Abstract : Topological Data Analysis (TDA), a relatively new field of data analysis, has proved highly useful in a variety of applications. Recently, much TDA research has been devoted to not only developing theories but also developing TDA compatible in machine learning workflow. This workshop will bring together researchers working on the areas of TDA and machine learning and provide an opportunity where they present their recent research and share ideas both in theory and applications. Further, this workshop will also provide recent progresses of computational tools developed for TDA combined with machine learning in various applications.
Organizer(s) : Jae-Hun Jung, Shizuo Kaji, Moo K. Chung
[03334] Topological Data Analysis of Spatial Systems
Format : Talk at Waseda University
Author(s) :
Mason Alexander Porter (UCLA)
Abstract : I discuss several applications of topological data analysis to spatial systems. I will consider examples from voting, city streets, the spread of COVID-19.
[03341] Bigraded persistence barcodes and their stability
Format : Talk at Waseda University
Author(s) :
Anthony Bahri (Rider University)
Ivan Limonchenko (Higher School of Economics)
Taras Evgenievich Panov (Moscow State University)
Jongbaek Song (Pusan National University)
Donald Stanley (University of Regina)
Abstract : We define the bigraded persistent homology modules and the bigraded barcodes of a finite pseudo-metric space X using the ordinary and double homology of the moment-angle complex associated with the Vietoris-Rips filtration of X. Then we discuss the stability for the bigraded persistent double homology modules and corresponding bigraded barcodes.
[03617] Learning visual representation with homological labels
Format : Talk at Waseda University
Author(s) :
Shizuo Kaji (Kyushu University)
Yohsuke Watanabe (ZOZO inc.)
Abstract : We propose a new scheme for convolutional neural networks to learn visual representation with synthetic images and mathematically-defined labels that capture topological information. Our scheme can be viewed as a type of self-supervised learning, where the regression of vectorised persistent homology of an image is learned. We show that the acquired visual representation supplements the one obtained by the usual supervised learning with manually-defined labels by confirming an improved convergence in training for image classification. Our method provides a simple way to encourage the model to learn global features through a specifically designed task based on topology. It requires no real images nor manual labels and can be utilised at a minimal extra cost.
[03647] Exact multi-parameter persistent homology of time-series data: Fast and variable one-dimensional reduction of multi-parameter persistence theory
Format : Talk at Waseda University
Author(s) :
Keunsu Kim (POSTECH)
Jae-Hun Jung (POSTECH)
Abstract : Time-series data can be inferred as a periodic signal, enabling a continuous approximation with discrete Fourier transform to reveal the relation between its Fourier modes and topology of the data. We introduce an exact multi-parameter persistent homology construction utilizing the fast Fourier transform and Künneth formula, which computes and interprets the corresponding persistent barcode in a fast and efficient manner. This work is based on https://arxiv.org/abs/2211.03337
[03657] Generic transitions for flows on surfaces with or without constraints
Format : Talk at Waseda University
Author(s) :
Tomoo Yokoyama (Saitama University)
Abstract : This talk describes the generic time evolutions of gradient flows and Hamiltonian flows on surfaces with or without physical constraints. Moreover, we show the non-contractibility of connected components of the spaces of such flows, respectively, under the non-existence of creations and annihilations of singular points by using combinatorics and simple homotopy theory.
[04624] Topological learning for multiscale biology
Format : Talk at Waseda University
Author(s) :
Heather Harrington (University of Oxford)
Abstract : Biological processes are multi-scale. Spatial structures and patterns vary across levels of organisation, from molecular to multi-cellular to multi-organism. With more sophisticated mechanistic models and data available, quantitative tools are needed to study their evolution in space and time. The most prominent tool in topological data analysis is persistent homology (PH), which provides a multi-scale summary of data. Here we present extensions to the PH pipeline and highlight its utility with concrete case studies.
[04715] Topological Data Analysis for Biological Images and Video
Format : Online Talk on Zoom
Author(s) :
Peter Bubenik (University of Florida)
Abstract : I will present the results of two projects applying topological data analysis (TDA) and machine learning (ML) to biological data. In the first, we have developed a tool, TDAExplore, that combines TDA and ML to both classify biological images and to provide a visualization that is biologically informative. In the second, we use TDA and ML to classify quasi-periodic biological videos and we apply TDA to such a video to produce synthetic periodic videos.
