We study a class of optimization problems including matrix scaling, matrix balancing, multidimensional array scaling, operator scaling, and tensor scaling that arise frequently in theory and in practice. Some of these problems, such as matrix and array scaling, are convex in the Euclidean sense, but others such as operator scaling and tensor scaling are geodesically convex on a different Riemannian manifold. Trust region methods, which include box-constrained Newton's method, are known to produce high precision solutions very quickly for matrix scaling and matrix balancing (Cohen et. al., FOCS 2017, Allen-Zhu et. al. FOCS 2017), and result in polynomial time algorithms for some geodesically convex problems like operator scaling (Garg et. al. STOC 2018, B\"urgisser et. al. FOCS 2019). One is led to ask whether these guarantees also hold for multidimensional array scaling and tensor scaling. We show that this is not the case by exhibiting instances with exponential diameter bound: we construct polynomial-size instances of 3-dimensional array scaling and 3-tensor scaling whose approximate solutions all have doubly exponential condition number. Moreover, we study convex-geometric notions of complexity known as margin and gap, which are used to bound the running times of all existing optimization algorithms for such problems. We show that margin and gap are exponentially small for several problems including array scaling, tensor scaling and polynomial scaling. Our results suggest that it is impossible to prove polynomial running time bounds for tensor scaling based on diameter bounds alone. Therefore, our work motivates the search for analogues of more sophisticated algorithms, such as interior point methods, for geodesically convex optimization that do not rely on polynomial diameter bounds.

翻译：我们研究一类优化问题，包括矩阵缩放、矩阵平衡、多维数组缩放、算子缩放和张量缩放，这些问题在理论和实践中频繁出现。其中一些问题（如矩阵和数组缩放）在欧几里得意义上是凸的，但其他问题（如算子缩放和张量缩放）则在不同的黎曼流形上具有测地凸性。已知信赖域方法（包括盒约束牛顿法）能够为矩阵缩放和矩阵平衡问题快速生成高精度解（Cohen等人，FOCS 2017；Allen-Zhu等人，FOCS 2017），并为某些测地凸问题（如算子缩放）产生多项式时间算法（Garg等人，STOC 2018；Bürgisser等人，FOCS 2019）。这自然引出一个问题：这些保证是否同样适用于多维数组缩放和张量缩放？我们通过展示具有指数直径界的实例证明情况并非如此：我们构造了三维数组缩放和三阶张量缩放的、多项式规模的实例，其所有近似解均具有双指数条件数。此外，我们研究了被称为裕度（margin）和间隙（gap）的凸几何复杂度概念，这些概念被用于界定所有现有优化算法在此类问题上的运行时间。我们证明对于数组缩放、张量缩放和多项式缩放等多个问题，裕度和间隙均呈指数级小。我们的结果表明，仅基于直径界为张量缩放证明多项式运行时间界限是不可能的。因此，我们的工作推动了对更复杂算法（如内点法）的类比研究，以用于不依赖于多项式直径界的测地凸优化。