Descrizione del progetto
I nuovi approcci per le analisi di probabilità sono una certezza
La probabilità è un’area importante della matematica che ci aiuta ad analizzare eventi casuali (fenomeni fortuiti) in modo logico, permettendoci di prevedere la probabilità che un evento si verifichi. Il conteggio e la campionatura sono entrambi importanti per creare le distribuzioni di probabilità che forniscono informazioni sull’eventualità che si verifichi un dato evento. La classica curva a campana o distribuzione normale è un esempio: si avvicina bene a molti fenomeni naturali, inclusi i risultati del lancio di due dadi e del lancio di una moneta, nonché l’altezza e il peso alla nascita nella popolazione generale. Il progetto NACS, finanziato dall’UE, ha deciso di rivedere e aggiornare gli approcci classici relativi al conteggio e alla campionatura basati sugli ultimi sviluppi emergenti. Si prevede che i risultati avranno impatti di vasta portata sui modelli probabilistici in campi che vanno dalla teoria della probabilità e dall’informatica quantistica all’apprendimento automatico.
Obiettivo
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.
Campo scientifico
- natural sciencesmathematicspure mathematicsmathematical analysis
- natural sciencesmathematicspure mathematicsgeometry
- natural sciencesmathematicspure mathematicsdiscrete mathematicscombinatorics
- natural sciencesmathematicsapplied mathematicsstatistics and probability
- natural sciencescomputer and information sciencesartificial intelligencemachine learning
Programma(i)
Argomento(i)
Meccanismo di finanziamento
ERC-STG - Starting GrantIstituzione ospitante
EH8 9YL Edinburgh
Regno Unito