Related papers: Distribution Theory by Riemann Integrals
We consider the one-dimensional quantum mechanical problem of defining interactions concentrated at a single point in the framework of the theory of distributions. The often ill-defined product which describes the interaction term in the…
$f$-divergences, which quantify discrepancy between probability distributions, are ubiquitous in information theory, machine learning, and statistics. While there are numerous methods for estimating $f$-divergences from data, a limit…
This tutorial provides a comprehensive survey of methods for fine-tuning diffusion models to optimize downstream reward functions. While diffusion models are widely known to provide excellent generative modeling capability, practical…
As deep learning applications continue to become more diverse, an interesting question arises: Can general problem solving arise from jointly learning several such diverse tasks? To approach this question, deep multi-task learning is…
A differential algebra of nonlinear generalized functions is presented as a tool for a wide range of nonsmooth nonlinear problems. The power of the differential algebra is used to do mathematical calculations or proofs; then the final…
This paper studies the master-worker distributed linearly separable computation problem, where the considered computation task, referred to as linearly separable function, is a typical linear transform model widely used in cooperative…
Dunkl theory is a far reaching generalization of Fourier analysis and special function theory related to root systems. During the sixties and seventies, it became gradually clear that radial Fourier analysis on rank one symmetric spaces was…
Drawing inspiration from Dirac's work on functions of non commuting observables, we develop a fresh approach to phase space descriptions of operators and the Wigner distribution in quantum mechanics. The construction presented here is…
In these lecture notes we present an introduction to non-standard analysis especially written for the community of mathematicians, physicists and engineers who do research on J. F. Colombeau' theory of new generalized functions and its…
Distributed learning is the problem of inferring a function in the case where training data is distributed among multiple geographically separated sources. Particularly, the focus is on designing learning strategies with low computational…
Diffusion Models (DMs), as a leading class of generative models, offer key advantages for reinforcement learning (RL), including multi-modal expressiveness, stable training, and trajectory-level planning. This survey delivers a…
In the framework of distributionally generalized quantum theory, the object $H\psi$ is defined as a distribution. The mathematical significance is a mild generalization for the theory of para- and pseudo-differential operators (as well as a…
In this pedagogical text aimed at those wanting to start thinking about or brush up on probabilistic inference, I review the rules by which probability distribution functions can (and cannot) be combined. I connect these rules to the…
We focus on the distribution regression problem: regressing to vector-valued outputs from probability measures. Many important machine learning and statistical tasks fit into this framework, including multi-instance learning and point…
The aim of the lectures is to introduce first-year Ph.D. students and research workers to the theory of the Dirac operator, spinor techniques, and their relevance for the theory of eigenvalues in Riemannian geometry. Topics: differential…
In analogy with the Poisson summation formula, we identify when the fractional Fourier transform, applied to a Dirac comb in dimension one, gives a discretely supported measure. We describe the resulting series of complex multiples of delta…
These lecture notes introduce some topics of classical statistical physics, particularly those that are relevant for neural networks and deep learning. Statistical physics is treated as a branch of probability theory or statistics, with the…
When expressing a distribution in Euclidean space in spherical co-ordinates, derivation with respect to the radial and angular co-ordinates is far from trivial. Exploring the possibilities of defining a radial derivative of the…
This article investigates signal estimation in wireless transmission (i.e., receive combining) from the perspective of statistical machine learning, where the transmit signals may be from an integrated sensing and communication system; that…
Bayesian graphical modeling provides an appealing way to obtain uncertainty estimates when inferring network structures, and much recent progress has been made for Gaussian models. These models have been used extensively in applications to…