Related papers: Demuth's path to randomness
We consider a self-decimated generator of pseudorandom numbers and examine the preperiod $\lambda$ and the period $\mu$ of its state sequence. We obtain the expectations and variances of $\lambda$ and $\mu$ for the case when decimation…
In this report we consider the following problem: Given a trained model that is partially faulty, can we correct its behaviour without having to train the model from scratch? In other words, can we ``debug" neural networks similar to how we…
We consider the problem of learning the structure of ferromagnetic Ising models Markov on sparse Erdos-Renyi random graph. We propose simple local algorithms and analyze their performance in the regime of correlation decay. We prove that an…
Can an algorithm create original and compelling fashion designs to serve as an inspirational assistant? To help answer this question, we design and investigate different image generation models associated with different loss functions to…
We study mechanisms to characterize how the asymptotic convergence of backpropagation in deep architectures, in general, is related to the network structure, and how it may be influenced by other design choices including activation type,…
Randomness is a crucial resource for a broad range of important applications, such as Monte Carlo simulation and computation, generative artificial intelligence and cryptography. But what is randomness? A widely accepted definition has…
We present a new method for nonlinear prediction of discrete random sequences under minimal structural assumptions. We give a mathematical construction for optimal predictors of such processes, in the form of hidden Markov models. We then…
Probabilistic circuits (PCs) represent a probability distribution as a computational graph. Enforcing structural properties on these graphs guarantees that several inference scenarios become tractable. Among these properties, structured…
We present a Markov-chain analysis of blockwise-stochastic algorithms for solving partially block-separable optimization problems. Our main contributions to the extensive literature on these methods are statements about the Markov operators…
Delorme suggested that the set of all complete intersection numerical semigroups can be computed recursively. We have implemented this algorithm, and particularized it to several subfamilies of this class of numerical semigroups: free and…
This article explores the model-dependent nature of set cardinality, emphasizing that cardinality is not absolute but varies across different axiomatic frameworks. Although Cantor's diagonal argument shows the real numbers are…
We present a new method for online prediction and learning of tensors ($N$-way arrays, $N >2$) from sequential measurements. We focus on the specific case of 3-D tensors and exploit a recently developed framework of structured tensor…
Economic theory is a mathematically rich field in which there are opportunities for the formal analysis of singularities and catastrophes. This article looks at the historical context of singularities through the work of two eminent…
This is a survey of several approaches to the framework for working with infinitesimals and infinite numbers, originally developed by Abraham Robinson in the 1960s, and their constructive engagement with the Cantor-Dedekind postulate and…
In this work we study randomised reduction strategies,a notion already known in the context of abstract reduction systems, for the $\lambda$-calculus. We develop a simple framework that allows us to prove a randomised strategy to be…
In La G\'eom\'etrie, Descartes proposed a balance between geometric constructions and symbolic manipulation with the introduction of suitable ideal machines. In modern terms, that is a balance between analog and symbolic computation.…
This short course offers a new perspective on randomized algorithms for matrix computations. It explores the distinct ways in which probability can be used to design algorithms for numerical linear algebra. Each design template is…
Existing studies on provably efficient algorithms for Markov games (MGs) almost exclusively build on the "optimism in the face of uncertainty" (OFU) principle. This work focuses on a different approach of posterior sampling, which is…
Termination analysis of linear loops plays a key r\^{o}le in several areas of computer science, including program verification and abstract interpretation. Already for the simplest variants of linear loops the question of termination…
This paper considers a class of reinforcement-based learning (namely, perturbed learning automata) and provides a stochastic-stability analysis in repeatedly-played, positive-utility, finite strategic-form games. Prior work in this class of…