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Fourier Series is the second of monographs we present on harmonic analysis. Harmonic analysis is one of the most fascinating areas of research in mathematics. Its centrality in the development of many areas of mathematics such as partial…
In this paper the theory of semi-bounded rationality is proposed as an extension of the theory of bounded rationality. In particular, it is proposed that a decision making process involves two components and these are the correlation…
Given a positive definite covariance matrix $\widehat \Sigma$, we strive to construct an optimal \emph{approximate} factor analysis model $HH^\top +D$, with $H$ having a prescribed number of columns and $D>0$ diagonal. The optimality…
According to a mainstream position in contemporary cognitive science and philosophy, the use of abstract compositional concepts is both a necessary and a sufficient condition for the presence of genuine thought. In this article, we show how…
The objective function of a matrix factorization model usually aims to minimize the average of a regression error contributed by each element. However, given the existence of stochastic noises, the implicit deviations of sample data from…
Estimating prevalence, the fraction of a population with a certain medical condition, is fundamental to epidemiology. Traditional methods rely on classification of test samples taken at random from a population. Such approaches to…
In this paper we propose a general spectral theory for tensors. Our proposed factorization decomposes a tensor into a product of orthogonal and scaling tensors. At the same time, our factorization yields an expansion of a tensor as a…
Scaling problems have a rich and diverse history, and thereby have found numerous applications in several fields of science and engineering. For instance, the matrix scaling problem has had applications ranging from theoretical computer…
Variable selection for models including interactions between explanatory variables often needs to obey certain hierarchical constraints. The weak or strong structural hierarchy requires that the existence of an interaction term implies at…
Generative diffusion models showed high success in many fields with a powerful theoretical background. They convert the data distribution to noise and remove the noise back to obtain a similar distribution. Many existing reviews focused on…
Factorization models express a statistical object of interest in terms of a collection of simpler objects. For example, a matrix or tensor can be expressed as a sum of rank-one components. However, in practice, it can be challenging to…
We frame the problem of selecting an optimal audio encoding scheme as a supervised learning task. Through uniform convergence theory, we guarantee approximately optimal codec selection while controlling for selection bias. We present…
This paper is a brief overview of the concepts involved in measuring the degree of contextuality and detecting contextuality in systems of binary measurements of a finite number of objects. We discuss and clarify the main concepts and…
Regression problems are traditionally analyzed via univariate characteristics like the regression function, scale function and marginal density of regression errors. These characteristics are useful and informative whenever the association…
Structured variational inference constitutes a core methodology in modern statistical applications. Unlike mean-field variational inference, the approximate posterior is assumed to have interdependent structure. We consider the natural…
This note presents a method that provides optimal monotone conditional error functions for a large class of adaptive two stage designs. The presented method builds on a previously developed general theory for optimal adaptive two stage…
The theory of large deviations deals with the probabilities of rare events (or fluctuations) that are exponentially small as a function of some parameter, e.g., the number of random components of a system, the time over which a stochastic…
The perturbation theory based on typicality introduced in Ref. [1] and further refined in Refs. [2, 3] provides a powerful tool since it is intended to be applicable to a wide range of scenarios while relying only on a few parameters. Even…
It is known that the common factors in a large panel of data can be consistently estimated by the method of principal components, and principal components can be constructed by iterative least squares regressions. Replacing least squares…
Standard statistical theory has arguably proved to be unsuitable as a basis for constructing a satisfactory completely general framework for performing statistical inference. For example, frequentist theory has never come close to providing…