Low-Tubal-Rank Tensor Recovery via Factorized Gradient Descent (2401.11940v2)

Abstract: This paper considers the problem of recovering a tensor with an underlying low-tubal-rank structure from a small number of corrupted linear measurements. Traditional approaches tackling such a problem require the computation of tensor Singular Value Decomposition (t-SVD), that is a computationally intensive process, rendering them impractical for dealing with large-scale tensors. Aim to address this challenge, we propose an efficient and effective low-tubal-rank tensor recovery method based on a factorization procedure akin to the Burer-Monteiro (BM) method. Precisely, our fundamental approach involves decomposing a large tensor into two smaller factor tensors, followed by solving the problem through factorized gradient descent (FGD). This strategy eliminates the need for t-SVD computation, thereby reducing computational costs and storage requirements. We provide rigorous theoretical analysis to ensure the convergence of FGD under both noise-free and noisy situations. Additionally, it is worth noting that our method does not require the precise estimation of the tensor tubal-rank. Even in cases where the tubal-rank is slightly overestimated, our approach continues to demonstrate robust performance. A series of experiments have been carried out to demonstrate that, as compared to other popular ones, our approach exhibits superior performance in multiple scenarios, in terms of the faster computational speed and the smaller convergence error.

• The paper introduces a novel factorized gradient descent approach that bypasses t-SVD to efficiently recover low-tubal-rank tensors.
• It employs a decomposition into two smaller factors, achieving linear convergence in noise-free settings and sub-linear rates when the rank is overestimated.
• Experimental results confirm robust performance with minimal relative error and faster computation compared to traditional tensor recovery methods.

Introduction to Factorized Gradient Descent for Tensor Recovery

The field of data recovery, be it in imaging, video compression or sensor networks, frequently revolves around tensors. The effective manipulation of tensor representations, especially for recovery scenarios often leverages the notion of low-rank structures within tensors. Traditional methods for low-rank tensor recovery, however, bear significant computational overhead due to the necessity of tensor Singular Value Decomposition (t-SVD). To ameliorate this, Liu et al.'s research presents a novel factorization-based approach for efficient tensor recovery, eschewing the t-SVD computation entirely.

Factorization-Based Tensor Recovery

The core proposition of the paper revolves around decomposing a tensor of interest into a product of two smaller factor tensors and applying a Factorized Gradient Descent (FGD) to solve the recovery problem. This factorization is conceptualized in line with the low-tubal-rank tensor recovery (LTRTR) undertaken within the t-SVD framework. By steering clear of the precise estimation of the tensor's tubal-rank, the methodology demonstrates resilience, showing robust performance even when the rank is slightly overestimated.

Convergence and Performance Analysis

The paper meticulously details the convergence scenarios of the proposed FGD approach. In noise-free environments, the convergence is linear for exact rank and sub-linear for overestimated rank, both exhibiting a reliable recovery performance. In the presence of noise, the analysis unfolds a deterministic error convergence with a sub-linear rate. The iteration complexity presented puts the proposed method at an advantage compared to other established LTRTR methods, delivering a fast convergence speed and a smaller relative convergence error.

Experimental Validation

The experimental section reinforces the theoretical contributions—demonstrating that the new approach not only records the fastest convergence in absence of noise but also withstands the presence of noise effectively. A series of experiments under various scenarios are conducted: examining convergence rates under exact and over rank conditions; assessing performance in noiseless contexts against the state-of-the-art methods; and noise robustness. Across all experiment sets, FGD consistently achieves minimum relative recovery error and requires the least computational time, substantiating the paper's claims of efficiency and effectiveness.

Conclusion and Future Work

In conclusion, this approach to tensor recovery offers a computationally attractive alternative to t-SVD-based methods with solid theoretical guarantees and excellent empirical results. Looking ahead, the paper paves the way for exploring asymmetrical tensor recovery scenarios, enhancing convergence rates in overestimated rank conditions, and diving into over-parameterized tensor recovery under more practical settings. Further explorations include adapting techniques such as small initialization or preconditioning to better optimize convergence behavior and computational scope.