Higher-order tensors can represent scores in a rating system, frames in a video, and images of the same subject. In practice, the measurements are often highly quantized due to the sampling strategies or the quality of devices. Existing works on tensor recovery have focused on data losses and random noises. Only a few works consider tensor recovery from quantized measurements but are restricted to binary measurements. This paper, for the first time, addresses the problem of tensor recovery from multi-level quantized measurements by leveraging the low CANDECOMP/PARAFAC (CP) rank property. We study the recovery of both general low-rank tensors and tensors that have tensor singular value decomposition (TSVD) by solving nonconvex optimization problems. We provide the theoretical upper bounds of the recovery error, which diminish to zero when the sizes of dimensions increase to infinity. We further characterize the fundamental limit of any recovery algorithm and show that our recovery error is nearly order-wise optimal. A tensor-based alternating proximal gradient descent algorithm with a convergence guarantee and a TSVD-based projected gradient descent algorithm are proposed to solve the nonconvex problems. Our recovery methods can also handle data losses and do not necessarily need the information of the quantization rule. The methods are validated on synthetic data, image datasets, and music recommender datasets.

高阶张量可以表示评分系统中的分数，视频中的帧以及同一主题的图像。在实践中，由于采样策略或设备质量的原因，通常会高度量化测量结果。关于张量恢复的现有工作集中在数据丢失和随机噪声上。只有少数工作考虑了从量化测量中恢复张量，但仅限于二进制测量。本文首次利用低CANDECOMP / PARAFAC（CP）等级属性，解决了多级量化测量中的张量恢复问题。我们通过解决非凸优化问题来研究一般低阶张量和具有张量奇异值分解（TSVD）的张量的恢复。我们提供了恢复误差的理论上限，当尺寸尺寸增加到无穷大时，它减小为零。我们进一步表征了任何恢复算法的基本极限，并表明我们的恢复误差几乎是顺序最优的。为了解决非凸问题，提出了一种具有收敛保证的基于张量的交替近端梯度下降算法和基于TSVD的投影梯度下降算法。我们的恢复方法还可以处理数据丢失，并且不一定需要量化规则的信息。该方法在合成数据，图像数据集和音乐推荐者数据集上得到验证。为了解决非凸问题，提出了一种具有收敛保证的基于张量的交替近端梯度下降算法和基于TSVD的投影梯度下降算法。我们的恢复方法还可以处理数据丢失，并且不一定需要量化规则的信息。该方法在合成数据，图像数据集和音乐推荐者数据集上得到验证。为了解决非凸问题，提出了一种具有收敛保证的基于张量的交替近端梯度下降算法和基于TSVD的投影梯度下降算法。我们的恢复方法还可以处理数据丢失，并且不一定需要量化规则的信息。该方法在合成数据，图像数据集和音乐推荐者数据集上得到验证。