Overview
This paper addresses the challenge of efficient low-rank tensor recovery from memory-efficient linear measurements with tensorial structure. It proposes local trimming techniques and two novel iterative hard thresholding (IHT) algorithms designed to overcome limitations of standard recovery methods when applied to tensor-structured measurements.
Research Context
Standard iterative low-rank recovery methods, such as iterative hard thresholding (IHT), typically rely on model assumptions regarding the measurement operator, specifically the restricted isometry property (RIP). The RIP ensures that the measurement operator preserves the norms of low-rank tensors sufficiently well. However, tensor-structured random linear maps, while convenient for application and memory-efficient, generally lack strong restricted isometry properties. This deficiency means they do not adequately preserve the norms of low-rank tensors, presenting a barrier to the theoretical guarantees underpinning many recovery algorithms.
Approach
To mitigate the limitations of tensor-structured maps concerning RIP, the researchers developed local trimming techniques. These techniques are designed to restore point-wise geometry-preservation properties of the tensor-structured maps, bringing their performance closer to that of unstructured sub-Gaussian measurements. Following the development of these trimming techniques, the paper introduces two new versions of tensor IHT algorithms:
- An adaptive gradient trimming algorithm.
- A randomized Kaczmarz-based IHT algorithm.
The proposed methods aim to efficiently recover low-rank tensors from linear measurements. Theoretical guarantees for these novel methods are provided, and their practical efficiency is evaluated through numerical experiments.
Findings
- Local trimming techniques can provably restore point-wise geometry-preservation properties of tensor-structured maps, making them comparable to unstructured sub-Gaussian measurements.
- The proposed adaptive gradient trimming algorithm and randomized Kaczmarz-based IHT algorithm are designed to efficiently recover low-rank tensors from linear measurements.
- Numerical experiments on real and synthetic data indicate that the novel algorithms demonstrate efficiency improvements over the original TensorIHT for low HOSVD and CP-rank tensors.
Why This Matters
Tensor-structured measurements are memory-efficient and convenient to apply in various contexts. Enhancing the ability to perform efficient low-rank tensor recovery from these measurements addresses a fundamental challenge stemming from their inherent lack of strong restricted isometry properties. This development contributes to improving the reliability and performance of tensor-based data processing and analysis.