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To enable the study of open sets in computational approaches to mathematics, lots of extra data and structure on these sets is assumed. For both foundational and mathematical reasons, it is then a natural question, and the subject of this…
This article is meant to provide an additional point of view, applying known knowledge, to supply keys that have a series of non-repeating digits, in a manner that is not usually thought of. Traditionally, prime numbers are used in…
Differentially private (DP) tabular data synthesis generates artificial data that preserves the statistical properties of private data while safeguarding individual privacy. The emergence of diverse algorithms in recent years has introduced…
We build a longitudinally smooth differentiable groupoid associated to any manifold with corners. The pseudodifferential calculus on this groupoid coincides with the pseudodifferential calculus of Melrose (also called b-calculus). We also…
Bayesian statistical inverse problems are often solved with Markov chain Monte Carlo (MCMC)-type schemes. When the problems are governed by large-scale discrete nonlinear partial differential equations (PDEs), they are computationally…
Differentially private weighted prefix sum under continual observation is a crucial component in the production-level deployment of private next-word prediction for Gboard, which, according to Google, has over a billion users. More…
A formalism of arithmetic partial differential equations (PDEs) is being developed in which one considers several arithmetic differentiations at one fixed prime. In this theory solutions can be defined in algebraically closed p-adic fields.…
We view the penalty algorithm of Ceperley and Dewing (1999), a Markov chain Monte Carlo (MCMC) algorithm for Bayesian inference, in the context of data privacy. Specifically, we study differential privacy of the penalty algorithm and…
In this paper, we find an elementary approach for double sums where the inner sum is binomial but incomplete. We apply our core identity and its relatives to double sums involving famous numbers such as harmonic numbers, Fibonacci numbers,…
The Spencer operator, introduced by D.C. Spencer fifty years ago, is rarely used in mathematics today and, up to our knowledge, has never been used in engineering applications or mathematical physics. The main purpose of this paper, an…
Cross-attention has emerged as a cornerstone module in modern artificial intelligence, underpinning critical applications such as retrieval-augmented generation (RAG), system prompting, and guided stable diffusion. However, this is a rising…
The aim of this paper is to define and study $\mathcal{B}$-open sets and related properties. A $\mathcal{B}$-open set is, roughly speaking, a generalization of a $b$-open set, which is in turn a generalization of a pre-open set and a…
Training machine learning models with differential privacy (DP) limits an adversary's ability to infer sensitive information about the training data. It can be interpreted as a bound on adversary's capability to distinguish two adjacent…
Deep ensemble learning has been shown to improve accuracy by training multiple neural networks and averaging their outputs. Ensemble learning has also been suggested to defend against membership inference attacks that undermine privacy. In…
This chapter surveys advances in the field of Bayesian computation over the past twenty years, with missing data. It also contains some novel computational entries on the double-exponential model that may be of interest per se.
1. Understanding the mechanisms underlying biological systems, and ultimately, predicting their behaviours in a changing environment requires overcoming the gap between mathematical models and experimental or observational data.…
Private computation, which includes techniques like multi-party computation and private query execution, holds great promise for enabling organizations to analyze data they and their partners hold while maintaining data subjects' privacy.…
The mathematical software \texttt{GAP} (Groups, Algorithms, Programming) offers a powerful set of tools to investigate computationally group theory. Using this software package we investigate a variation of a well-known problem in…
Bayesian inference is an important technique throughout statistics. The essence of Beyesian inference is to derive the posterior belief updated from prior belief by the learned information, which is a set of differentially private answers…
Computability on uncountable sets has no standard formalization, unlike that on countable sets, which is given by Turing machines. Some of the approaches to define computability in these sets rely on order-theoretic structures to translate…