Related papers: Dimensional Analysis: A Centenary Update
A number of topics in analysis are discussed, with emphasis on basic principles. There is some overlap with "Elements of linear and real analysis" (arXiv:math/0108030), with numerous changes in content and presentation since then.
I describe a version of so-called naive dimensional analysis, a rule for estimating the sizes of terms in an effective theory below the scale of chiral symmetry breaking induced by a strong gauge interaction. The rule is simpler and more…
A mathematical relation between elements of one- and multi-dimensional discrete Fourier transforms (DFT) is found. A method of analysing the multi-dimensional data by their single one-dimensional (1-D) DFT is offered. An experiment of…
This paper presents an extension and an elaboration of the theory of differential similarity, which was originally proposed in arXiv:1401.2411 [cs.LG]. The goal is to develop an algorithm for clustering and coding that combines a geometric…
We derive a well-defined renormalized version of mutual information that allows to estimate the dependence between continuous random variables in the important case when one is deterministically dependent on the other. This is the situation…
Traditional statistical inference considers relatively small data sets and the corresponding theoretical analysis focuses on the asymptotic behavior of a statistical estimator when the number of samples approaches infinity. However, many…
Dimensionality is an important aspect for analyzing and understanding (high-dimensional) data. In their 2006 ICDM paper Tatti et al. answered the question for a (interpretable) dimension of binary data tables by introducing a normalized…
Convexity is an important notion in non linear optimization theory as well as in infinite dimensional functional analysis. As will be seen below, very simple and powerful tools will be derived from elementary duality arguments (which are…
A plethora of dimension reduction methods have been developed to visualize high-dimensional data in low dimensions. However, different dimension reduction methods often output different and possibly conflicting visualizations of the same…
Relevance is generally understood as a multi-level and multi-dimensional relationship between an information need and an information object. However, traditional IR evaluation metrics naively assume mono-dimensionality. We ask: How to deal…
We introduce a new class of multiplications of distributions in one dimension merging together two different regularizations of distributions. Some of the features of these multiplications are discussed in a certain detail. We use our…
We consider partially observed multiscale diffusion models that are specified up to an unknown vector parameter. We establish for a very general class of test functions that the filter of the original model converges to a filter of reduced…
This article is due to appear in the Handbook of Statistics, Vol. 43, Elsevier/North-Holland, Amsterdam, edited by Arni S. R. Srinivasa Rao and C. R. Rao. In modern day analytics, there is ever growing need to develop statistical models to…
The growing complexity and information content of data, together with the need to understand both the complex structures, relationships, and phenomena present in these data spaces, compounded with the emerging need to understand the results…
Astronomical researchers often think of analysis and visualization as separate tasks. In the case of high-dimensional data sets, though, interactive exploratory data visualization can give far more insight than an approach where data…
Multi-collinearity is a wide-spread phenomenon in modern statistical applications and when ignored, can negatively impact model selection and statistical inference. Classic tools and measures that were developed for "$n>p$" data are not…
Intrinsic dimension and differential entropy estimators are studied in this paper, including their systematic bias. A pragmatic approach for joint estimation and bias correction of these two fundamental measures is proposed. Shared steps on…
Incrementality is ubiquitous in human-human interaction and beneficial for human-computer interaction. It has been a topic of research in different parts of the NLP community, mostly with focus on the specific topic at hand even though…
The advent of high resolution imaging has made data on surface shape widespread. Methods for the analysis of shape based on landmarks are well established but high resolution data require a functional approach. The starting point is 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…