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Two-dimensional Delaunay triangulation is a fundamental aspect of computational geometry. This paper presents a novel algorithm that is specifically designed to ensure the correctness of 2D Delaunay triangulation, namely the Polygonal…

Modeling data as being sampled from a union of independent subspaces has been widely applied to a number of real world applications. However, dimensionality reduction approaches that theoretically preserve this independence assumption have…

Proportional representation (PR) is often discussed in voting settings as a major desideratum. For the past century or so, it is common both in practice and in the academic literature to jump to single transferable vote (STV) as the…

We study the scaling limits of stochastic gradient descent (SGD) with constant step-size in the high-dimensional regime. We prove limit theorems for the trajectories of summary statistics (i.e., finite-dimensional functions) of SGD as the…

Direct Preference Optimisation (DPO) has emerged as a powerful method for aligning Large Language Models (LLMs) with human preferences, offering a stable and efficient alternative to approaches that use Reinforcement learning via Human…

Datasets such as images, text, or movies are embedded in high-dimensional spaces. However, in important cases such as images of objects, the statistical structure in the data constrains samples to a manifold of dramatically lower…

Classical spatial models of two-party competition typically predict convergence to the median voter, yet real-world party systems often exhibit persistent and asymmetric polarization. We develop a spatial model of two-party competition in…

Over three decades of scientific endeavors to realize programmable matter, a substance that can change its physical properties based on user input or responses to its environment, there have been many advances in both the engineering of…

Selection bias poses a critical challenge for fairness in machine learning, as models trained on data that is less representative of the population might exhibit undesirable behavior for underrepresented profiles. Semi-supervised learning…

Motion planning and control problems are embedded and essential in almost all robotics applications. These problems are often formulated as stochastic optimal control problems and solved using dynamic programming algorithms. Unfortunately,…

Stochastic computing has a long history as an alternative method of performing arithmetic on a computer. While it can be considered an unbiased estimator of real numbers, it has a variance and MSE on the order of $\Omega(\frac{1}{N})$. On…

Voting is a commonly applied method for the aggregation of the preferences of multiple agents into a joint decision. If preferences are binary, i.e., "yes" and "no", every voting system can be described by a (monotone) Boolean function…

The support vector machine (SVM) and minimum Euclidean norm least squares regression are two fundamentally different approaches to fitting linear models, but they have recently been connected in models for very high-dimensional data through…

Dimensionless numbers and scaling laws provide elegant insights into the characteristic properties of physical systems. Classical dimensional analysis and similitude theory fail to identify a set of unique dimensionless numbers for a…

Empirically it has been observed that the performance of deep neural networks steadily improves as we increase model size, contradicting the classical view on overfitting and generalization. Recently, the double descent phenomena has been…

We introduce a minor variant of the approximate D-optimal design of experiments with a more general information matrix that takes into account the representation of the design space S. The main motivation (and result) is that if S in R^d is…

We consider the optimization problem of minimizing an objective functional, which admits a variational form and is defined over probability distributions on the constrained domain, which poses challenges to both theoretical analysis and…

We propose a new sufficient dimension reduction approach designed deliberately for high-dimensional classification. This novel method is named maximal mean variance (MMV), inspired by the mean variance index first proposed by Cui, Li and…

Dimension reduction is often the first step in statistical modeling or prediction of multivariate spatial data. However, most existing dimension reduction techniques do not account for the spatial correlation between observations and do not…

The focus of this paper is on stochastic variational inequalities (VI) under Markovian noise. A prominent application of our algorithmic developments is the stochastic policy evaluation problem in reinforcement learning. Prior…