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This paper gives a review and synthesis of methods of evaluating dimensionality reduction techniques. Particular attention is paid to rank-order neighborhood evaluation metrics. A framework is created for exploring dimensionality reduction…
In ordinary Dimensionality Reduction (DR), each data instance in a high dimensional space (original space), or on a distance matrix denoting original space distances, is mapped to (projected onto) one point in a low dimensional space…
This paper provides an overview of modern digital geometry and topology through mathematical principles, algorithms, and measurements. It also covers recent developments in the applications of digital geometry and topology including image…
Sufficient dimension reduction [J. Amer. Statist. Assoc. 86 (1991) 316-342] has long been a prominent issue in multivariate nonparametric regression analysis. To uncover the central dimension reduction space, we propose in this paper an…
Over the last two decades, significant advances have been made in the design and analysis of fixed-parameter algorithms for a wide variety of graph-theoretic problems. This has resulted in an algorithmic toolbox that is by now…
We investigate metric projections and distance functions referring to convex bodies in finite-dimensional normed spaces. For this purpose we identify the vector space with its dual space by using, instead of the usual identification via the…
The aim of this paper is to provide a coherent framework for transforming boundary pairs of digital images from one resolution to another without knowledge of the full images. It is intended to facilitate the simultaneous usage of…
Measurement is an integral part of modern science, providing the fundamental means for evaluation, comparison, and prediction. In the context of visualization, several different types of measures have been proposed, ranging from approaches…
We are on the way to obtaining multi-dimensional "pictures" of the proton. The field is bursting with activities, both from the theoretical and experimental side. A brief selection of important achievements of the last years and open…
We review the current status of dimensions, as the result of a long and controversial history that includes input from philosophy and physics. Our conclusion is that they are subjective but essential concepts which provide a kind of…
We propose a new objective numerical figure of merit to aid in the evaluation and comparison of tissue-selective images generated from dual-energy radiography systems. A metric is developed through identification of the requirements of a…
The quest for simplification in physics drives the exploration of concise mathematical representations for complex systems. This Dissertation focuses on the concept of dimensionality reduction as a means to obtain low-dimensional…
Cognitive Dimensions is a framework for analyzing human-computer interaction. It is used for meta-analysis, that is, for talking about characteristics of systems without getting bogged down in details of a particular implementation. In this…
Time is an important dimension of brain function, but little is still known about the underlying cognitive principles and neurobiological mechanisms. The field of timing and time perception has witnessed rapid growth and multidisciplinary…
Dimension reduction (DR) is inherently non-unique: multiple embeddings can preserve the structure of high-dimensional data equally well while differing in layout or geometry. In this paper, we formally define the Rashomon set for DR -- the…
The realm of textiles spans clothing, households, healthcare, sports, and industrial applications. The deformable nature of these objects poses unique challenges that prior work on rigid objects cannot fully address. The increasing interest…
The optical resolution of a digital camera is one of its most crucial parameters with broad relevance for consumer electronics, surveillance systems, remote sensing, or medical imaging. However, resolution is physically limited by the…
We propose a new dimensionality reduction toolkit designed to address some of the challenges faced by traditional methods like UMAP and tSNE such as loss of global structure and computational efficiency. Built on the JAX framework, DiRe…
Historically, scalability has been a major challenge to the successful application of semidefinite programming in fields such as machine learning, control, and robotics. In this paper, we survey recent approaches for addressing this…
The objective of this paper is to improve the customary definition of redundancy by providing quantitative measures in its place, which we coin upper and lower redundancies, that match better with an intuitive understanding of redundancy…