Probability Theory seminars

Daniel Szyld extends randomized point and block Gauss-Seidel and Gauss-Southwell convergence results from Hermitian positive-definite matrices to certain non-Hermitian classes, including overlapping variables in domain decomposition. The analysis treats a range of sampling probabilities and greedy selection strategies and identifies choices that optimize the bounds. The best expected convergence bounds for randomized methods match those of more expensive deterministic Gauss-Southwell algorithms. These results establish a convergence rate for asynchronous iterations. Joint work with Andreas Frommer.

Optimal transport is a mathematical method to define a distance between probability distributions. This is particularly useful in various domains, including physics, biology, machine learning, and economics, among others. After introducing the Optimal Transport (OT) problem at finite temperature, we show how it can be formulated as a statistical physics problem. This approach allows us to derive very efficient algorithms to effectively compute the distance between two probability distributions. The a priori unrelated Schrödinger bridge (SB) problem is presented, and it is shown to be a dynamical version of the optimal transport problem. Indeed, the Schrodinger bridge looks for the most probable path in probability distribution space, which connects two given probabilities. The Schrodinger bridge problem, originally devised for freely diffusing particles, can be generalized to the case of interacting particles. It can be formulated in terms of functional integrals over bosonic fields, which allows us to derive partial differential equations that characterize the most probable paths in probability space. CARL VAN VREESWIJK MEMORIAL LECTURE 2026. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2026-04-15. Recording duration: 00:55:05.

David Woodruff develops sampling bounds for estimating the spectrum of symmetric matrices with bounded entries. Principal-submatrix sampling achieves epsilon times n additive error using roughly 1/epsilon² samples, eliminating dependence on n and improving prior epsilon dependence up to logarithmic factors. Squared row-norm sampling gives epsilon times the Frobenius norm accuracy with roughly 1/epsilon² samples, improving a previous 1/epsilon⁸ bound. For bounded-entry positive-semidefinite matrices, O(1/epsilon) sampled columns permit nonadaptive approximation of the leading eigenvector with epsilon times n additive error. Applications include faster dense-matrix spectral sketches and improved sample complexity. Joint work with William Swartworth.

Joel Tropp studies structured dimension reduction through the injectivity of random maps, motivated by fast low-rank approximation and least-squares regression. This viewpoint sharpens guarantees for sparse maps and gives exponential improvements for tensor-product dimension reduction. Experiments assess the resulting structured random matrices on synthetic problems and scientific applications. Joint work with Chris Camaño, Ethan Epperly, and Raphael Meyer, available as arXiv:2508.21189.

Charles Bordenave revisits von Neumann and Goldstine's proposal to model numerical computing errors as probabilistic noise. This perspective helps explain large errors in computed eigenvalues of badly conditioned matrices. Toeplitz matrices provide a case study: they occur throughout mathematics, physics, and engineering, are highly sensitive to small perturbations, and possess a rich spectral theory that supports precise descriptions under random perturbations. The talk is based on joint work with Mireille Capitaine and François Chapon.

Doron Puder studies probability distributions on finite groups obtained by evaluating a free-group word on uniformly random group elements. Earlier work connected such distributions on symmetric groups with the partially ordered set of finitely generated free-group subgroups. The talk introduces free groups and free-group algebras, then develops new connections between word distributions on matrix groups over finite fields and free-group algebras. It compares these findings with the symmetric-group case and presents related conjectures. Joint work with Danielle Ernst-West and Matan Seidel.

Nassim Nicholas Taleb examines three technical problems in applied probability. First, he considers how errors in probabilities derived from fat-tailed survival functions can translate into disproportionately large, potentially unbounded errors in thresholds. He relates this to uncertainty in disease growth rates and forecasting. Second, he discusses what he regards as fundamental probability errors in psychology papers. Third, he examines how correlation and relative distances can mislead in the geometry of information, proposing heuristics for applying entropy-based methods to genetic distances. The lecture connects these examples through the consequences of using probabilistic quantities outside the conditions under which they are informative.

Visual perceptual learning (VPL) is defined as improved performance on a visual task due to visual experience. It was once argued that attention to a visual feature is necessary for VPL of the feature to occur. Contrary to this view, a phenomenon called task-irrelevant VPL demonstrated that VPL can occur due to exposure to a feature which is sub-threshold and task-irrelevant, and therefore, unattended. A series of findings based on task-irrelevant VPL has indicated the following two mechanisms. First, attention to a feature facilitates VPL of the feature while inhibiting VPL of unattended and supra-threshold features. Second, reward paired with a feature enables VPL of the feature irrespective of whether the feature is attended or not. However, we recently found an additional twist; VPL of a task-irrelevant and supra-threshold feature embedded in a natural scene is not subject to the inhibition of attention. This new finding suggests a need to revise the current view or add a new mechanism as to how VPL occurs.

Nassim Nicholas Taleb examines distributions whose behaviour is dominated by extremes and tail events. He classifies these distributions and identifies circumstances in which familiar statistical tools become unreliable, including slow or problematic convergence of sample averages under the law of large numbers. The lecture questions the robustness of commonly used statistical procedures in fat-tailed settings, the reliability of frequency-based forecasting, and the use of past averages as guides to future outcomes. Taleb then develops implications for decision-making and the changes in method needed when extreme observations have disproportionate influence.

Nassim Nicholas Taleb develops two approaches to reasoning about extreme risk. In the first part, on the law of large numbers in the real world, he defines fat-tailed distributions and examines why conventional statistical procedures can perform poorly for economic variables. He stresses the much larger data requirements and the choice of estimators when rare observations dominate outcomes. The second part considers how to detect fragility in portfolios. Taleb describes fragility as sensitivity to volatility and argues for analysing the shape of an exposure and its response to shocks when reliable numerical estimates of tail probabilities are unavailable. The lecture connects limitations of statistical estimation with practical heuristics for identifying vulnerability.