Curvature and optimal transport
Lead Research Organisation:
Imperial College London
Department Name: Mathematics
Abstract
We study curvature of coarse objects, such as graphs, and their embeddings in smooth surfaces using the theory of optimal transport.
Curvature in the setting of smooth surfaces has been an extensively studied subject since the times of C. F. Gauss and it is the main object of study of modern Riemannian geometry. On the other hand, one of the first studies of curvature in non-smooth spaces was carried out only towards the end of the 1940's by A. D. Alexandrov. Alexandrov's generalization of curvature to non-smooth spaces relied on the observation that triangles in spaces of positive sectional curvature have larger sum of internal angles than triangles in plane.
The study of coarse curvatures by optimal transport was initiated in the early 2000s by a series of insights about the evolution of the entropy functional along geodesics in Wasserstein spaces. In 2003, J. Lott and C. Villani and independently K.-T. Sturm initiated their work on Ricci curvature by displacement convexity of the entropy functional along geodesics in the 2-Wasserstein space of probability measures on a general metric space equipped with a measure. This notion is independent of any smooth structure on the base space considered and yields lower curvature bounds that are consistent with the smooth case.
One inconvenience of this theory was that its definition of curvature was not applicable to discrete spaces, such as weighted graphs. Hence, two alternative notions of coarse curvature by optimal transport were introduced later. Y. Ollivier proposed a new definition of coarse curvature in 2007, defining curvature between two points as the deviation of optimal transport distance of two balls centred at the two points from the metric distance of the two points. On the other hand, J. Maas introduced in 2012 an adapted definition of Lott and Villani that is valid on discrete spaces and translated existing results to the discrete setting.
Overall, coarse curvature by optimal transport is a young, flourishing field of mathematics with plenitude of open problems while already enjoying a solid mathematical grounding which was mostly established in the last 20 years. We intend to bring contribution to this field by studying relationships between existing notions of curvature and bringing innovation of our own. Moreover, the potential impact outside the immediate scope of the project are novel methods in geometric analysis of graphs and graph embeddings that may have practical applications.
Curvature in the setting of smooth surfaces has been an extensively studied subject since the times of C. F. Gauss and it is the main object of study of modern Riemannian geometry. On the other hand, one of the first studies of curvature in non-smooth spaces was carried out only towards the end of the 1940's by A. D. Alexandrov. Alexandrov's generalization of curvature to non-smooth spaces relied on the observation that triangles in spaces of positive sectional curvature have larger sum of internal angles than triangles in plane.
The study of coarse curvatures by optimal transport was initiated in the early 2000s by a series of insights about the evolution of the entropy functional along geodesics in Wasserstein spaces. In 2003, J. Lott and C. Villani and independently K.-T. Sturm initiated their work on Ricci curvature by displacement convexity of the entropy functional along geodesics in the 2-Wasserstein space of probability measures on a general metric space equipped with a measure. This notion is independent of any smooth structure on the base space considered and yields lower curvature bounds that are consistent with the smooth case.
One inconvenience of this theory was that its definition of curvature was not applicable to discrete spaces, such as weighted graphs. Hence, two alternative notions of coarse curvature by optimal transport were introduced later. Y. Ollivier proposed a new definition of coarse curvature in 2007, defining curvature between two points as the deviation of optimal transport distance of two balls centred at the two points from the metric distance of the two points. On the other hand, J. Maas introduced in 2012 an adapted definition of Lott and Villani that is valid on discrete spaces and translated existing results to the discrete setting.
Overall, coarse curvature by optimal transport is a young, flourishing field of mathematics with plenitude of open problems while already enjoying a solid mathematical grounding which was mostly established in the last 20 years. We intend to bring contribution to this field by studying relationships between existing notions of curvature and bringing innovation of our own. Moreover, the potential impact outside the immediate scope of the project are novel methods in geometric analysis of graphs and graph embeddings that may have practical applications.
Planned Impact
Probabilistic modelling permeates the Financial services, healthcare, technology and other Service industries crucial to the UK's continuing social and economic prosperity, which are major users of stochastic algorithms for data analysis, simulation, systems design and optimisation. There is a major and growing skills shortage of experts in this area, and the success of the UK in addressing this shortage in cross-disciplinary research and industry expertise in computing, analytics and finance will directly impact the international competitiveness of UK companies and the quality of services delivered by government institutions.
By training highly skilled experts equipped to build, analyse and deploy probabilistic models, the CDT in Mathematics of Random Systems will contribute to
- sharpening the UK's research lead in this area and
- meeting the needs of industry across the technology, finance, government and healthcare sectors
MATHEMATICS, THEORETICAL PHYSICS and MATHEMATICAL BIOLOGY
The explosion of novel research areas in stochastic analysis requires the training of young researchers capable of facing the new scientific challenges and maintaining the UK's lead in this area. The partners are at the forefront of many recent developments and ideally positioned to successfully train the next generation of UK scientists for tackling these exciting challenges.
