论文标题
部分可观测时空混沌系统的无模型预测
Estimating Higher-Order Mixed Memberships via the $\ell_{2,\infty}$ Tensor Perturbation Bound
论文作者
论文摘要
高阶多向数据在机器学习和统计数据中无处不在，并且经常表现出类似社区的结构，其中每个不同模式的每个组件（节点）都具有与之相关的社区成员资格。在本文中，我们提出了张量混合会员块模型，对张量块模型的概括认为会员资格不必离散，而是潜在社区的凸组合。我们建立了模型的可识别性，并提出了一个基于高阶正交迭代算法（HOOI）的计算有效估计过程，该量子张张量SVD由单纯形转角找到算法组成。然后，我们通过提供每个节点误差绑定来证明我们的估计过程的一致性，该误差显示了高阶结构对估计精度的影响。为了证明我们的一致性结果，我们开发了$ \ ell_ {2，\ infty} $ tensor扰动，在独立，异性恋，subgaussian噪声下绑定了HOOI，可能具有独立的利益。我们的分析对迭代使用了一种新颖的遗留结构，我们的边界仅取决于在几乎最佳的信噪比条件下，基础低级张量的光谱特性，因此张量SVD在计算上是可行的。最后，我们将方法应用于真实和模拟数据，证明了具有离散社区成员资格的模型无法识别的某些效果。
Higher-order multiway data is ubiquitous in machine learning and statistics and often exhibits community-like structures, where each component (node) along each different mode has a community membership associated with it. In this paper we propose the tensor mixed-membership blockmodel, a generalization of the tensor blockmodel positing that memberships need not be discrete, but instead are convex combinations of latent communities. We establish the identifiability of our model and propose a computationally efficient estimation procedure based on the higher-order orthogonal iteration algorithm (HOOI) for tensor SVD composed with a simplex corner-finding algorithm. We then demonstrate the consistency of our estimation procedure by providing a per-node error bound, which showcases the effect of higher-order structures on estimation accuracy. To prove our consistency result, we develop the $\ell_{2,\infty}$ tensor perturbation bound for HOOI under independent, heteroskedastic, subgaussian noise that may be of independent interest. Our analysis uses a novel leave-one-out construction for the iterates, and our bounds depend only on spectral properties of the underlying low-rank tensor under nearly optimal signal-to-noise ratio conditions such that tensor SVD is computationally feasible. Finally, we apply our methodology to real and simulated data, demonstrating some effects not identifiable from the model with discrete community memberships.
