Related papers: Optimization on the Extended Tensor-Train Manifold…

In this paper, we analyze the fundamental conditions for low-rank tensor completion given the separation or tensor-train (TT) rank, i.e., ranks of unfoldings. We exploit the algebraic structure of the TT decomposition to obtain the…

The tensor-train (TT) decomposition expresses a tensor in a data-sparse format used in molecular simulations, high-order correlation functions, and optimization. In this paper, we propose four parallelizable algorithms that compute the TT…

To account for phenomenological theories and a set of invariants, stress and strain are usually decomposed into a pair of pressure and deviatoric stress and a pair of volumetric strain and deviatoric strain. However, the conventional…

The tensor decomposition addressed in this paper may be seen as a generalisation of Singular Value Decomposition of matrices. We consider general multilinear and multihomogeneous tensors. We show how to reduce the problem to a truncated…

We present an iterative algorithm, called the symmetric tensor eigen-rank-one iterative decomposition (STEROID), for decomposing a symmetric tensor into a real linear combination of symmetric rank-1 unit-norm outer factors using only…

Transformers have attained superior performance in natural language processing and computer vision. Their self-attention and feedforward layers are overparameterized, limiting inference speed and energy efficiency. Tensor decomposition is a…

Higher-order tensors are becoming prevalent in many scientific areas such as computer vision, social network analysis, data mining and neuroscience. Traditional tensor decomposition approaches face three major challenges: model selecting,…

Low-rank tensor decompositions (TDs) provide an effective framework for multiway data analysis. Traditional TD methods rely on predefined structural assumptions, such as CP or Tucker decompositions. From a probabilistic perspective, these…

We consider the problem of learning a low-rank matrix, constrained to lie in a linear subspace, and introduce a novel factorization for modeling such matrices. A salient feature of the proposed factorization scheme is it decouples the…

Recent approaches to the tensor completion problem have often overlooked the nonnegative structure of the data. We consider the problem of learning a nonnegative low-rank tensor, and using duality theory, we propose a novel factorization of…

Tensor decomposition is a popular technique for tensor completion, However most of the existing methods are based on linear or shallow model, when the data tensor becomes large and the observation data is very small, it is prone to over…

We study the decomposability and the subdifferential of the tensor nuclear norm. Both concepts are well understood and widely applied in matrices but remain unclear for higher-order tensors. We show that the tensor nuclear norm admits a…

High-dimensional partial differential equations (PDEs) are ubiquitous in economics, science and engineering. However, their numerical treatment poses formidable challenges since traditional grid-based methods tend to be frustrated by the…

Tensor decomposition plays a key role in identifying common features across a collection of matrices in many areas of science. A fundamental need in big data research is to process data tabulated as large-scale matrices using eigenvectors.…

Scientific problems require resolving multi-scale phenomena across different resolutions and learning solution operators in infinite-dimensional function spaces. Neural operators provide a powerful framework for this, using…

The advancement of sensing technology has driven the widespread application of high-dimensional data. However, issues such as missing entries during acquisition and transmission negatively impact the accuracy of subsequent tasks. Tensor…

Compressed sensing extends from the recovery of sparse vectors from undersampled measurements via efficient algorithms to the recovery of matrices of low rank from incomplete information. Here we consider a further extension to the…

Factorization of matrices where the rank of the two factors diverges linearly with their sizes has many applications in diverse areas such as unsupervised representation learning, dictionary learning or sparse coding. We consider a setting…

Advanced tensor decomposition, such as Tensor train (TT) and Tensor ring (TR), has been widely studied for deep neural network (DNN) model compression, especially for recurrent neural networks (RNNs). However, compressing convolutional…

This work considers a computationally and statistically efficient parameter estimation method for a wide class of latent variable models---including Gaussian mixture models, hidden Markov models, and latent Dirichlet allocation---which…