We present a certain geometrical interpretation of the natural numbers, where these numbers appear as joint products of 5and 3-multiples located at specified positions in a revolving chamber. Numbers without factors 2, 3 or 5 appear at eight such positions, and any prime number larger than 7 manifests at one of these eight positions after a specified amount of rotations of the chamber. Our appr...

An important task in computational statistics and machine learning is to approximate a posterior distribution p(x) with an empirical measure supported on a set of representative points {xi}i=1. This paper focuses on methods where the selection of points is essentially deterministic, with an emphasis on achieving accurate approximation when n is small. To this end, we present Stein Points. The i...

Discussion Papers are a series of manuscripts in their draft form. They are not intended for circulation or distribution except as indicated by the author. For that reason Discussion Papers may not be reproduced or distributed without the written consent of the author. Summary Our investigation concerns the estimation of predictive densities and a study of efficiency as measured by the frequent...

Financial Disclosure(s): The author(s) have no proprietary or commercial interest in any materials discussed in this article. Financial support received from the National Eye Institute K23 Mentored Clinician Scientist Award (1K23EY019511, to J.D.S.) and a Research to Prevent Blindness “Physician Scientist” Award. Author Contributions: Conception and design: Elam, Blachley, Stein Data collection...

This paper is devoted to adaptive signal denoising in the context of Graph Signal Processing (GSP) using Spectral Wavelet Transform (SGWT). issue addressed via a data-driven thresholding process transformed domain by optimizing parameters sense Mean Square Error (MSE) Stein’s Unbiased Risk Estimator (SURE). The SGWT considered built upon partition unity making transform semi-orthogonal so that ...

The James-Stein estimator is an of the multivariate normal mean and dominates maximum likelihood (MLE) under squared error loss. original work inspired great interest in developing shrinkage estimators for a variety problems. Nonetheless, research on estimation manifold-valued data scarce. In this paper, we propose parameters Log-Normal distribution defined manifold $N \times N$ symmetric posit...

Is a complex space X which is the union of an increasing sequence X1 ⊂ X2 ⊂ X3 ⊂ · · · of open Stein subspaces itself a Stein space ? From the begining this question has held great interest in Stein theory. The special case when {Xj}j≥1 is a sequence of Stein domains in I C n had been proved long time ago by Behnke and Stein [2]. In 1956, Stein [13] answered positively the question under the ad...

Large-scale kernel approximation is an important problem in machine learning research. Approaches using random Fourier features have become increasingly popular [Rahimi and Recht, 2007], where kernel approximation is treated as empirical mean estimation via Monte Carlo (MC) or Quasi-Monte Carlo (QMC) integration [Yang et al., 2014]. A limitation of the current approaches is that all the feature...

We consider a priori generalization bounds developed in terms of cross-validation estimates and the stability of learners. In particular, we first derive an exponential Efron-Stein type tail inequality for the concentration of a general function of n independent random variables. Next, under some reasonable notion of stability, we use this exponential tail bound to analyze the concentration of ...