论文标题
基于序数观测值的张量降解和完成
Tensor denoising and completion based on ordinal observations
论文作者
论文摘要
高阶张量经常出现在神经影像，推荐系统，社交网络分析和心理学研究等应用中。我们考虑了可能不完整的序数值观测值的低级张量估计问题。研究了两个相关的问题，一个关于张量DENOISING，另一个在张量完成。我们提出了一个多线性累积链路模型，开发了受cank限制的M估计器，并获得理论准确性保证。我们的平方平方误差限制的速度比以前的结果更快，我们表明所提出的估计器在低级别模型类别下是最佳的。此外，开发的过程是一种有效的完成方法，可以保证仅使用$ \ tilde {\ tilde {\ natercal {o}}（o}}（kd）$ noisy，量化的观察，仅使用$ \ tilde {\ tilde {\ tilde {\ tilde {\ tilde {\ tilde {\ tilde {\ tilde {\ tilde {\ tilde {\ tilde {\ tilde {\ tilde {我们证明了与以前有关聚类和协作过滤任务的方法的表现。
Higher-order tensors arise frequently in applications such as neuroimaging, recommendation system, social network analysis, and psychological studies. We consider the problem of low-rank tensor estimation from possibly incomplete, ordinal-valued observations. Two related problems are studied, one on tensor denoising and the other on tensor completion. We propose a multi-linear cumulative link model, develop a rank-constrained M-estimator, and obtain theoretical accuracy guarantees. Our mean squared error bound enjoys a faster convergence rate than previous results, and we show that the proposed estimator is minimax optimal under the class of low-rank models. Furthermore, the procedure developed serves as an efficient completion method which guarantees consistent recovery of an order-$K$ $(d,\ldots,d)$-dimensional low-rank tensor using only $\tilde{\mathcal{O}}(Kd)$ noisy, quantized observations. We demonstrate the outperformance of our approach over previous methods on the tasks of clustering and collaborative filtering.
