Tensor recovery from quantized measurements based on modewise operators

The problem of tensor recovery from quantized measurements aims to reconstruct a low-rank tensor based on its quantized linear inner-product measurements, which has diverse applications in achieving compressed representation or efficient transmission of high-order tensors. Existing methodologies for this problem rely on computing the product between the vectorization of the underlying tensor and a scale-matched random Gaussian measurement matrix that is significantly larger than the original tensor. The challenge lies in addressing both storage and transmission problems associated with such a random measurement matrix, which may exceed those posed by the original tensor. This study introduces a multi-stage modewise measurement strategy into quantized measurements, thereby facilitating the utilization of multiple smaller sized measurement matrices to effectively alleviate this issue. An iterative projected back projection recovery algorithm is proposed to match such quantized multi-stage modewise measurements within the framework of higher order singular value decomposition. By developing the tensor limited projection distortion property and combining it with the restricted isometry property, we establish sufficient conditions on both the linear sampling operator and quantizer to ensure that our approach enables reconstruction of low-rank tensors. Specifically, we have demonstrated that several multi-stage modewise measurement mappings, such as those derived from sub-Gaussian and subsampled orthogonal ensembles like discrete Fourier measurements, satisfy these conditions. Experiments conducted on both synthetic and real-world data have verified the credibility of our theory and the superiority of our algorithm.