标题： 学习部分二进制测量中的张量 显示英文标题

标题： Learning tensors from partial binary measurements

摘要： 本文中，我们将1比特矩阵补全问题推广到高阶张量。我们证明当 $r=O(1)$ 是一个秩界为$r$、阶为$d$ 的张量时，在$\mathbb{R}^{N} \times \mathbb{R}^{N} \times \cdots \times \mathbb{R}^{N}$ 中的张量$T$ 可以通过正则化其最大q范数和M范数（作为秩的替代量）仅用$m=O(Nd)$ 次二值测量高效地估计出来。我们证明类似于矩阵情形（即当$d=2$时），从张量部分元素的1比特测量恢复低秩张量的样本复杂度与从非量化测量恢复相同。此外，我们理论和数值上都展示了使用1比特张量补全相比矩阵化的优势。具体而言，我们展示了如何将1比特测量模型用于上下文感知推荐系统。 显示英文摘要

摘要： In this paper we generalize the 1-bit matrix completion problem to higher order tensors. We prove that when $r=O(1)$ a bounded rank-$r$, order-$d$ tensor $T$ in $\mathbb{R}^{N} \times \mathbb{R}^{N} \times \cdots \times \mathbb{R}^{N}$ can be estimated efficiently by only $m=O(Nd)$ binary measurements by regularizing its max-qnorm and M-norm as surrogates for its rank. We prove that similar to the matrix case, i.e., when $d=2$, the sample complexity of recovering a low-rank tensor from 1-bit measurements of a subset of its entries is the same as recovering it from unquantized measurements. Moreover, we show the advantage of using 1-bit tensor completion over matricization both theoretically and numerically. Specifically, we show how the 1-bit measurement model can be used for context-aware recommender systems.