AI 中文总结
本文提出Hadamard镜像下降框架，在矩多胞形上高效优化凸函数，首次实现量子泛函等张量参数的一阶高效计算。
AI 中文摘要
多胞形上的凸优化出现在科学的许多领域。当多胞形是隐式给出或具有指数多个顶点和面时，标准方法可能不适用或无效。矩多胞形（如纠缠多胞形）就是这种情况，它们在量子信息和代数复杂性中起着基础性作用。它们引出了重要的纠缠度量和张量参数，如量子泛函，然而计算这些量的通用有效方法一直难以捉摸。在本文中，我们解决了这一挑战。我们开发了一个称为Hadamard镜像下降的一阶框架，用于在矩多胞形上优化合适的凸函数，更一般地，在测地凸函数的梯度集上优化。它在局部运行，不依赖于多胞形的任何显式描述。我们的框架将镜像下降（一种有效且广泛使用的凸优化框架）从欧几里得设置扩展到Hadamard流形，并受到Hirai最近工作的启发，我们将其解释为镜像流的Hadamard版本。将该框架应用于纠缠多胞形，产生了第一个高效的一阶算法来计算量子泛函、对称量子泛函和G-稳定秩，以及一种用于非交换秩的新直接算法。
英文摘要
Convex optimization on polytopes arises in many areas of science. When the polytope is given implicitly or has exponentially many vertices and facets, standard methods may not apply or be ineffective. This is the case for moment polytopes, such as the entanglement polytopes, which play a foundational role in quantum information and algebraic complexity. They give rise to important entanglement measures and tensor parameters such as the quantum functionals, yet general effective methods for computing these quantities have been elusive. In this paper we address this challenge. We develop a first-order framework called Hadamard mirror descent to optimize suitable convex functions over moment polytopes and, more generally, the gradient sets of geodesically convex functions. It operates locally and does not rely on any explicit description of the polytope. Our framework extends mirror descent, an effective and widely used framework for convex optimization, from the Euclidean setting to Hadamard manifolds, and is motivated by a recent work by Hirai, which we interpret as a Hadamard version of mirror flow. Applying the framework to entanglement polytopes yields the first efficient first-order algorithms to compute the quantum functionals, the symmetric quantum functional, and the G-stable ranks, as well as a new direct algorithm for the non-commutative rank.
Comments81 pages
