Related papers: Approximating Continuous Functions on Persistence …
Summarizing topological information from datasets and maps defined on them is a central theme in topological data analysis. \textsf{Mapper}, a tool for such summarization, takes as input both a possibly high dimensional dataset and a map…
Sparse systems are usually parameterized by a tuning parameter that determines the sparsity of the system. How to choose the right tuning parameter is a fundamental and difficult problem in learning the sparse system. In this paper, by…
Finding an optimal parameter of a black-box function is important for searching stable material structures and finding optimal neural network structures, and Bayesian optimization algorithms are widely used for the purpose. However, most of…
Exploiting geometric structure to improve the asymptotic complexity of discrete assignment problems is a well-studied subject. In contrast, the practical advantages of using geometry for such problems have not been explored. We implement…
This paper presents the first approach to visualize the importance of topological features that define classes of data. Topological features, with their ability to abstract the fundamental structure of complex data, are an integral…
An Important tool in the field topological data analysis is known as persistent Homology (PH) which is used to encode abstract representation of the homology of data at different resolutions in the form of persistence diagram (PD). In this…
In urgent decision making applications, ensemble simulations are an important way to determine different outcome scenarios based on currently available data. In this paper, we will analyze the output of ensemble simulations by considering…
We develop fixed-point algorithms for the approximation of structured matrices with rank penalties. In particular we use these fixed-point algorithms for making approximations by sums of exponentials, or frequency estimation. For the basic…
Deep neural networks have emerged as powerful tools for learning operators defined over infinite-dimensional function spaces. However, existing theories frequently encounter difficulties related to dimensionality and limited…
FeFETs hold strong potential for advancing memory and logic technologies, but their inherent randomness arising from both operational cycling and fabrication variability poses significant challenges for accurate and reliable modeling.…
This work is motivated by the following question in data-driven study of dynamical systems: given a dynamical system that is observed via time series of persistence diagrams that encode topological features of solutions snapshots, what…
Dynamic community detection methods often lack effective mechanisms to ensure temporal consistency, hindering the analysis of network evolution. In this paper, we propose a novel deep graph clustering framework with temporal consistency…
Persistent homology is a fundamental tool in Topological Data Analysis. The associated algebraic structure is the persistence module, a sequence of vector spaces connected by linear maps. Persistence modules admit a complete and…
The time delay (or Sliding Window) embedding is a technique from dynamical systems to reconstruct attractors from time series data. Recently, descriptors from Topological Data Analysis (TDA) -- specifically, persistence diagrams -- have…
Graph embeddings have emerged as a powerful tool for representing complex network structures in a low-dimensional space, enabling the use of efficient methods that employ the metric structure in the embedding space as a proxy for the…
Let $n$ be a positive integer. We provide an explicit geometrically motivated $1$-Lipschitz map from the space of persistence diagrams on $n$ points (equipped with the Bottleneck distance) into the Hilbert space $\ell^2$. Such maps are a…
Motivated by the rapidly increasing relevance of virtual material design in the domain of materials science, it has become essential to assess whether topological properties of stochastic models for a spatial tessellation are in accordance…
A topological shape analysis is proposed and utilized to learn concepts that reflect shape commonalities. Our approach is two-fold: i) a spatial topology analysis of point cloud segment constellations within objects. Therein constellations…
One of the most elusive challenges within the area of topological data analysis is understanding the distribution of persistence diagrams. Despite much effort, this is still largely an open problem. In this paper, we present a series of…
We propose a general technique for extracting a larger set of stable information from persistent homology computations than is currently done. The persistent homology algorithm is usually viewed as a procedure which starts with a filtered…