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Persistent homology is a widely-used tool in topological data analysis (TDA) for understanding the underlying shape of complex data. By constructing a filtration of simplicial complexes from data points, it captures topological features…
Sequence models, and particularly Linear Recurrent Neural Networks (LRNNs) of the form $\mathbf{h}_{k+1} = \mathbf{W} \mathbf{h}_{k} + \mathbf{y}_k + \mathbf{b}$, are widely applicable in time-series analysis for dynamical systems, yet, as…
A general method for constructing simplicial complex from observed time series of dynamical systems based on the delay coordinate reconstruction procedure is presented. The obtained simplicial complex preserves all pertinent topological…
In this paper, we use topological data analysis (TDA) tools such as persistent homology, persistent entropy and bottleneck distance, to provide a {\it TDA-based summary} of any given set of texts and a general method for computing a…
The latent space model is one of the well-known methods for statistical inference of network data. While the model has been much studied for a single network, it has not attracted much attention to analyze collectively when multiple…
Making accurate forecasts for a complex system is a challenge in various practical applications. The major difficulty in solving such a problem concerns nonlinear spatiotemporal dynamics with time-varying characteristics. Takens' delay…
One of the most important problems arising in time series analysis is that of bifurcation, or change point detection. That is, given a collection of time series over a varying parameter, when has the structure of the underlying dynamical…
We provide a method to identify system parameters of dynamical systems, called ID-ODE -- Inference by Differentiation and Observing Delay Embeddings. In this setting, we are given a dataset of trajectories from a dynamical system with…
High-dimensional reduction methods are powerful tools for describing the main patterns in big data. One of these methods is the topological data analysis (TDA), which modeling the shape of the data in terms of topological properties. This…
This paper presents a new clustering algorithm for space-time data based on the concepts of topological data analysis and in particular, persistent homology. Employing persistent homology - a flexible mathematical tool from algebraic…
Topological Data Analysis (TDA) is a discipline that applies algebraic topology techniques to analyze complex, multi-dimensional data. Although it is a relatively new field, TDA has been widely and successfully applied across various…
Topological data analysis is an approach to study shape of a data set by means of topology. Its main object of study is the persistence diagram, which represents the topological features of the data set at different spatial resolutions.…
Topological data analysis (TDA) has emerged as an effective approach in data science, with its key technique, persistent homology, rooted in algebraic topology. Although alternative approaches based on differential topology, geometric…
Time series are ubiquitous in our data rich world. In what follows I will describe how ideas from dynamical systems and topological data analysis can be combined to gain insights from time-varying data. We will see several applications to…
Topological data analysis (TDA) is gaining prominence across a wide spectrum of machine learning tasks that spans from manifold learning to graph classification. A pivotal technique within TDA is persistent homology (PH), which furnishes an…
Topological Data Analysis (TDA) has been successfully used for various tasks in signal/image processing, from visualization to supervised/unsupervised classification. Often, topological characteristics are obtained from persistent homology…
In various scientific and engineering fields, the primary research areas have revolved around physics-based dynamical systems modeling and data-driven time series analysis. According to the embedding theory, dynamical systems and time…
With the rapid development of machine learning applications on time-series data, accurately assessing the value of training samples has become essential for data selection, noise detection, and model optimization. However, traditional data…
Evolving temporal networks serve as the abstractions of many real-life dynamic systems, e.g., social network and e-commerce. The purpose of temporal network embedding is to map each node to a time-evolving low-dimension vector for…
Spherically embedded time series are time series with values naturally residing on or can be equivalently mapped to the sphere. Despite their ubiquity in diverse scientific fields, these data frequently exhibit complex non-stationarity…