Related papers: Machine Learning of the Prime Distribution
This paper presents the formulation and analysis of a novel distributed maximum likelihood algorithm that utilizes a first-order optimization scheme. The proposed approach utilizes a static average consensus algorithm to reach agreement on…
We present a new statistical learning paradigm for Boltzmann machines based on a new inference principle we have proposed: the latent maximum entropy principle (LME). LME is different both from Jaynes maximum entropy principle and from…
The work continues the author's many-year research in theory of maximal branching processes, which are obtained from classical branching processes by replacing the summation of descendant numbers with taking the maximum. One can say that in…
Probabilistic models for the distribution of primes in the natural numbers are constructed in the article. The author found and proved the probabilistic estimates of the deviation $R(x)=|\pi(x)- Li(x)|$. The author has analyzed the…
We propose a distributionally robust approach to learning hyperparameters for first-order methods in convex optimization. Given a dataset of problem instances, we minimize a Wasserstein distributionally robust version of the performance…
Power-law distributions are typical macroscopic features occurring in almost all complex systems observable in nature. As a result, researchers in quantitative analyses must often generate random synthetic variates obeying power-law…
Consider a collection of m competing machine learning algorithms. Given their performance on a benchmark of datasets, we would like to identify the best performing algorithm. Specifically, which algorithm is most likely to ``win'' (rank…
We determine for what proportion of integers $h$ one now knows that there are infinitely many prime pairs $p,\ p+h$ as a consequence of the Zhang-Maynard-Tao theorem. We consider the natural generalization of this to $k$-tuples of integers,…
Ordinary differential equations (ODEs), via their induced flow maps, provide a powerful framework to parameterize invertible transformations for the purpose of representing complex probability distributions. While such models have achieved…
The aim of this paper is to provide several novel upper bounds on the excess risk with a primal focus on classification problems. We suggest two approaches and the obtained bounds are represented via the distribution dependent local…
A common approach to statistical learning with big-data is to randomly split it among $m$ machines and learn the parameter of interest by averaging the $m$ individual estimates. In this paper, focusing on empirical risk minimization, or…
We introduce a new probabilistic model of the primes consisting of integers that survive the sieving process when a random residue class is selected for every prime modulus below a specific bound. From a rigorous analysis of this model, we…
We consider the problem of hypothesis testing for discrete distributions. In the standard model, where we have sample access to an underlying distribution $p$, extensive research has established optimal bounds for uniformity testing,…
The maximum entropy principle (MEP) is a method for obtaining the most likely distribution functions of observables from statistical systems, by maximizing entropy under constraints. The MEP has found hundreds of applications in ergodic and…
Optimization results are one method for understanding neural computation from Nature's perspective and for defining the physical limits on neuron-like engineering. Earlier work looks at individual properties or performance criteria and…
The Principle of Maximum Entropy is a rigorous technique for estimating an unknown distribution given partial information while simultaneously minimizing bias. However, an important requirement for applying the principle is that the…
We use the PAC-Bayesian theory for the setting of learning-to-optimize. To the best of our knowledge, we present the first framework to learn optimization algorithms with provable generalization guarantees (PAC-Bayesian bounds) and explicit…
Recently, the original storage prescription for the Hopfield model of neural networks -- as well as for its dense generalizations -- has been turned into a genuine Hebbian learning rule by postulating the expression of its Hamiltonian for…
Let $u(x)$ be a subpolynomial function in a Hardy field. We establish necessary and sufficient conditions for the weighted uniform distribution of the sequences $(u(n))_{n\in\mathbb{N}}$ and $(u(p_n))_{n\in\mathbb{N}}$, where $p_n$ denotes…
Research on the distribution of prime numbers has revealed a dual character: deterministic in definition yet exhibiting statistical behavior reminiscent of random processes. In this paper we show that it is possible to use an image-focused…