Project description
Navigating curved mathematical spaces
Most optimisation tasks occur in flat spaces. However, challenging problems in quantum computing and machine learning arise in curved spaces. Symmetry defines the structure of these worlds. Standard methods are not adapted to these structures. The ERC-funded SYMOPTIC project aims to develop a system called noncommutative group optimisation. This method extends standard optimisation techniques to spaces with complex symmetries. The project will apply group-theoretic mathematics, in which the order of operations affects the outcome. It will develop fast algorithms for statistical and quantum data. By establishing the foundations for this new method, it helps scientists to apply these tools to classical and quantum problems and advance how researchers explore the limits of modern computing.
Objective
Noncommutative group optimization is a powerful emerging paradigm, which has already led to the solution of outstanding problems in computational complexity, algebra, and statistics. Pioneered by the PI and collaborators, it generalizes convex optimization from Euclidean space to the far more general setting of curved spaces with symmetries. Its unfamiliar kind of convexity has recently received much attention in statistics and machine learning. The symmetries are realized by noncommutative groups and imply a high degree of algebraic structure. This combination of symmetry and optimization promises to be key to fast algorithms and deep structural insight. Noncommutative group optimization connects important problems across a wide range of disciplines that appear unrelated at first glance: program testing and derandomization in computer science, estimation problems in statistics, isomorphism problems in algebra, the P vs NP problem and circuit lower bounds in complexity theory, optimal transport in machine learning, marginal and entanglement problems in quantum information, and optimization on quantum computers. This list contains both discrete and continuous problems, theoretical and applied ones, for classical as well as for quantum computers. They have been studied separately over many years by many authors. Here they are brought together in a new innovative way.
This project aims to develop the theoretical and algorithmic foundations of noncommutative group optimization and apply it to longstanding theoretical problems and practical applications. This has high potential for long-lasting impact at several frontiers of computation: in addition to contributing a new paradigm and widely-applicable methods to optimization, we aim to give efficient algorithmic solutions to difficult problems in algebra, make progress on the limits of efficient computation, and unlock the potential of quantum computers for optimization.
Fields of science (EuroSciVoc)
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Keywords
Project’s keywords as indicated by the project coordinator. Not to be confused with the EuroSciVoc taxonomy (Fields of science)
Project’s keywords as indicated by the project coordinator. Not to be confused with the EuroSciVoc taxonomy (Fields of science)
- Scaling algorithms
- Operator scaling
- Convex optimization
- Geodesic convexity
- Non-commutative groups
- Polynomial identity testing
- Invariant theory
- Orbit problems
- Moment polytopes
- Non-commutative algebra
- Algebraic complexity theory
- Geometric complexity theory
- Quantum convex optimization
- Quantum marginal problem
- Quantum entanglement
- Quantum information
Programme(s)
Multi-annual funding programmes that define the EU’s priorities for research and innovation.
Multi-annual funding programmes that define the EU’s priorities for research and innovation.
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HORIZON.1.1 - European Research Council (ERC)
MAIN PROGRAMME
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Topic(s)
Calls for proposals are divided into topics. A topic defines a specific subject or area for which applicants can submit proposals. The description of a topic comprises its specific scope and the expected impact of the funded project.
Calls for proposals are divided into topics. A topic defines a specific subject or area for which applicants can submit proposals. The description of a topic comprises its specific scope and the expected impact of the funded project.
Funding Scheme
Funding scheme (or “Type of Action”) inside a programme with common features. It specifies: the scope of what is funded; the reimbursement rate; specific evaluation criteria to qualify for funding; and the use of simplified forms of costs like lump sums.
Funding scheme (or “Type of Action”) inside a programme with common features. It specifies: the scope of what is funded; the reimbursement rate; specific evaluation criteria to qualify for funding; and the use of simplified forms of costs like lump sums.
HORIZON-ERC - HORIZON ERC Grants
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Call for proposal
Procedure for inviting applicants to submit project proposals, with the aim of receiving EU funding.
Procedure for inviting applicants to submit project proposals, with the aim of receiving EU funding.
(opens in new window) ERC-2021-STG
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Net EU financial contribution. The sum of money that the participant receives, deducted by the EU contribution to its linked third party. It considers the distribution of the EU financial contribution between direct beneficiaries of the project and other types of participants, like third-party participants.
80539 Planegg
Germany
The total costs incurred by this organisation to participate in the project, including direct and indirect costs. This amount is a subset of the overall project budget.