We develop new tools to study landscapes in nonconvex optimization. Given one optimization problem, we pair it with another by smoothly parametrizing the domain. This is either for practical purposes (e.g., to use smooth optimization algorithms with good guarantees) or for theoretical purposes (e.g., to reveal that the landscape satisfies a strict saddle property). In both cases, the central question is: how do the landscapes of the two problems relate? More precisely: how do desirable points such as local minima and critical points in one problem relate to those in the other problem? A key finding in this paper is that these relations are often determined by the parametrization itself, and are almost entirely independent of the cost function. Accordingly, we introduce a general framework to study parametrizations by their effect on landscapes. The framework enables us to obtain new guarantees for an array of problems, some of which were previously treated on a case-by-case basis in the literature. Applications include: optimizing low-rank matrices and tensors through factorizations; solving semidefinite programs via the Burer-Monteiro approach; training neural networks by optimizing their weights and biases; and quotienting out symmetries.

翻译：我们开发了新工具来研究非凸优化中的景观。给定一个优化问题，我们通过平滑参数化其定义域将其与另一个问题配对。这种做法或出于实际目的（例如，使用具有良好保证的平滑优化算法），或出于理论目的（例如，揭示景观满足严格鞍点性质）。在这两种情况下，核心问题是：两个问题的景观如何关联？更精确地说：一个问题中的局部极小点和临界点等理想点如何与另一个问题中的相应点相关联？本文的一个关键发现是，这些关系通常由参数化本身决定，且几乎完全独立于代价函数。据此，我们引入了一个通用框架，通过参数化对景观的影响来研究参数化。该框架使我们能够为一系列问题获得新保证，其中部分问题在文献中此前仅以个案方式处理。应用包括：通过分解优化低秩矩阵和张量；基于Burer-Monteiro方法求解半定规划；通过优化权重和偏置训练神经网络；以及商化对称性。