  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. Leveraging the low-rank property of the tensor, this paper
proposes a nonconvex optimization problem for tensor recovery. We provide a
theoretical upper bound of the recovery error, which diminishes to zero when
the sizes of dimensions increase to infinity. Our error bound significantly
improves over the existing results in one-bit tensor recovery and quantized
matrix recovery. A tensor-based alternating proximal gradient descent algorithm
with a convergence guarantee is proposed to solve the nonconvex problem. Our
recovery method can handle data losses and do not need the information of the
quantization rule. The method is validated on synthetic data, image datasets,
and music recommender datasets.
