Applied Harmonic Analysis Methods for Non-Convex Optimizations and Low-Rank Matrix Analysis

该项目通过扩大应用调和分析在数学和应用方面的领域来促进科学的进步。该研究解决了当前超级计算机难以解决的复杂优化问题。这些项目拓宽了数学与工程和计算机科学之间的双向交流，同时促进了教学、培训和学习。PI将通过在行业和政府实验室实习，为具有全球竞争力的STEM力量培养研究生。该项目增加了与工业界的现有合作伙伴关系，同时提供了探索现实世界应用数学的机会，并为现有问题创造了新的解决方案。第一个推力发展同伦方法的非凸优化。特别是，该项目将集中在低秩矩阵估计，如在相位恢复问题的情况下，和二次分配问题，如在图匹配问题。同伦方法通过一个连续参数扩展了原始搜索空间（相空间），该连续参数在目标非凸目标函数和精心选择的凸惩罚项之间进行交换。路径跟踪器在全局优化器处初始化，并逐渐向非凸目标函数的全局最优值演变。该研究将获得最优性的保证，这种方法。第二个推力涉及低秩半正定矩阵的几何和功能分析。该程序研究这些度量空间的各种度量和Lipschitz嵌入。初步结果表明，这个半代数品种的度量的丰富和复杂的集合。特别是，一个这样的措施是有关的最佳扩展到非负秩一矩阵的总和。事实证明，这种分解直接关系到在某些调制空间中具有核的紧致积分算子的分析。该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

This project promotes the progress of science by expanding the area of applied harmonic analysis both in mathematics and in applications. The research addresses complex optimization problems that are intractable by current super-computers. The projects broaden the two-way communication between mathematics on one side and engineering and computer science on the other side while promoting teaching, training, and learning. The PI will train graduate students for a globally competitive STEM force through internships in industry and government labs. The project increases the existing partnerships with industry while offering opportunities to explore mathematics of real-world applications and to create novel solutions to existing problems.The project contains two thrusts. The first thrust develops homotopic methods for non-convex optimizations. In particular, the project will focus on low-rank matrix estimation, such as the case in the phase retrieval problem, and quadratic assignment problems, as in graph matching problems. The homotopic method extends the original search space (the phase space) by one continuous parameter that trades between the target non-convex objective function and a carefully chosen convex penalty term. A path tracker is initialized at the global optimizer and gradually evolved towards the global optimum of the non-convex objective function. The research will obtain guarantees of optimality for this method. The second thrust relates to geometric and functional analysis of low-rank positive semi-definite matrices. The program studies various metrics and Lipschitz embeddings of these metric spaces. Preliminary results show a rich and complex collection of metrics on this semi-algebraic variety. In particular, one such measure is related to optimal expansions into sums of nonnegative rank-one matrices. It turns out this decomposition is directly related to the analysis of compact integral operators with kernels in certain modulation spaces. These results will be extended to finite-dimensional settings.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.