CAREER: AF: Fast Algorithms for Riemannian Optimization

黎曼优化是研究黎曼流形上成本函数的最小化，由于其在现代统计学和机器学习中的许多应用而日益突出。这些流行应用的一个小样本包括度量学习、混合模型参数估计、协方差估计和子空间恢复以及矩阵补全。在非统计领域，黎曼优化正成为扩散张量成像的重要工具集，扩散张量成像是一种利用磁共振成像来描绘人类大脑的新技术，也可用于解决支持现实世界物体三维成像的旋转同步问题。该奖项的首要目标是构建以最快的计算速度解决黎曼优化问题的新方法。该奖项的实际收益将包括轻松解决黎曼优化问题的新软件包，向本科生和研究生介绍这一令人兴奋的领域的新教材，以及为研究生在一个主要由代表性不足的少数民族组成的研究小组提供资金。在更细粒度的层面上解释，该奖项旨在构建最优速率方法，其中这些速率在调用神谕中被量化，这些神谕产生微分信息，以最小化黎曼流形上的成本函数。该奖项的每个组成项目都将导致算法的发展，其复杂性与其在欧几里得空间子集上的优化类似，如梯度下降法和牛顿法。为此，该奖项侧重于两大类问题：测地线凸优化问题和测地线非凸优化问题。自然地，测地线凸优化是凸优化对流形设置的推广。受Nesterov关于加速梯度下降的著名工作的启发，该奖项特别关注将Nesterov式动量纳入基于一阶和二阶微分信息的现有黎曼优化方法中。为了使黎曼优化的现实实践受益，该奖项将进一步资助可部署软件包的开发，其中包括在本研究期间构建的最优速率算法。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Riemannian optimization, the study of minimizing a cost function over a Riemannian manifold, is surging in prominence due to its many applications in modern statistics and machine learning. A small sample of these popular applications includes metric learning, mixture model parameter estimation, covariance estimation and subspace recovery, and matrix completion. In the non-statistical realm, Riemannian optimization is becoming an important toolset for diffusion tensor imaging, a novel technology for using magnetic resonance imaging to profile the human brain, as well as for solving synchronization of rotation problems that support 3-D imaging of real-world objects. This award's overarching goal is to construct new methods for solving Riemannian optimization problems with the fastest possible computational speed. Tangible benefits of this award will include new software packages for easily solving Riemannian optimization problems, new educational materials that introduce this exciting field to undergraduate and graduate students, and funding for graduate students in a research group predominantly comprised of underrepresented minorities. Explained on a more granular level, the award aims to construct optimal rate methods, where such rates are quantified in calls to oracles that produce differential information for minimizing a cost function over a Riemannian manifold. Each of the award's constituent projects will lead to the development of algorithms whose complexity matches their analogs for optimization over a subset of a Euclidean space, such as gradient descent and Newton's method. To this end, the award focuses on two broad classes of problems: geodesically convex optimization problems and geodesically non-convex optimization problems. Naturally, geodesically convex optimization is the generalization of convex optimization to the manifold setting. Inspired by Nesterov's famous work on accelerated gradient descent, the award pays particular attention to the incorporation of Nesterov-style momentum in existing Riemannian optimization methods based on first- and second-order differential information. To benefit the real-world practice of Riemannian optimization, the award will further fund the development of deployable software packages that include the optimal rate algorithms built during this research.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.