arXiv subjects

Alan Legg

Publications and source records attributed to Alan Legg.

3 recordsLinked to original sources

arXiv2026

On Constrained Riesz Minimum Energy Problems

The constrained equilibrium problem for the logarithmic potential was introduced by Rakhmanov (1996) as he realized that the asymptotic distribution of the zeros of discrete orthogonal polynomials in a compact interval could be described in terms of the equilibrium measure of this interval in a class of measures subject to a certain constraint. Other authors such as Dragnev and Saff (1997), and Kuijlaars and Van Assche (1999), extended this approach to more general settings. These problems have proven to be useful for describing asymptotic distributions in different settings. In the current paper, we consider constrained equilibrium problems in the Riesz setting, that is, for $s$-Riesz potentials in the hyperplane $\mathbb{R}^d,$ with $d\geq 1$ and $\max (0,d-2) < s < d$. Along with some general results, an illustrative example consisting of the solution of a constrained equilibrium problem in the unit ball is studied in detail. This is the main part of the paper. Finally, a number of numerical experiments show how the constrained equilibrium measure, the solution of the problem, may be discretized using the so-called 'constrained Leja points' introduced by Coroian and Dragnev (2001).

math.CA↗

arXiv2025

Overspecified Mixture Discriminant Analysis: Exponential Convergence, Statistical Guarantees, and Remote Sensing Applications

This study explores the classification error of Mixture Discriminant Analysis (MDA) in scenarios where the number of mixture components exceeds those present in the actual data distribution, a condition known as overspecification. We use a two-component Gaussian mixture model within each class to fit data generated from a single Gaussian, analyzing both the algorithmic convergence of the Expectation-Maximization (EM) algorithm and the statistical classification error. We demonstrate that, with suitable initialization, the EM algorithm converges exponentially fast to the Bayes risk at the population level. Further, we extend our results to finite samples, showing that the classification error converges to Bayes risk with a rate $n^{-1/2}$ under mild conditions on the initial parameter estimates and sample size. This work provides a rigorous theoretical framework for understanding the performance of overspecified MDA, which is often used empirically in complex data settings, such as image and text classification. To validate our theory, we conduct experiments on remote sensing datasets.

stat.ML↗

arXiv2025

Learning Overspecified Gaussian Mixtures Exponentially Fast with the EM Algorithm

We investigate the convergence properties of the EM algorithm when applied to overspecified Gaussian mixture models -- that is, when the number of components in the fitted model exceeds that of the true underlying distribution. Focusing on a structured configuration where the component means are positioned at the vertices of a regular simplex and the mixture weights satisfy a non-degeneracy condition, we demonstrate that the population EM algorithm converges exponentially fast in terms of the Kullback-Leibler (KL) distance. Our analysis leverages the strong convexity of the negative log-likelihood function in a neighborhood around the optimum and utilizes the Polyak-{\L}ojasiewicz inequality to establish that an $\epsilon$-accurate approximation is achievable in $O(\log(1/\epsilon))$ iterations. Furthermore, we extend these results to a finite-sample setting by deriving explicit statistical convergence guarantees. Numerical experiments on synthetic datasets corroborate our theoretical findings, highlighting the dramatic acceleration in convergence compared to conventional sublinear rates. This work not only deepens the understanding of EM's behavior in overspecified settings but also offers practical insights into initialization strategies and model design for high-dimensional clustering and density estimation tasks.

stat.ML↗