In this work, we consider the optimization formulation for symmetric tensor decomposition recently introduced in the Subspace Power Method (SPM) of Kileel and Pereira. Unlike popular alternative functionals for tensor decomposition, the SPM objective function has the desirable properties that its maximal value is known in advance, and its global optima are exactly the rank-1 components of the tensor when the input is sufficiently low-rank. We analyze the non-convex optimization landscape associated with the SPM objective. Our analysis accounts for working with noisy tensors. We derive quantitative bounds such that any second-order critical point with SPM objective value exceeding the bound must equal a tensor component in the noiseless case, and must approximate a tensor component in the noisy case. For decomposing tensors of size $D^{\times m}$, we obtain a near-global guarantee up to rank $\widetilde{o}(D^{\lfloor m/2 \rfloor})$ under a random tensor model, and a global guarantee up to rank $\mathcal{O}(D)$ assuming deterministic frame conditions. This implies that SPM with suitable initialization is a provable, efficient, robust algorithm for low-rank symmetric tensor decomposition. We conclude with numerics that show a practical preferability for using the SPM functional over a more established counterpart.
翻译:在这项工作中,我们考虑最近在基尔尔和佩雷拉的子空间动力法(SPM)中引入的对称强分分分解优化配方。 与热分解的流行替代功能不同, SPM目标功能具有其最大值为预知的可取性, 而其全球调制正是输入量足够低时, 其压强的一等成分。 我们分析与SPM目标相关的非康氏优化景观。 我们的分析记录了与吵闹的振动器合作的情况。 我们得出的定量界限是, 任何具有SPM目标值超过约束值的第二等临界点, 必须在无噪音的情况下等于一个发声器组件, 并且必须在吵闹的案例中接近一个发声器组件。 为了分解大小为$D ⁇ times m}的变色器, 我们得到了近全球的保证, 最高为$( lobilte{o} (Düldrop m/2 rloom}) 。 我们的分析记录了在随机高压模式下的工作, 以及一个全球保证最高为 $\mathcal {O} 全球保证在无噪音情况下, 假设一个高效的SMMMMMMFillable 以合适的初始状态下, 这意味着我们可以做出一个更精确的模拟的模拟的模拟的状态。