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Optimization with orthogonality constraints frequently arises in various fields such as machine learning. Riemannian optimization offers a powerful framework for solving these problems by equipping the constraint set with a Riemannian…

An optimization-based approach for the Tucker tensor approximation of parameter-dependent data tensors and solutions of tensor differential equations with low Tucker rank is presented. The problem of updating the tensor decomposition is…

Low-rank tensor completion problem aims to recover a tensor from limited observations, which has many real-world applications. Due to the easy optimization, the convex overlapping nuclear norm has been popularly used for tensor completion.…

Low-rank tensor completion (LRTC) aims to recover a complete low-rank tensor from incomplete observed tensor, attracting extensive attention in various practical applications such as image processing and computer vision. However, current…

We consider the problem of learning low-rank tensors from partial observations with structural constraints, and propose a novel factorization of such tensors, which leads to a simpler optimization problem. The resulting problem is an…

Low-rank tensor completion aims to recover a tensor from partially observed entries, and it is widely applicable in fields such as quantum computing and image processing. Due to the significant advantages of the tensor train (TT) format in…

In recent studies, the tensor ring (TR) rank has shown high effectiveness in tensor completion due to its ability of capturing the intrinsic structure within high-order tensors. A recently proposed TR rank minimization method is based on…

This paper studies a recovery task of finding a low multilinear-rank tensor that fulfills some linear constraints in the general settings, which has many applications in computer vision and graphics. This problem is named as the low…

We study extensions of compressive sensing and low rank matrix recovery (matrix completion) to the recovery of low rank tensors of higher order from a small number of linear measurements. While the theoretical understanding of low rank…

We consider stochastic gradient methods under the interpolation regime where a perfect fit can be obtained (minimum loss at each observation). While previous work highlighted the implicit regularization of such algorithms, we consider an…

In the realm of tensor optimization, the low-rank Tucker decomposition is crucial for reducing the number of parameters and for saving storage. We explore the geometry of Tucker tensor varieties -- the set of tensors with bounded Tucker…

The problem of recovering a low $n$-rank tensor is an extension of sparse recovery problem from the low dimensional space (matrix space) to the high dimensional space (tensor space) and has many applications in computer vision and graphics…

Rank minimization (RM) is a wildly investigated task of finding solutions by exploiting low-rank structure of parameter matrices. Recently, solving RM problem by leveraging non-convex relaxations has received significant attention. It has…

The low-rank matrix completion problem can be solved by Riemannian optimization on a fixed-rank manifold. However, a drawback of the known approaches is that the rank parameter has to be fixed a priori. In this paper, we consider the…

In this paper we focus on the problem of completion of multidimensional arrays (also referred to as tensors) from limited sampling. Our approach is based on a recently proposed tensor-Singular Value Decomposition (t-SVD) [1]. Using this…

In intelligent transportation systems, traffic data imputation, estimating the missing value from partially observed data is an inevitable and challenging task. Previous studies have not fully considered traffic data's multidimensionality…

In this paper, we propose a new approaches for low rank approximation of quaternion tensors \cite{chen2019low,zhang1997quaternions,hamilton1866elements}. The first method uses quasi-norms to approximate the tensor by a low-rank tensor using…

We consider the problem of recovering a low-multilinear-rank tensor from a small amount of linear measurements. We show that the Riemannian gradient algorithm initialized by one step of iterative hard thresholding can reconstruct an…

The tensor train (TT) format enjoys appealing advantages in handling structural high-order tensors. The recent decade has witnessed the wide applications of TT-format tensors from diverse disciplines, among which tensor completion has drawn…

Higher-order low-rank tensors naturally arise in many applications including hyperspectral data recovery, video inpainting, seismic data recon- struction, and so on. We propose a new model to recover a low-rank tensor by simultaneously…