Plenary Talk
이강주 (Seoul National University)
Title: Applied Topology and Network Science
Abstract: A graph is a fundamental object in topological data analysis and network analysis, serving as a bridge between these two fields. In this talk, I will explore how TDA and network analysis are interconnected. Furthermore, I will introduce generalizations to simplicial complexes and applications of combinatorial Laplacians.
Speakers
허은우 (POSTECH)
Title: Persistent homology of featured time series data and its applications
Abstract: Constructing graphs based on the frequency of occurrence in time series observations is a straightforward method that effectively reflects information of time series data. However, this simplicity often leads to significant information loss during the data transformation process. We introduce a new concept and research methodology that preserves the advantages of graph transformations using frequency, while controlling for information loss. Featured time series refers to time series that are enhanced with features to mitigate information loss. We prove a theorem demonstrating that an influence vector, which represents the impact of features, maintains stability properties throughout the analysis process. Based on this theorem, we introduce a research methodology focused on finding influence vectors that minimize information loss for improved time series analysis. We sought influence vectors that effectively detect anomalies in stock data, such as the Lehman bankruptcy and the dotcom crash. Conversely, we also present a methodology that sets the desired analysis target as a feature. We analyzed the impact of musical notation on overall time series data by observing changes in the influence vector.
유준원 (POSTECH)
Title: Persistent homology for link prediction
Abstract: In graph data, the links, representing vital interconnections and relationships, play a pivotal role in enhancing our understanding and analysis of complex systems, ranging from social networks to biological interactions. In this talk, I will introduce latest research developments in the fields focusing on Link Prediction Based on Graph Neural Networks(Zhang et al., 2018) and Neural Link Prediction with Walk Pooling (Pan et al., 2021). Furthermore, we propose Persistent Homology for Link Prediction (PHLP) and Multi-Angle PHLP with close to state-of-the-art performance. To the best of our knowledge, this is the first method which only use persistent homology for link prediction. Also, our model can be added any other methods simply.
김세훈 (Seoul National University)
Title: Neural Collapse through the lens of persistent homology
Abstract: Neural collapse (NC) refers to the unique geometric configuration of the last hidden layer's features in deep neural network training. In this talk, we examine NC through the lens of persistent homology. Our analysis contributes to the growing body of evidence indicating that the relationship between NC and a model's generalization performance is not substantial. Additionally, we explore the evolution of the topological complexity of a dataset as it propagates through the network layers, providing an insight into the training dynamics of neural networks.
정찬규 (Seoul National University)
Title: Invariant sets in the restricted three-body problem
Abstract: The restricted three-body problem describes the motion of a massless object under the gravitational pull of two heavy bodies. It is a classical example of a non-integrable Hamiltonian system, whose dynamics contains a mixture of periodic and quasiperiodic orbits but also chaos. To understand its global dynamics, the study of invariant sets such as periodic orbits and invariant tori are essential. In this talk, we introduce the three-body problem and describe how invariant sets play a crucial role in understanding its dynamics. Motivated by this perspective, we describe a method which uses computational homology tools to detect the invariant sets and related stable periodic orbits.
염시진 (POSTECH)
Title: Introduction to Quiver Theory
Abstract: Zomoroidan and Carlsson (2005) described persistence modules in terms of representations, highlighting a one-to-one correspondence between persistence modules and graded modules over the polynomial ring k[t] by utilizing the structure theorem for finitely generated modules over a principal ideal domain. Carlsson and de Silva (2009) introduce zigzag modules, linking them to representations of A-type quivers. They propose a new constructive proof of Gabriel's theorem in the special case of A-type quivers, providing a practical algorithm for computing decompositions of zigzag modules. As a Topological Data Analysis community member, studying quiver theory appears inevitable due to its connection to persistence modules. In this talk, I will introduce the quiver theory and explain what Gabriel did.
소병창 (Seoul National University)
Title: Application of weight measure to periodic time series feature extraction
Abstract: Periodic time series arises in data from various domains including health data, IoT sensor data, etc. In order to draw information from each periodic time series numeric descriptors, or features, can be calculated to be used in subsequent tasks inside entire data analysis pipeline. In case of online analysis the pipeline is applied to short segments, which in general contain incomplete period and therefore yield values vary upon segments, even if they are segmented from the same periodic time series. There are several solutions such as periodic segmentation, time delay embedding and deep-learning based approach etc. each of which entails respective shortcomings.
This talk suggests a method to extract features from periodic time series, which is robust to selection of segments. Key ingredients are time-delay embedding and weight measure, which makes the suggested method insensitive to the position and length of the segment, respectively. In specific, even under imbalance among different portions of a period (i.e. window spans non-integer multiple of period), weight measure is able to flatten out such imbalance. Efficacy of suggest method will be applied to real-world data, namely gait data from Physionet.