Related papers: Topological Data Analysis of Neural Network Layer …

When training transformers on graph-structured data, incorporating information about the underlying topology is crucial for good performance. Topological masking, a type of relative position encoding, achieves this by upweighting or…

We study the persistent homology of both functional data on compact topological spaces and structural data presented as compact metric measure spaces. One of our goals is to define persistent homology so as to capture primarily properties…

Recently, persistent homology has had tremendous success in biomolecular data analysis. It works by examining the topological relationship or connectivity of a group of atoms in a molecule at a variety of scales, then rendering a family of…

Graph neural networks (GNNs) have demonstrated a significant success in various graph learning tasks, from graph classification to anomaly detection. There recently has emerged a number of approaches adopting a graph pooling operation…

Persistent homology is a popular computational tool for analyzing the topology of point clouds, such as the presence of loops or voids. However, many real-world datasets with low intrinsic dimensionality reside in an ambient space of much…

Neural networks excel across a wide range of tasks, yet remain black boxes. In particular, how their internal representations are shaped by the complexity of the input data and the problems they solve remains obscure. In this work, we…

Persistent Homology (PH) has been successfully used to train networks to detect curvilinear structures and to improve the topological quality of their results. However, existing methods are very global and ignore the location of topological…

Topological features play an essential role in ensuring geometric plausibility and structural consistency in image analysis tasks such as segmentation and skeletonization. However, integrating topology-preserving learning based on simple…

We describe a new methodology for studying persistence of topological features across a family of spaces or point-cloud data sets, called zigzag persistence. Building on classical results about quiver representations, zigzag persistence…

Understanding the decision-making processes of large language models is critical given their widespread applications. To achieve this, we aim to connect a formal mathematical framework - zigzag persistence from topological data analysis -…

We start with a simple introduction to topological data analysis where the most popular tool is called a persistent diagram. Briefly, a persistent diagram is a multiset of points in the plane describing the persistence of topological…

Real data is often given as a point cloud, i.e. a finite set of points with pairwise distances between them. An important problem is to detect the topological shape of data --- for example, to approximate a point cloud by a low-dimensional…

We propose PLLay, a novel topological layer for general deep learning models based on persistence landscapes, in which we can efficiently exploit the underlying topological features of the input data structure. In this work, we show…

We consider the slow and athermal deformations of amorphous solids and show how the ensuing sequence of discrete plastic rearrangements can be mapped onto a directed network. The network topology reveals a set of highly connected regions…

Topological methods can provide a way of proposing new metrics and methods of scrutinising data, that otherwise may be overlooked. In this work, a method of quantifying the shape of data, via a topic called topological data analysis will be…

Phase separation mechanisms can produce a variety of complicated and intricate microstructures, which often can be difficult to characterize in a quantitative way. In recent years, a number of novel topological metrics for microstructures…

Topological landscape is introduced for networks with functions defined on the nodes. By extending the notion of gradient flows to the network setting, critical nodes of different indices are defined. This leads to a concise and…

Time-series prediction is an active area of research across various fields, often challenged by the fluctuating influence of short-term and long-term factors. In this study, we introduce a feature engineering method that enhances the…

We present a way to apply topological data analysis for classifying encrypted bits into distinct classes. Persistent homology is applied to generate topological features of a point cloud obtained from sets of encryptions. We see that this…

Manifold learning aims to discover and represent low-dimensional structures underlying high-dimensional data while preserving critical topological and geometric properties. Existing methods often fail to capture local details with global…