Project description
Novel approaches for probability analyses are a sure thing
Probability is an important area of mathematics that helps us analyse chance events (random phenomena) in a logical way, enabling us to predict the likelihood that an event will occur. Counting and sampling are both important to creating the probability distributions that provide insight into the likelihood of events happening. The classic bell curve or normal distribution is one example; it approximates many natural phenomena well, including the outcomes of rolling two dice and tossing a coin as well as heights and birth weights in the general population. The EU-funded NACS project has set out to revisit and update classic approaches related to counting and sampling based on the latest emerging developments. Outcomes are expected to have far-reaching impacts on probabilistic models in fields ranging from probability theory and quantum computation to machine learning.
Objective
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.
Fields of science
- natural sciencesmathematicspure mathematicsmathematical analysis
- natural sciencesmathematicspure mathematicsgeometry
- natural sciencesmathematicspure mathematicsdiscrete mathematicscombinatorics
- natural sciencesmathematicsapplied mathematicsstatistics and probability
- natural sciencescomputer and information sciencesartificial intelligencemachine learning
Programme(s)
Topic(s)
Funding Scheme
ERC-STG - Starting GrantHost institution
EH8 9YL Edinburgh
United Kingdom