[05453] Topological data analysis of music data and AI composition
Format : Talk at Waseda University
Author(s) :
Jae-Hun Jung (POSTECH)
Mai Lan Tran (POSTECH)
Dongjin Lee (POSTECH)
Abstract : We employ topological data analysis to analyze music. Initially, the provided music data is transformed into a graph and we identify embedded cycles within the music using persistent homology. We elucidate how the cycle structure changes based on the metric definition between music nodes, with theoretical justifications. Then, we introduce the overlap matrix, which shows the interconnectedness of these cycles. We explain an AI algorithm utilizing the overlap matrix to facilitate new music compositions.
[05456] GRIL: A 2-parameter Persistence Based Vectorization for Machine Learning
Format : Talk at Waseda University
Author(s) :
Soham Mukherjee (PhD Student)
Tamal Krishna Dey (Purdue University)
Abstract : $1$-parameter persistent homology, a cornerstone in Topological Data Analysis (TDA), studies the evolution of topological features such as connected components and cycles hidden in data. It has been applied to enhance the representation power of deep learning models, such as Graph Neural Networks (GNNs). To enrich the representations of topological features, here we propose to study $2$-parameter persistence modules induced by bi-filtration functions. In order to incorporate these representations into machine learning models, we introduce a novel vector representation called Generalized Rank Invariant Landscape (GRIL) for $2$-parameter persistence modules. We show that this vector representation is $1$-Lipschitz stable and differentiable with respect to underlying filtration functions and can be easily integrated into machine learning models to augment encoding topological features. We present an algorithm to compute the vector representation efficiently. We also test our methods on synthetic and benchmark graph datasets, and compare the results with previous vector representations of $1$-parameter and $2$-parameter persistence modules.
[05457] Topological Classification of Zero Sum Games
Format : Talk at Waseda University
Author(s) :
Alexander Strang (University of Chicago)
Abstract : Zero-sum two-player games are widely used to model competitive interactions in biology, economics, and reinforcement learning. Unlike classical game theory, which focuses on optima, empirical game theory studies the structure of games and decision problems via observations of play by a population. We study a classification scheme for games based on their topology after embedding into a latent space. Using observed interactions, we infer the spectrum of the payout function when treated as the kernel of an integral operator. The eigenfunctions of the operator can be used to embed agents. The embedded agents form a scatter cloud whose topology provides a natural framework for classifying games. We study the classification of a series of randomly generated extensive form games and decision problems.
[05458] Topological Embedding of Brain Networks for Differentiating Temporal Lobe Epilepsy
Format : Talk at Waseda University
Author(s) :
Moo K Chung (University of Wisconsin-Madison)
Abstract : In this study, we approach the discrimination of functional brain networks in temporal lobe epilepsy patients from those of healthy controls through persistent homology. Starting with a weighted graph, we perform a graph filtration, yielding the birth-death decomposition. This process allows us to uniquely decompose each graph into two subgraphs characterized by 0D and 1D topology. The 0D subgraph arises from the birth of connected components, whereas the 1D subgraph manifests through the death of 1-cycles during the filtration. The distinguishing features of each graph are thus represented by the sorted birth and death values. To compare multiple weighted graphs, we propose a topological version of multidimensional scaling, which embeds these graphs into a 2D plane. This technique offers potential insights for resting-state functional magnetic resonance imaging (rs-fMRI) studies, particularly in distinguishing the functional brain networks associated with temporal lobe epilepsy. This presentation draws upon the findings from the paper available on arXiv:2302.06673.
[05459] Barcodes and Kernels for multiparameter persistence
Format : Online Talk on Zoom
Author(s) :
Mathieu Carrière (Centre Inria d'Université Côte d'Azur)
Abstract : Multiparameter persistence is a generalization of persistent homology that allows for more than a single filtration function. Such constructions arise naturally when considering data with outliers or variations in density, time-varying data, or functional data.
In single-parameter persistence, the barcode is equivalent to the “rank invariant”: the function that associates the rank of the corresponding linear map to every pair of comparable points. However, nearly all of the tools developed in persistent homology are based on the barcode. This is because it is a concise and geometric descriptor that lends well to data analysis and visualization. Therefore, it is crucial, and perhaps imperative, to construct a generalized barcode to work with the rank-invariant for multiparameter persistence efficiently.
Perhaps surprisingly, recent work has shown that if we allow the elements of the barcode to be signed intervals, then such a generalization is possible. I will discuss how one can use homological algebra to obtain a signed barcode in a stable manner. Furthermore, I will discuss how signed barcodes can be used in machine learning pipelines and report on recent computational results obtained using generalizations of the so-called sliced Wasserstein kernel to such signed barcodes.
