Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
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2063 research outputs found
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Prony’s method: an old trick for new problems
No full textIn 1795, French mathematician Gaspard de Prony invented
an ingenious trick to solve a recovery problem,
aiming at reconstructing functions from their values
at given points, which arose from a specific application
in physical chemistry. His technique became
later useful in many different areas, such as signal
processing, and it relates to the concept of sparsity
that gained a lot of well-deserved attention recently.
Prony’s contribution, therefore, has developed into a
very modern mathematical concept
Matrix Estimation Meets Statistical Network Analysis: Extracting low-dimensional structures in high dimension
No full textThe study of complex relationships among the elements of a large collection of random variables lead to the development of a number of areas in probability and statistics such as probabilistic network analysis or random matrix theory. The aim of the workshop was to address the challenge to develop a coherent mathematical framework within which these areas can be integrated, for a successful analysis of massive and complicated data sets
Classical Algebraic Geometry
No full textProgress in algebraic geometry often comes through the introduction of new tools and ideas to tackle the classical problems in the development of the field. Examples include new invariants that capture some aspect of geometry in a novel way, such as the derived category, and the extension of the class of geometric objects considered to allow constructions not previously possible, such as the transition from varieties to schemes or from schemes to stacks. Many famous old problems and outstanding conjectures have been resolved in this way over the last 50 years. While the new theories are sometimes studied for their own sake, they are in the end best understood in the context of the classical questions they illuminate. The goal of the workshop was to study new developments in algebraic geometry, with a view toward their application to the classical problems
MOSES: A Streaming Algorithm for Linear Dimensionality Reduction
No full textThis paper introduces Memory-limited Online Subspace Estimation Scheme (MOSES) for both estimating the principal components of data and reducing its dimension. More specifically, consider a scenario where the data vectors are presented sequentially to a user who has limited storage and processing time available, for example in the context of sensor networks. In this scenario, MOSES maintains an estimate of leading principal components of the data that has arrived so far and also reduces its dimension. In terms of its origins, MOSES slightly generalises the popular incremental Singular Value Decomposition (SVD) to handle thin blocks of data. This simple generalisation is in part what allows us to complement MOSES with a comprehensive statistical analysis that is not available for incremental SVD, despite its empirical success. This generalisationalso enables us to concretely interpret MOSES as an approximate solver for the underlying non-convex optimisation program. We also find that MOSES shows state-of-the-art performance in our numerical experiments with both synthetic and real-world datasets
Combinatorial Optimization
No full textCombinatorial Optimization is an active research area that developed from the rich interaction among many mathematical areas, including combinatorics, graph theory, geometry, optimization, probability, theoretical computer science, and many others. It combines algorithmic and complexity analysis with a mature mathematical foundation and it yields both basic research and applications in manifold areas such as, for example, communications, economics, traffic, network design, VLSI, scheduling, production, computational biology, to name just a few. Through strong inner ties to other mathematical fields it has been contributing to and benefiting from areas such as, for example, discrete and convex geometry, convex and nonlinear optimization, algebraic and topological methods, geometry of numbers, matroids and combinatorics, and mathematical programming. Moreover, with respect to applications and algorithmic complexity, Combinatorial Optimization is an essential link between mathematics, computer science and modern applications in data science, economics, and industry
Mini-Workshop: Gibbs Measures for Nonlinear Dispersive Equations
No full textIn this mini-workshop we brought together leading experts working on the application of Gibbs measures to the study of nonlinear PDEs. This framework is a powerful tool in the probabilistic study of solutions to nonlinear dispersive PDEs, in many ways alternative or complementary to deterministic methods. Among the special topics discussed were the construction of the measures, applications to dynamics, as well as the microscopic derivation of Gibbs measures from many-body quantum mechanics
Mathematical General Relativity
No full textGeneral Relativity is one of the triumphs of twentieth century physics. Its spectacular predictions include gravitational waves, black holes, and spacetime singularities. The mathematical study of this theory leads to deep problems connecting the areas of partial differential equations, geometry and topology with physics. The talks of the workshop illustrated the rapid progress in this subject over the last few years
Herbrand’s Theorem as Higher Order Recursion
No full textResearch in Pairs 2016We provide a means to compute Herbrand disjunctions directly from sequent calculus proofs with cuts. Our approach associates to a first-order classical proof , where is quantifier free, an acyclic higher order recursion scheme whose language is finite and yields a Herbrand disjunction for . More generally, we show that the language of contains the Herbrand disjunction implicit in any cut-free proof obtained from via a sequence of Gentzen-style cut reductions that always reduce the weak side of a cut before the strong side
Mini-Workshop: Deep Learning and Inverse Problems
No full textMachine learning and in particular deep learning offer several data-driven methods to amend the typical shortcomings of purely analytical approaches. The mathematical research on these combined models is presently exploding on the experimental side but still lacking on the theoretical point of view. This workshop addresses the challenge of developing a solid mathematical theory for analyzing deep neural networks for inverse problems
Statistical Inference for Structured High-dimensional Models
No full textHigh-dimensional statistical inference is a newly emerged direction of statistical science in the 21 century. Its importance is due to the increasing dimensionality and complexity of models needed to process and understand the modern real world data. The main idea making possible meaningful inference about such models is to assume suitable lower dimensional underlying structure or low-dimensional approximations, for which the error can be reasonably controlled. Several types of such structures have been recently introduced including sparse high-dimensional regression, sparse and/or low rank matrix models, matrix completion models, dictionary learning, network models (stochastic block model, mixed membership models) and more. The workshop focused on recent developments in structured sequence and regression models, matrix and tensor estimation, robustness, statistical learning in complex settings, network data, and topic models