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Modeling data as being sampled from a union of independent subspaces has been widely applied to a number of real world applications. However, dimensionality reduction approaches that theoretically preserve this independence assumption have…
The real-life data have a complex and non-linear structure due to their nature. These non-linearities and the large number of features can usually cause problems such as the empty-space phenomenon and the well-known curse of dimensionality.…
We use the trimming transformations to study the tight span of a metric space.
Dimension reduction is the process of embedding high-dimensional data into a lower dimensional space to facilitate its analysis. In the Euclidean setting, one fundamental technique for dimension reduction is to apply a random linear map to…
Multidimensional scaling (MDS) is a popular technique for mapping a finite metric space into a low-dimensional Euclidean space in a way that best preserves pairwise distances. We study a notion of MDS on infinite metric measure spaces,…
Dimensionality reduction represents the process of generating a low dimensional representation of high dimensional data. Motivated by the formation control of mobile agents, we propose a nonlinear dynamical system for dimensionality…
Any three-dimensional Riemannian metric can be locally obtained by deforming a constant curvature metric along one direction. The general interest of this result, both in geometry and physics, and related open problems are stressed.
The magnetic hard disc drive industry continues to face serious challenges in its quest for ever decreasing bit size. This review summarizes recent advances and promising new technology which have foundations in fundamental physical…
This article presents an overview of neutrino physics research, with highlights on the physics goals, results and interpretations of the current neutrino experiments and future directions and program. It is not meant to be a comprehensive…
This paper identifies the challenges associated with coordinating the development of new research methodologies and an accelerated pace of new discoveries in materials science with slower-evolving textbooks and curricula. The target…
Over the past two decades, the field of high-dimensional statistics has experienced substantial progress, driven largely by technological advances that have dramatically reduced the cost and effort for data collection and storage across a…
The aim of this paper is to provide an overview of recent development related to Bregman distances outside its native areas of optimization and statistics. We discuss approaches in inverse problems and image processing based on Bregman…
Selecting the dimensionality reduction technique that faithfully represents the structure is essential for reliable visual communication and analytics. In reality, however, practitioners favor projections for other attractions, such as…
Scalability of statistical estimators is of increasing importance in modern applications and dimension reduction is often used to extract relevant information from data. A variety of popular dimension reduction approaches can be framed as…
There has been a lot of interest in sufficient dimension reduction (SDR) methodologies as well as nonlinear extensions in the statistics literature. In this note, we use classical results regarding metric spaces and positive definite…
In this article, we obtain several new weighted bounds for the numerical radius of a Hilbert space operator. The significance of the obtained results is the way they generalize many existing results in the literature; where certain values…
Linear dimensionality reduction methods are a cornerstone of analyzing high dimensional data, due to their simple geometric interpretations and typically attractive computational properties. These methods capture many data features of…
One of the most stimulating recent ideas in particle physics involves a possibility that our universe has additional compactified spatial dimensions, perhaps as large as 1 mm. In this mini-review, we discuss the results of recent…
Approximate Bayesian computation (ABC) methods make use of comparisons between simulated and observed summary statistics to overcome the problem of computationally intractable likelihood functions. As the practical implementation of ABC…
The article provides a brief survey of the mathematics of some of the newly being developed so called "hybrid" (also called "multi-physics" or "multi-wave") imaging techniques.