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M\"obius transformations have been studied over the field of complex numbers. In this paper, we investigate M\"obius transformations over two rings which are not fields: the ring of double numbers and the ring of dual numbers. We give types…

The large N generating functions for the counting of chiral operators in $\mathcal{N}=1$, four-dimensional quiver gauge theories have previously been obtained in terms of the weighted adjacency matrix of the quiver diagram. We introduce the…

Deep learning models have become a cornerstone of modern AI research, yet their initializations and learning rates may at times be set in an opaque or ad-hoc fashion due to the high cost of hyperparameter sweeps. The $\mu$-Parameterization…

The Bayesian approach to machine learning amounts to computing posterior distributions of random variables from a probabilistic model of how the variables are related (that is, a prior distribution) and a set of observations of variables.…

Model-free deep reinforcement learning algorithms have been shown to be capable of learning a wide range of robotic skills, but typically require a very large number of samples to achieve good performance. Model-based algorithms, in…

The M{\"o}bius transform is a crucial transformation into the Boolean world; it allows to change the Boolean representation between the True Table and Algebraic Normal Form. In this work, we introduce a new algebraic point of view of this…

We propose to interpret machine learning functions as physical observables, opening up the possibility to apply "standard" statistical-mechanical methods to outputs from neural networks. This includes histogram reweighting and finite-size…

Our goal is to provide a review of deep learning methods which provide insight into structured high-dimensional data. Rather than using shallow additive architectures common to most statistical models, deep learning uses layers of…

Many analyses in particle and nuclear physics use simulations to infer fundamental, effective, or phenomenological parameters of the underlying physics models. When the inference is performed with unfolded cross sections, the observables…

For years the model performance in machine learning obeyed a power-law relationship with the model size. For the consideration of parameter efficiency, recent studies focus on increasing model depth rather than width to achieve better…

We develop new transfer learning algorithms to accelerate prediction of material properties from ab initio simulations based on density functional theory (DFT). Transfer learning has been successfully utilized for data-efficient modeling in…

This paper investigates the learnability of the nonlinearity property of Boolean functions using neural networks. We train encoder style deep neural networks to learn to predict the nonlinearity of Boolean functions from examples of…

In this paper we describe spectral transformation algorithms for the computation of eigenvalues with positive real part of sparse nonsymmetric matrix pencils $(J,L)$, where $L$ is of the form $\pmatrix{M&0\cr 0&0}$. For this we define a…

Let $g:\mathbb{N}\to\{-1,1\}$ be a completely multiplicative function, $\mu$ be the M\"obius function and $\mu_2^2(n)$ be the indicator that $n$ is cubefree. We prove that $f=\mu^2g$ and $f=\mu_2^2g$ have unbounded partial sums. Our proofs…

An elementary recursive relation for M$\ddot{\mathrm{o}}$bius function $\mu (n)$ is introduced by two simple ways. With this recursive relation, $\mu (n)$ can be calculated without directly knowing the factorization of the $n$. $\mu (1)…

In this review we cover the basics of efficient nonparametric parameter estimation (also called functional estimation), with a focus on parameters that arise in causal inference problems. We review both efficiency bounds (i.e., what is the…

We use machine learning to classify examples of braids (or flat braids) as trivial or non-trivial. Our ML takes form of supervised learning using neural networks (multilayer perceptrons). When they achieve good results in classification, we…

While model selection is a well-studied topic in parametric and nonparametric regression or density estimation, selection of possibly high-dimensional nuisance parameters in semiparametric problems is far less developed. In this paper, we…

Machine unlearning (MU) aims to remove the influence of particular data points from the learnable parameters of a trained machine learning model. This is a crucial capability in light of data privacy requirements, trustworthiness, and…

This paper aims to develop an effective model-free inference procedure for high-dimensional data. We first reformulate the hypothesis testing problem via sufficient dimension reduction framework. With the aid of new reformulation, we…