Descripción del proyecto
Los nuevos enfoques de los análisis de probabilidad son una apuesta segura
La probabilidad es un área importante de las matemáticas que ayuda a analizar eventos fortuitos (fenómenos aleatorios) de forma lógica, lo que permite predecir la probabilidad de que se produzca un evento. Los conteos y muestreos son importantes para crear las distribuciones de probabilidad que informan sobre la verosimilitud de que se produzcan los eventos. Un ejemplo es la campana de Gauss o distribución normal: hace buenas predicciones de muchos fenómenos naturales, incluidos los resultados de un lanzamiento de dos dados o de una moneda, así como las alturas y los pesos al nacer de la población general. El proyecto NACS, financiado con fondos europeos, se ha propuesto revisar y actualizar los enfoques clásicos relativos a los conteos y los muestreos basándose en los últimos avances. Se espera que los resultados tengan un impacto de gran calado en los modelos probabilísticos de diferentes ámbitos, desde la teoría de la probabilidad y la computación cuántica hasta el aprendizaje automático.
Objetivo
Probabilistic models have been adopted in almost all scientific disciplines. Many related computational problems arise, involving estimating the probability of certain events, or drawing samples from a desired distribution. These problems, technically known as counting and sampling problems, are the main focus of the NACS project, with an emphasis on rigorous mathematical analysis.
Markov chain Monte Carlo algorithms have been designed and applied for counting and sampling long before the computational complexity theory, and they are among the oldest randomised algorithms. There had been dramatic progress in the rigorous analysis of Markov chains in the golden era of early 90s, which has lead to many landmark results. Nevertheless, after about 30 years' development, the complexity of many fundamental problems remains open. In the last few years, a number of exciting approaches have emerged, resulting from new perspectives and surprising connections. Consequently, a lot of old challenges start to crumble. It is now the right time to revisit some of the oldest problems in counting complexity.
The emerging techniques include brand new sampling frameworks, such as partial rejection sampling, as well as new ways to analyse traditional algorithms, especially Markov chain algorithms. They can be classified under two themes: those connected with the Lovasz local lemma and those with the geometry of polynomials. The goal of this project is to unleash the full power of the new approaches, to establish novel algorithmic paradigms, and to attack major open problems, such as sampling perfect matchings in general graphs. Due to the fundamental nature of counting and sampling problems, success of the project will also benefit a number of related areas, ranging from combinatorial optimisation, machine learning, and randomised algorithms, to combinatorics, probability theory, statistical physics, and quantum computation.
Ámbito científico
- natural sciencesmathematicspure mathematicsmathematical analysis
- natural sciencesmathematicspure mathematicsgeometry
- natural sciencesmathematicspure mathematicsdiscrete mathematicscombinatorics
- natural sciencesmathematicsapplied mathematicsstatistics and probability
- natural sciencescomputer and information sciencesartificial intelligencemachine learning
Programa(s)
Régimen de financiación
ERC-STG - Starting GrantInstitución de acogida
EH8 9YL Edinburgh
Reino Unido