用于对对称导分解和通用五氯苯甲醚的子空间动力法 (Subspace power method for symmetric tensor decomposition and generalized PCA)

from arxiv, 58 pages, 7 figures, 2 tables. v3: merged introduction and literature review; clarified convergence proof; increased the number of trials in the numerical experiments; and other improvements

We introduce the Subspace Power Method (SPM) for calculating the CP decomposition of low-rank even-order real symmetric tensors. This algorithm applies the tensor power method of Kolda-Mayo to a certain modified tensor, constructed from a matrix flattening of the original tensor, and then uses deflation steps. Numerical simulations indicate SPM is roughly one order of magnitude faster than state-of-the-art algorithms, while performing robustly for low-rank tensors subjected to additive noise. We obtain rigorous guarantees for SPM regarding convergence and global optima, for tensors of rank up to roughly the square root of the number of tensor entries, by drawing on results from classical algebraic geometry and dynamical systems. In a second contribution, we extend SPM to compute De Lathauwer's symmetric block term tensor decompositions. As an application of the latter decomposition, we provide a method-of-moments for generalized principal component analysis.

翻译：我们引入了子空间动力法(SPM)来计算低级平级均分分分分解的低级实对称振压。这一算法将科尔达-梅奥的高压功率法应用到从原高压平坦的矩阵平坦中建造的某种修改的抗拉,然后使用通缩步骤。数字模拟显示SPM比最新算法快于最先进的一个数量级,同时对受添加噪音影响的低级压强者进行强力操作。我们为SPM在趋同和全球opima方面得到了严格的保证,对高压条目数级数级数级数级数级数级数级数级数级数至约平方根的严格保障,我们借鉴了古典代代数几何地测量和动态系统的结果。在第二项贡献中,我们将SPM扩大SPM,将De Lathauwer的对称区区区段术语 " 数控分解 " 。作为后一个分解法的应用,我们为通用主要部件分析提供了一种方法。