SYMOPTIC · Symmetry and Optimization at the Frontiers of Computation

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.