Tensor Recovery from Noisy and Multi-Level Quantized Measurements

TL;DR

This paper introduces a novel approach for tensor recovery from multi-level quantized measurements, providing theoretical guarantees and an efficient algorithm, advancing beyond binary quantization and improving error bounds.

Contribution

It is the first to address tensor recovery from multi-level quantized data, proposing a nonconvex optimization method with convergence guarantees and superior error bounds.

Findings

Theoretical error bounds decrease to zero as tensor dimensions grow.

The proposed algorithm effectively recovers tensors from quantized data.

Method outperforms existing binary and matrix quantized recovery techniques.

Abstract

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.