Robust Low-Tubal-Rank Tensor Recovery From Binary Measurements.

Area of Science:

• Applied Mathematics
• Signal Processing
• Machine Learning

Background:

• Low-rank tensor recovery (LRTR) extends low-rank matrix recovery to high-dimensional data.
• Existing LRTR methods often overlook quantization errors, limiting their use with low-level quantization.
• Extreme quantization can reduce measurements to binary sign information.

Purpose of the Study:

• To develop robust low-rank tensor recovery methods that account for extreme quantization effects.
• To reconstruct tensors from binary measurements, specifically sign information.
• To analyze and improve the convergence rates of these recovery methods.

Main Methods:

• Utilizing the tensor Singular Value Decomposition (t-SVD) framework.
• Proposing a tensor hard singular tube thresholding method.
• Developing a constrained tensor nuclear norm minimization method.
• Introducing a new quantization scheme to accelerate convergence.

Main Results:

• Recovery of an n1×n2×n3 tensor with tubal rank r from m random Gaussian binary measurements.
• Achieving error decay at a polynomial speed dependent on the oversampling factor λ.
• Accelerating the convergence rate to an exponential function of λ with a novel quantization scheme.
• Validation of theoretical results through numerical experiments.

Conclusions:

• The proposed methods effectively recover tensors from binary measurements under extreme quantization.
• The new quantization scheme significantly enhances the convergence rate.
• The methods show promising performance for real-world data applications in tensor recovery.