Related papers: Algebraic Topology for Data Analysis
Big data analytics is one of the most promising areas of new research and development in computer science, enterprises, e-commerce, and defense. For many organizations, big data is considered one of their most important strategic assets.…
Modern applications of algebraic topology to point cloud data analysis have motivated active investigation of combinatorial clique complexes -- high-dimensional extensions of combinatorial graphs. We show that meaningful invariants of such…
The rise of Artificial Intelligence (AI) recently empowered researchers to investigate hard mathematical problems which eluded traditional approaches for decades. Yet, the use of AI in Universal Algebra (UA) -- one of the fields laying the…
Recent years have witnessed a fast growth in mathematical artificial intelligence (AI). One of the most successful mathematical AI approaches is topological data analysis (TDA) via persistent homology (PH) that provides explainable AI (xAI)…
In topological data analysis (TDA), one often studies the shape of data by constructing a filtered topological space, whose structure is then examined using persistent homology. However, a single filtered space often does not adequately…
Topological data analysis (TDA) is an emergent field of data analysis. The critical step of TDA is computing the persistent Betti numbers. Existing classical algorithms for TDA are limited if we want to learn from high-dimensional…
Analyzing data in non-Euclidean spaces, such as bioinformatics, biology, and geology, where variables represent directions or angles, poses unique challenges. This type of data is known as circular data in univariate cases and can be termed…
Topological Data Analysis (TDA) is an approach to handle with big data by studying its shape. A main tool of TDA is the persistence diagram, and one can use it to compare data sets. One approach to learn on the similarity between two…
Compositional data analysis is concerned with multivariate data that have a constant sum, usually 1 or 100\%. These are data often found in biochemistry and geochemistry, but also in the social sciences, when relative values are of interest…
Arithmetic combinatorics is often concerned with the problem of bounding the behaviour of arbitrary finite sets in a group or ring with respect to arithmetic operations such as addition or multiplication. Similarly, combinatorial geometry…
An algorithm is presented that generates sets of size equal to the degree of a given variety defined by a homogeneous ideal. This algorithm suggests a versatile framework to study various problems in combinatorial algebraic geometry and…
In the last quarter of a century, algebraic statistics has established itself as an expanding field which uses multilinear algebra, commutative algebra, computational algebra, geometry, and combinatorics to tackle problems in mathematical…
Improvements in experimental and computational technologies have led to significant increases in data available for analysis. Topological data analysis (TDA) is an emerging area of mathematical research that can identify structures in these…
Understanding the topological characteristics of data is important to many areas of research. Recent work has demonstrated that synthetic 4D image-type data can be useful to train 4D convolutional neural network models to see topological…
Biclustering numerical data became a popular data-mining task in the beginning of 2000's, especially for analysing gene expression data. A bicluster reflects a strong association between a subset of objects and a subset of attributes in a…
The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear…
Algebraic statistics is concerned with the study of probabilistic models and techniques for statistical inference using methods from algebra and geometry. This article presents a list of open mathematical problems in this emerging field,…
The main goal of this paper is to explore latent topic analysis (LTA), in the context of quantum information retrieval. LTA is a valuable technique for document analysis and representation, which has been extensively used in information…
Data augmentation (DA) turns seemingly intractable computational problems into simple ones by augmenting latent missing data. In addition to computational simplicity, it is now well-established that DA equipped with a deterministic…
The data augmentation (DA) algorithm is a simple and powerful tool in statistical computing. In this note basic information theory is used to prove a nontrivial convergence theorem for the DA algorithm.