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This paper proposes a novel approach to tensor completion, which recovers missing entries of data represented by tensors. The approach is based on the tensor train (TT) rank, which is able to capture hidden information from tensors thanks…
Tensor train (TT) decomposition has drawn people's attention due to its powerful representation ability and performance stability in high-order tensors. In this paper, we propose a novel approach to recover the missing entries of incomplete…
High-dimensional, higher-order tensor data are gaining prominence in a variety of fields, including but not limited to computer vision and network analysis. Tensor factor models, induced from noisy versions of tensor decompositions or…
We introduce a new tensor norm, the average spectrum norm, to study sample complexity of tensor completion problems based on the canonical polyadic decomposition (CPD). Properties of the average spectrum norm and its dual norm are…
In this work, we consider the optimization formulation for symmetric tensor decomposition recently introduced in the Subspace Power Method (SPM) of Kileel and Pereira. Unlike popular alternative functionals for tensor decomposition, the SPM…
Vector autoregressions (VARs) are popular model for analyzing multivariate economic time series. However, VARs can be over-parameterized if the numbers of variables and lags are moderately large. Tensor VAR, a recent solution to…
In datasets where the number of parameters is fixed and the number of samples is large, principal component analysis (PCA) is a powerful dimension reduction tool. However, in many contemporary datasets, when the number of parameters is…
We propose a message passing algorithm, based on variational Bayesian inference, for low-rank tensor completion with automatic rank determination in the canonical polyadic format when additional side information (SI) is given. The SI comes…
Tensor decompositions are promising tools for big data analytics as they bring multiple modes and aspects of data to a unified framework, which allows us to discover complex internal structures and correlations of data. Unfortunately most…
The Canonical Polyadic decomposition (CPD) is a convenient and intuitive tool for tensor factorization; however, for higher-order tensors, it often exhibits high computational cost and permutation of tensor entries, these undesirable…
In NMR spectroscopy, undersampling in the indirect dimensions causes reconstruction artifacts whose size can be bounded using the so-called {\it coherence}. In experiments with multiple indirect dimensions, new undersampling approaches were…
We consider a framework for the construction of iterative schemes for operator equations that combine low-rank approximation in tensor formats and adaptive approximation in a basis. Under fairly general assumptions, we obtain a rigorous…
In the last decades, tensors have emerged as the right tool to represent multidimensional data in a compact yet informative manner. Moreover, it is well-known that by performing low-rank factorizations of such tensors one is often able to…
We consider simultaneous Waring decompositions: Given forms $ f_d $ of degrees $ kd $, $ (d = 2,3 )$, which admit a representation as $ d $-th power sums of $ k $-forms $ q_1,\ldots,q_m $, when is it possible to reconstruct the addends $…
High-dimensional tensors or multi-way data are becoming prevalent in areas such as biomedical imaging, chemometrics, networking and bibliometrics. Traditional approaches to finding lower dimensional representations of tensor data include…
Tensor data often suffer from missing value problem due to the complex high-dimensional structure while acquiring them. To complete the missing information, lots of Low-Rank Tensor Completion (LRTC) methods have been proposed, most of which…
The rapid expansion in the size of new datasets has created a need for fast and efficient parameter-learning techniques. Compressive learning is a framework that enables efficient processing by using random, non-linear features to project…
Using the matrix product state (MPS) representation of the recently proposed tensor ring decompositions, in this paper we propose a tensor completion algorithm, which is an alternating minimization algorithm that alternates over the factors…
Tensor train (TT) decomposition is a powerful representation for high-order tensors, which has been successfully applied to various machine learning tasks in recent years. However, since the tensor product is not commutative, permutation of…
In recent years, the application of tensors has become more widespread in fields that involve data analytics and numerical computation. Due to the explosive growth of data, low-rank tensor decompositions have become a powerful tool to…