Related papers: Topological Data Analysis in Information Space
Homology-based invariants can be used to characterize the geometry of datasets and thereby gain some understanding of the processes generating those datasets. In this work we investigate how the geometry of a dataset changes when it is…
It is shown that space-time may possess the differentiability properties of manifolds as well as the ultraviolet finiteness properties of lattices. Namely, if a field's amplitudes are given on any sufficiently dense set of discrete points…
Conventional statistics begins with a model, and assigns a likelihood of obtaining any particular set of data. The opposite approach, beginning with the data and assigning a likelihood to any particular model, is explored here for the case…
The idea of slicing divergences has been proven to be successful when comparing two probability measures in various machine learning applications including generative modeling, and consists in computing the expected value of a `base…
A methodology is proposed for inferring the topology underlying point cloud data. The approach employs basic elements of Morse Theory, and is capable of producing not only a point estimate of various topological quantities (e.g., genus),…
The concept of $typed$ $topology$ is introduced. In a typed topological space, some open sets are assigned "types", and topological concepts such as closure, connectedness can be defined using types. A finite data set in $R^2$ is a…
In this work, we Extend Pawlak approximation spaces by topological spaces. Also, Rough Membership, equality and inclusion relations are extended using topological near open sets. In addition, new extended measures of accuracy and quality of…
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…
Interacting, self-propelled particles such as epithelial cells can dynamically self-organize into complex multicellular patterns, which are challenging to classify without a priori information. Classically, different phases and phase…
Assume that a finite set of points is randomly sampled from a subspace of a metric space. Recent advances in computational topology have provided several approaches to recovering the geometric and topological properties of the underlying…
As data structures and mathematical objects used for complex systems modeling, hypergraphs sit nicely poised between on the one hand the world of network models, and on the other that of higher-order mathematical abstractions from algebra,…
We study a general framework of distributional computational graphs: computational graphs whose inputs are probability distributions rather than point values. We analyze the discretization error that arises when these graphs are evaluated…
In computer vision and medical imaging, the problem of matching structures finds numerous applications from automatic annotation to data reconstruction. The data however, while corresponding to the same anatomy, are often very different in…
To what extent can we distinguish one probability distribution from another? Are there quantitative measures of distinguishability? The goal of this tutorial is to approach such questions by introducing the notion of the "distance" between…
Topological methods have the potential of exploring data clouds without making assumptions on their the structure. Here we propose a hierarchical topological clustering algorithm that can be implemented with any distance choice. The…
The central goal of this thesis is to develop methods to experimentally study topological phases. We do so by applying the powerful toolbox of quantum simulation techniques with cold atoms in optical lattices. To this day, a complete…
We present a new tool for data analysis: persistence discrete homology, which is well-suited to analyze filtrations of graphs. In particular, we provide a novel way of representing high-dimensional data as a filtration of graphs using…
Topological Data Analysis (TDA) is an emergent field that aims to discover topological information hidden in a dataset. TDA tools have been commonly used to create filters and topological descriptors to improve Machine Learning (ML)…
In 20th century mathematics, the field of topology, which concerns the properties of geometric objects under continuous transformation, has proved surprisingly useful in application to the study of discrete mathematics, such as…
This report concerns the problem of dimensionality reduction through information geometric methods on statistical manifolds. While there has been considerable work recently presented regarding dimensionality reduction for the purposes of…