The theory of regularity structures, pioneered by Hairer (Imperial), has generated a ground-breaking approach to singular stochastic partial differential equations (SPDEs) and opened the way to solve longstanding problems in physics of random interface growth and quantum field theory, spearheaded by Hairer's group at Imperial. The theory of rough paths, initiated by TJ Lyons (Oxford), is undergoing a renewal spurred by applications in Data Science and systems control, led by the Oxford group in conjunction with Cass (Imperial). Pathwise methods and infinite dimensional methods in stochastic analysis with applications to robust modelling in finance and control have been developed by both groups.
Applications of probabilistic modelling in population genetics, mathematical ecology and precision healthcare, are active areas in which our groups have recognized expertise.
FINANCIAL SERVICES and GOVERNMENT
The large-scale computerisation of financial markets and retail finance and the advent of massive financial data sets are radically changing the landscape of financial services, requiring new profiles of experts with strong analytical and computing skills as well as familiarity with Big Data analysis and data-driven modelling, not matched by current MSc and PhD programs. Financial regulators (Bank of England, FCA, ECB) are investing in analytics and modelling to face this challenge. We will develop a novel training and research agenda adapted to these needs by leveraging the considerable expertise of our teams in quantitative modelling in finance and our extensive experience in partnerships with the financial institutions and regulators.
DATA SCIENCE:
Probabilistic algorithms, such as Stochastic gradient descent and Monte Carlo Tree Search, underlie the impressive achievements of Deep Learning methods. Stochastic control provides the theoretical framework for understanding and designing Reinforcement Learning algorithms. Deeper understanding of these algorithms can pave the way to designing improved algorithms with higher predictability and 'explainable' results, crucial for applications.
We will train experts who can blend a deeper understanding of algorithms with knowledge of the application at hand to go beyond pure data analysis and develop data-driven models and decision aid tools
There is a high demand for such expertise in technology, healthcare and finance sectors and great enthusiasm from our industry partners. Knowledge transfer will be enhanced through internships, co-funded studentships and paths to entrepreneurs
By training highly skilled experts equipped to build, analyse and deploy probabilistic models, the CDT in Mathematics of Random Systems will contribute to
- sharpening the UK's research lead in this area and
- meeting the needs of industry across the technology, finance, government and healthcare sectors
MATHEMATICS, THEORETICAL PHYSICS and MATHEMATICAL BIOLOGY
The explosion of novel research areas in stochastic analysis requires the training of young researchers capable of facing the new scientific challenges and maintaining the UK's lead in this area. The partners are at the forefront of many recent developments and ideally positioned to successfully train the next generation of UK scientists for tackling these exciting challenges.
The theory of regularity structures, pioneered by Hairer (Imperial), has generated a ground-breaking approach to singular stochastic partial differential equations (SPDEs) and opened the way to solve longstanding problems in physics of random interface growth and quantum field theory, spearheaded by Hairer's group at Imperial. The theory of rough paths, initiated by TJ Lyons (Oxford), is undergoing a renewal spurred by applications in Data Science and systems control, led by the Oxford group in conjunction with Cass (Imperial). Pathwise methods and infinite dimensional methods in stochastic analysis with applications to robust modelling in finance and control have been developed by both groups.
Applications of probabilistic modelling in population genetics, mathematical ecology and precision healthcare, are active areas in which our groups have recognized expertise.
FINANCIAL SERVICES and GOVERNMENT
The large-scale computerisation of financial markets and retail finance and the advent of massive financial data sets are radically changing the landscape of financial services, requiring new profiles of experts with strong analytical and computing skills as well as familiarity with Big Data analysis and data-driven modelling, not matched by current MSc and PhD programs. Financial regulators (Bank of England, FCA, ECB) are investing in analytics and modelling to face this challenge. We will develop a novel training and research agenda adapted to these needs by leveraging the considerable expertise of our teams in quantitative modelling in finance and our extensive experience in partnerships with the financial institutions and regulators.
DATA SCIENCE:
Probabilistic algorithms, such as Stochastic gradient descent and Monte Carlo Tree Search, underlie the impressive achievements of Deep Learning methods. Stochastic control provides the theoretical framework for understanding and designing Reinforcement Learning algorithms. Deeper understanding of these algorithms can pave the way to designing improved algorithms with higher predictability and 'explainable' results, crucial for applications.
We will train experts who can blend a deeper understanding of algorithms with knowledge of the application at hand to go beyond pure data analysis and develop data-driven models and decision aid tools
There is a high demand for such expertise in technology, healthcare and finance sectors and great enthusiasm from our industry partners. Knowledge transfer will be enhanced through internships, co-funded studentships and paths to entrepreneurs
Organisations
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/S023925/1 | 01/04/2019 | 30/09/2027 | |||
| 2215977 | Studentship | EP/S023925/1 | 01/10/2019 | 30/09/2023 | Benedikt Petko |