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We consider the problems of clustering, classification, and visualization of high-dimensional data when no straightforward Euclidean representation exists. Typically, these tasks are performed by first reducing the high-dimensional data to…

Machine Learning · Statistics 2009-09-29 Kevin M. Carter , Raviv Raich , William G. Finn , Alfred O. Hero

We consider Gibbs distributions, which are families of probability distributions over a discrete space $\Omega$ with probability mass function of the form $\mu^\Omega_\beta(\omega) \propto e^{\beta H(\omega)}$ for $\beta$ in an interval…

Data Structures and Algorithms · Computer Science 2025-04-04 David G. Harris , Vladimir Kolmogorov

We propose the use of low bit-depth Sigma-Delta and distributed noise-shaping methods for quantizing the Random Fourier features (RFFs) associated with shift-invariant kernels. We prove that our quantized RFFs -- even in the case of $1$-bit…

Machine Learning · Computer Science 2022-04-14 Jinjie Zhang , Harish Kannan , Alexander Cloninger , Rayan Saab

Consider a family of distributions $\{\pi_{\beta}\}$ where $X\sim\pi_{\beta}$ means that $\mathbb{P}(X=x)=\exp(-\beta H(x))/Z(\beta)$. Here $Z(\beta)$ is the proper normalizing constant, equal to $\sum_x\exp(-\beta H(x))$. Then…

Probability · Mathematics 2015-03-19 Mark Huber

We calculated the spatial distribution of reduced density and pair distribution function (PDF) of solvent hard spheres near a solute using three-dimensional Ornstein-Zernike (OZ) equations coupled with closures in which Percus-Yevick (PY)…

Soft Condensed Matter · Physics 2023-11-27 Mika Matsuo , Yuka Nakamura , Masahiro Kinoshita , Ryo Akiyama

This paper presents an extensive survey of regular distributions in natural and social sciences. The survey includes studies from a wide scope of academic disciplines, in order to create an inventory of the different mathematical functions…

Physics and Society · Physics 2015-07-14 L. Benguigui , M. Marinov

Data sets of multivariate normal distributions abound in many scientific areas like diffusion tensor imaging, structure tensor computer vision, radar signal processing, machine learning, just to name a few. In order to process those normal…

Machine Learning · Computer Science 2024-06-11 Frank Nielsen

We propose a fast algorithm for evaluating the moments of Bingham distribution. The calculation is done by piecewise rational approximation, where interpolation and Gaussian integrals are utilized. Numerical test shows that the algorithm…

Numerical Analysis · Mathematics 2017-11-23 Yixiang Luo , Jie Xu , Pingwen Zhang

Information-theoretic inequalities play a fundamental role in numerous scientific and technological areas as they generally express the impossibility to have a complete description of a system via a finite number of information measures. In…

Mathematical Physics · Physics 2017-10-03 Steeve Zozor , David Puertas-Centeno , Jesús S. Dehesa

In this paper, we introduce a new four-parameter generalized version of the Gompertz model which is called Beta-Gompertz (BG) distribution. It includes some well-known lifetime distributions such as beta-exponential and generalized Gompertz…

Statistics Theory · Mathematics 2014-07-04 Ali Akbar Jafari , Saeid Tahmasebi , Morad Alizadeh

We apply machine learning in the form of a nearest neighbor instance-based algorithm (NN) to generate full photometric redshift probability density functions (PDFs) for objects in the Fifth Data Release of the Sloan Digital Sky Survey (SDSS…

In many scientific applications, the target probability distribution cannot be evaluated in closed form or sampled from directly. Instead, it can often be decomposed into multiple components, some of which are accessible only through…

Methodology · Statistics 2026-03-10 Roxana Darvishi , David C. Stenning , Ted von Hippel , Owen G. Ward

The Slepian-Bangs formula provides a very convenient way to compute the Fisher information matrix (FIM) for Gaussian distributed data. The aim of this letter is to extend it to a larger family of distributions, namely elliptically contoured…

Methodology · Statistics 2015-06-16 Olivier Besson , Yuri I. Abramovich

We apply the holonomic gradient method to compute the distribution function of a weighted sum of independent noncentral chi-square random variables. It is the distribution function of the squared length of a multivariate normal random…

Statistics Theory · Mathematics 2015-09-01 Tamio Koyama , Akimichi Takemura

The probability density function (PDF) and cumulative distribution function of the sum of L independent but not necessarily identically distributed Gamma variates, applicable to the output statistics of maximal ratio combining (MRC)…

Information Theory · Computer Science 2012-07-19 Imran Shafique Ansari , Ferkan Yilmaz , Mohamed-Slim Alouini , Oğuz Kucur

Random processes play a crucial role in scientific research, often characterized by distribution functions or probability density functions (PDFs). These PDFs serve as essential approximations of the actual and frequently undisclosed…

Methodology · Statistics 2023-06-06 Nico Schick

We compute all the moments of a normalization of the function which counts unramified $H_{8}$-extensions of quadratic fields, where $H_{8}$ is the quaternion group of order 8, and show that the values of this function determine a constant…

Number Theory · Mathematics 2017-06-07 Brandon Alberts , Jack Klys

Fant\^omas is a C++ toolkit for exploring the parametrization dependence of parton distribution functions (PDFs) and other correlator functions in quantum chromodynamics (QCD). Fant\^omas facilitates the generation of adaptable polynomial…

High Energy Physics - Phenomenology · Physics 2025-08-01 Lucas Kotz , Aurore Courtoy , T. J. Hobbs , Pavel Nadolsky , Fredrick Olness , Maximiliano Ponce-Chavez , Varada Purohit

This note corrects a technical error in Guardiola (2020, Journal of Statistical Distributions and Applications), presents updated derivations, and offers an extended discussion of the properties of the spherical Dirichlet distribution.…

Methodology · Statistics 2025-06-06 Jose H Guardiola

One of the major problems for maximum likelihood estimation in the well-established directional models is that the normalising constants can be difficult to evaluate. A new general method of "score matching estimation" is presented here on…

Statistics Theory · Mathematics 2016-04-29 Kanti V Mardia , John T Kent , Arnab K Laha