A Riemannian Trust Region Method for the Canonical Tensor Rank Approximation Problem
针对规范张量秩逼近这一非凸非光滑优化问题，提出一种基于Segre流形参数化的黎曼高斯-牛顿信赖域方法，并引入基于截断高阶奇异值分解的回缩算子和热重启机制，数值实验表明求解成功时间比现有方法快三个数量级。
The canonical tensor rank approximation problem (TAP) consists of approximating a real-valued tensor by one of low canonical rank, which is a challenging nonlinear, nonconvex, constrained optimization problem, where the constraint set forms a nonsmooth semialgebraic set. We introduce a Riemannian Gauss--Newton method with trust region for solving small-scale, dense TAPs. The novelty of our approach is threefold. First, we parametrize the constraint set as the Cartesian product of Segre manifolds, hereby formulating the TAP as a Riemannian optimization problem, and we argue why this parametrization is theoretically a good choice. Second, an original sequentially truncated higher-order singular value decomposition (ST-HOSVD) based retraction operator is proposed. Third, we introduce a hot restart mechanism that efficiently detects when the optimization process is tending to an ill-conditioned tensor rank decomposition and that often yields a quick escape path from such spurious decompositions. Numerical experiments show improvements of up to three orders of magnitude in terms of the expected time to compute a successful solution over existing state-of-the-art methods.