The field of tensor decomposition and completion is witnessing significant advancements, with a focus on developing efficient and accurate methods for large-scale data. Researchers are exploring innovative approaches to improve the scalability and interpretability of tensor decomposition techniques, including the use of geometric perspectives, nonconvex modeling frameworks, and randomized algorithms. These advancements have the potential to impact various applications, such as image reconstruction, spectromicroscopy, and gene expression analysis. Notably, some papers have introduced novel frameworks, such as sparseGeoHOPCA, which eliminates the need for explicit covariance estimation, and proposed fast and accurate randomized algorithms for low-rank tensor approximation. The Alternating Steepest Descent method has also been extended to tensor completion in the low-rank M-product format, demonstrating improved reconstruction accuracy in spectromicroscopy applications. Some noteworthy papers include:

• sparseGeoHOPCA, which introduces a geometric perspective to high-dimensional tensor decomposition, enabling significant gains in computational efficiency and interpretability.
• Accelerating Large-Scale Regularized High-Order Tensor Recovery, which devises fast and accurate randomized algorithms for low-rank tensor approximation and establishes theoretical bounds on the accuracy of the approximation error estimate.