arXiv research

Corrects errors in Hans' pseudocovering spaces paper.

Study evaluates different mathematical models for three case studies using statistical fitting.

This paper provides a mathematical foundation for deep neural networks solving PDEs.

This paper provides a mathematical framework for understanding distribution learning models.

Develops statistical guarantees for neural networks with regularization.

Quantum codes linked to abelian varieties, providing mathematical rigor.

Improved approximations for rough Heston model reduce errors.

While computer and communication technologies have provided effective means to scale up many aspects of education, the submission and grading of assessments such as homework assignments and tests remains a weak link. In this paper, we study the problem of automatically grading the kinds of open response mathematical qu…

Recently, artificial neural networks (ANNs) in conjunction with stochastic gradient descent optimization methods have been employed to approximately compute solutions of possibly rather high-dimensional partial differential equations (PDEs). Very recently, there have also been a number of rigorous mathematical results …

Karl Menger's 1934 paper on the St. Petersburg paradox contains mathematical errors that invalidate his conclusion that unbounded utility functions, specifically Bernoulli's logarithmic utility, fail to resolve modified versions of the St. Petersburg paradox.

Analyzes deep neural networks training errors with SGD and random init.

VERAFI improves financial AI by verifying calculations and compliance.

Subset selection for multiple linear regression aims to construct a regression model that minimizes errors by selecting a small number of explanatory variables. Once a model is built, various statistical tests and diagnostics are conducted to validate the model and to determine whether the regression assumptions are me…

Study reconstructs Faber-Schauder coefficients from antiderivative observations.

Autonomy and adaptation of machines requires that they be able to measure their own errors. We consider the advantages and limitations of such an approach when a machine has to measure the error in a regression task. How can a machine measure the error of regression sub-components when it does not have the ground truth…

The theory of quantum computation can be constructed from the abstract study of anyonic systems. In mathematical terms, these are unitary topological modular functors. They underlie the Jones polynomial and arise in Witten-Chern-Simons theory. The braiding and fusion of anyonic excitations in quantum Hall electron liqu…

A mathematical framework connects neural networks and polynomial regression for better model understanding.

First, we analyze the variance of the Cross Validation (CV)-based estimators used for estimating the performance of classification rules. Second, we propose a novel estimator to estimate this variance using the Influence Function (IF) approach that had been used previously very successfully to estimate the variance of …

Paper bounds tensor decomposition's RLCT, aiding Bayesian inference.

Mathematical framework using Riemannian geometry for intelligence and consciousness.

We provide a mathematical definition of fragility and antifragility as negative or positive sensitivity to a semi-measure of dispersion and volatility (a variant of negative or positive "vega") and examine the link to nonlinear effects. We integrate model error (and biases) into the fragile or antifragile context. Unli…

LLMs will inevitably hallucinate due to their mathematical structure.

Survey of mathematical foundations for reinforcement learning.

Study improves understanding of why agentic theorem provers succeed.

In this paper we outline methodology to efficiently simulate (jump) diffusion bridge sample paths without discretisation error. We achieve this by considering the simulation of conditioned (jump) diffusion bridge sample paths in light of recent work developing a mathematical framework for simulating finite dimensional …

Subset selection in multiple linear regression aims to choose a subset of candidate explanatory variables that tradeoff fitting error (explanatory power) and model complexity (number of variables selected). We build mathematical programming models for regression subset selection based on mean square and absolute errors…

This paper analyzes hedge errors in Black-Scholes models using finite difference techniques.

In this paper, we introduce a new concept of stability for cross-validation, called the $\left( β, \varpi \right)$-stability, and use it as a new perspective to build the general theory for cross-validation. The $\left( β, \varpi \right)$-stability mathematically connects the generalization ability and the stability of…

This lecture presents recent advances in the theory of errors propagation. We first explain in which cases the propagation of errors may be performed with a first order differential calculus or needs a second order differential calculus. Then we point out the link between error propagation and the concept of second ord…

Language models help text classification tasks by predicting next words.

Efficient kernel method learns differential equations with fewer data.

Book introduces deep learning methods with math, theory, and applications.

This article analyzes the weak error of SGD optimization schemes.

This paper analyzes how errors accumulate in PCA's deflation method.

Optimizes group testing for COVID-19 to reduce test numbers.

Paper analyzes error exponent in agnostic PAC learning.

Learning reward functions can lead to poor policy performance despite low error.

Bounds on factual and counterfactual distributions under measurement error in discrete models.

Randomly sampled interpolators achieve zero generalization error with enough data.

The paper analyzes error propagation in dynamic programming for stochastic control and option pricing.

Paper introduces ILD algorithm to determine Bayes error for binary classification.

A spiral unibike track emerges from a mathematical construction.

Study on double descent behavior in two-layer neural networks for binary classification.

This work bounds the generalization error of private algorithms for discrete data.

This note is concerned with accurate and computationally efficient approximations of moments of Gaussian random variables passed through sigmoid or softmax mappings. These approximations are semi-analytical (i.e. they involve the numerical adjustment of parametric forms) and highly accurate (they yield 5% error at most…

New approach uses interpolation models and error bounds for verifiable scientific machine learning.

This work analyzes discrete diffusion models using stochastic integrals, providing error bounds and insights.

The paper uncovers the mathematical structure enabling value decomposition in multi-agent systems.