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1. Grassmannian optimization for online tensor completion and tracking with the t-svd

   https://doi.org/10.1109/TSP.2022.3164837  

   Gilman, Kyle; Tarzanagh, Davoud Ataee; Balzano, Laura (January 2022, IEEE transactions on signal processing)

   We propose a new fast streaming algorithm for the tensor completion problem of imputing missing entries of a lowtubal-rank tensor using the tensor singular value decomposition (t-SVD) algebraic framework. We show the t-SVD is a specialization of the well-studied block-term decomposition for third-order tensors, and we present an algorithm under this model that can track changing free submodules from incomplete streaming 2-D data. The proposed algorithm uses principles from incremental gradient descent on the Grassmann manifold of subspaces to solve the tensor completion problem with linear complexity and constant memory in the number of time samples. We provide a local expected linear convergence result for our algorithm. Our empirical results are competitive in accuracy but much faster in compute time than state-of-the-art tensor completion algorithms on real applications to recover temporal chemo-sensing and MRI data under limited sampling. 

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2. Online Tensor Completion and Free Submodule Tracking With The T-SVD

   https://doi.org/10.1109/ICASSP40776.2020.9053199  

   Gilman, Kyle; Balzano, Laura (May 2020, IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP))

   null (Ed.)

   We propose a new online algorithm, called TOUCAN, forthe tensor completion problem of imputing missing entriesof a low tubal-rank tensor using the tensor-tensor product (t-product) and tensor singular value decomposition (t-SVD) al-gebraic framework. We also demonstrate TOUCAN’s abilityto track changing free submodules from highly incompletestreaming 2-D data. TOUCAN uses principles from incre-mental gradient descent on the Grassmann manifold to solvethe tensor completion problem with linear complexity andconstant memory in the number of time samples. We com-pare our results to state-of-the-art batch tensor completion al-gorithms and matrix completion algorithms. We show our re-sults on real applications to recover temporal MRI data underlimited sampling. 

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3. A Tensor SVD-based Classification Algorithm Applied to fMRI Data

   Keegan, Katherine; Vishwanath, Tanvi; Xu, Yihua (October 2021, ArXivorg)

   To analyze the abundance of multidimensional data, tensor-based frameworks have been developed. Traditionally, the matrix singular value decomposition (SVD) is used to extract the most dominant features from a matrix containing the vectorized data. While the SVD is highly useful for data that can be appropriately represented as a matrix, this step of vectorization causes us to lose the high-dimensional relationships intrinsic to the data. To facilitate efficient multidimensional feature extraction, we utilize a projection-based classification algorithm using the t-SVDM, a tensor analog of the matrix SVD. Our work extends the t-SVDM framework and the classification algorithm, both initially proposed for tensors of order 3, to any number of dimensions. We then apply this algorithm to a classification task using the StarPlus fMRI dataset. Our numerical experiments demonstrate that there exists a superior tensor-based approach to fMRI classification than the best possible equivalent matrix-based approach. Our results illustrate the advantages of our chosen tensor framework, provide insight into beneficial choices of parameters, and could be further developed for classification of more complex imaging data. We provide our Python implementation at https://github.com/elizabethnewman/tensor-fmri 

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4. HOSVD-Based Algorithm for Weighted Tensor Completion

   https://doi.org/10.3390/jimaging7070110  

   Chao, Zehan; Huang, Longxiu; Needell, Deanna (July 2021, Journal of Imaging)

   null (Ed.)

   Matrix completion, the problem of completing missing entries in a data matrix with low-dimensional structure (such as rank), has seen many fruitful approaches and analyses. Tensor completion is the tensor analog that attempts to impute missing tensor entries from similar low-rank type assumptions. In this paper, we study the tensor completion problem when the sampling pattern is deterministic and possibly non-uniform. We first propose an efficient weighted Higher Order Singular Value Decomposition (HOSVD) algorithm for the recovery of the underlying low-rank tensor from noisy observations and then derive the error bounds under a properly weighted metric. Additionally, the efficiency and accuracy of our algorithm are both tested using synthetic and real datasets in numerical simulations. 

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5. Semi-parametric tensor factor analysis by iteratively projected singular value decomposition

   https://doi.org/10.1093/jrsssb/qkae001  

   Chen, Elynn Y; Xia, Dong; Cai, Chencheng; Fan, Jianqing (February 2024, Journal of the Royal Statistical Society Series B: Statistical Methodology)

   Abstract This paper introduces a general framework of Semi-parametric TEnsor Factor Analysis (STEFA) that focuses on the methodology and theory of low-rank tensor decomposition with auxiliary covariates. Semi-parametric TEnsor Factor Analysis models extend tensor factor models by incorporating auxiliary covariates in the loading matrices. We propose an algorithm of iteratively projected singular value decomposition (IP-SVD) for the semi-parametric estimation. It iteratively projects tensor data onto the linear space spanned by the basis functions of covariates and applies singular value decomposition on matricized tensors over each mode. We establish the convergence rates of the loading matrices and the core tensor factor. The theoretical results only require a sub-exponential noise distribution, which is weaker than the assumption of sub-Gaussian tail of noise in the literature. Compared with the Tucker decomposition, IP-SVD yields more accurate estimators with a faster convergence rate. Besides estimation, we propose several prediction methods with new covariates based on the STEFA model. On both synthetic and real tensor data, we demonstrate the efficacy of the STEFA model and the IP-SVD algorithm on both the estimation and prediction tasks. 

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