We introduce the Subspace Power Method (SPM) for calculating the CP decomposition of low-rank real symmetric tensors. This algorithm calculates one new CP component at a time, alternating between applying the shifted symmetric higher-order power method (SS-HOPM) to a certain modified tensor, constructed from a matrix flattening of the original tensor; and using appropriate deflation steps. We obtain rigorous guarantees for SPM regarding convergence and global optima for input tensors of dimension $d$ and order $m$ of rank up to $O(d^{\lfloor m/2\rfloor})$, via results in classical algebraic geometry and optimization theory. As a by-product of our analysis we prove that SS-HOPM converges unconditionally, settling a conjecture of Kolda-Mayo. Numerical experiments demonstrate that SPM is roughly one order of magnitude faster than state-of-the-art CP decomposition algorithms at moderate ranks. Furthermore, prior knowledge of the CP rank is not required by SPM.

翻译：本文提出子空间幂法（SPM），用于计算低秩实对称张量的CP分解。该算法每次计算一个新的CP分量，交替执行两个步骤：首先对原始张量的矩阵展开构造特定修正张量，应用平移对称高阶幂法（SS-HOPM）；随后执行适当的收缩操作。通过经典代数几何与优化理论的结果，我们为SPM建立了严格的收敛性保证与全局最优性证明，适用于维度为$d$、阶数为$m$、秩不超过$O(d^{\lfloor m/2\rfloor})$的输入张量。作为分析的副产品，我们证明了SS-HOPM具有无条件收敛性，解决了Kolda-Mayo猜想。数值实验表明，在中等秩条件下，SPM的计算速度比当前最先进的CP分解算法快约一个数量级。此外，SPM无需预先知道CP秩。