, Palande, Percival, Purvine, and Tymochko.
Daniel Leykam (National University of Singapore)
Title: Applications of persistent homology to quantum systems
Abstract: Persistent homology quantifies the shape of data given a suitable distance measure by computing its topological features over a range of scales. In physics the shape information provided by persistent homology has been applied to various classical and quantum systems to detect phase transitions, e.g. between ordered and amorphous phases. In these applications considerable effort and physical insight is often required to identify the most appropriate input data, distance measure, and means of summarizing the information encoded in persistence diagrams. This added complexity makes it important to identify areas where persistent homology can unveil novel phenomena inaccessible using conventional analysis methods. I will present two promising examples where persistent homology provides new insights: (1) Identifying order-disorder transitions in quantum condensed matter systems, specifically the Aubry-Andre-Harper model, and (2) detecting quantum chaos in a driven Kerr nonlinear cavity using single photon detection time series.
Sunhyuk Lim (Max Planck Institute for Mathematics in the Sciences)
Title: Vietoris-Rips persistent homology, injective metric spaces, and the filling radius
Abstract: In the applied algebraic topology community, the persistent homology induced by the Vietoris-Rips simplicial filtration is a standard method for capturing topological information from metric spaces. In this paper, we consider a different, more geometric way of generating persistent homology of metric spaces which arises by first embedding a given metric space into a larger space and then considering thickenings of the original space inside this ambient metric space. In the course of doing this, we construct an appropriate category for studying this notion of persistent homology and show that, in a category theoretic sense, the standard persistent homology of the Vietoris-Rips filtration is isomorphic to our geometric persistent homology provided that the ambient metric space satisfies a property called injectivity.
As an application of this isomorphism result we are able to precisely characterize the type of intervals that appear in the persistence barcodes of the Vietoris-Rips filtration of any compact metric space and also to give succinct proofs of the characterization of the persistent homology of products and metric gluings of metric spaces. Our results also permit proving several bounds on the length of intervals in the Vietoris-Rips barcode by other metric invariants, for example the notion of spread introduced by M. Katz.
As another application, we connect this geometric persistent homology to the notion of filling radius of manifolds introduced by Gromov and show some consequences related to (1) the homotopy type of the Vietoris-Rips complexes of spheres which follow from work of M. Katz and (2) characterization (rigidity) results for spheres in terms of their Vietoris-Rips persistence barcodes which follow from work of F. Wilhelm.
Finally, we establish a sharp version of Hausmann’s theorem for spheres which may be of independent interest.
German Magai (Higher School of Economics)
Title: Deep neural networks from the perspective of manifold learning
Abstract: Despite significant advances in the field of deep learning in applications to various fields, the explanation of the learning process of neural network models remains an important open question. The purpose of this study is a comprehensive comparison and description of neural network architectures in terms of geometry and topology. We focus on the internal representation of neural networks and on the dynamics of changes in the topology and geometry of dataset on different layers. In our study, we use the concepts of topological data analysis (TDA) and persistent homological fractal dimension. We present a wide range of experiments with different datasets and configurations of convolutional neural network (CNN) architectures and Transformers in CV and NLP tasks. We also propose a method for estimating the generalizing ability of deep models based on topological descriptors.
Chul Moon (Southern Methodist University)
Title: Statistical inference using persistent homology features of porous media
Abstract: We statistically infer porous materials' fluid flow and transport properties based on their geometry and connectivity. We summarize structure by persistent homology and then determine the similarity of structures using image analysis and statistics. We first compute persistent homology of binarized 3D images of material subvolume samples. The persistence parameter is the signed Euclidean distance from inferred material interfaces, which captures the distribution of sizes of pores and grains. We fit statistical models using the persistent homology features to estimate material permeability, tortuosity, and anisotropy. We also develop a Structural SIMilarity index to determine statistical representative elementary volumes.
Jongbaek Song (Korea Institute for Advanced Study)
Title: Persistent module of the double cohomology and its stability.
Abstract: Given a simplicial complex K, one can define a topological space Z(K) called the moment-angle complex. The cohomology of Z(K) is captured by the Tor-algebra corresponding to the face ring of K. In this talk, we introduce a certain differential the cohomology of Z(K) to make it a chain complex. This leads us to define a double cohomology of Z(K), which is a new combinatorial invariant of K. Then, we discuss how it comes in the standard pipeline of the topological data analysis (TDA). This is a joint work (in progress) with A. Bahri, I. Limonchenko, T. Panov and D. Stanley.
Kelin Xia (Nanyang Technological University)
Title: Mathematical AI for molecular data analysis
Abstract: Artificial intelligence (AI) based molecular data analysis has begun to gain momentum due to the great advancement in experimental data, computational power and learning models. However, a major issue that remains for all AI-based learning models is the efficient molecular representations and featurization. Here we propose advanced mathematics-based molecular representations and featurization (or feature engineering). Molecular structures and their interactions are represented as various simplicial complexes (Rips complex, Neighborhood complex, Dowker complex, and Hom-complex), hypergraphs, and Tor-algebra-based models. Molecular descriptors are systematically generated from various persistent invariants, including persistent homology, persistent Ricci curvature, persistent spectral, and persistent Tor-algebra. These features are combined with machine learning and deep learning models, including random forest, CNN, RNN, Transformer, BERT, and others. They have demonstrated great advantage over traditional models in drug design and material informatics.