Open research questions in Tensor decomposition and applications
98 gap statements mined from Tensor decomposition and applications papers in our 4.5M-paper local library, which holds 707 papers on the topic — drawn mostly from each paper's own stated research gap, future-work, challenge and limitation notes, and its abstract. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one.
What the literature leaves open
Existing methods have limitations in detecting both spectral and spatial anomalies. There is a need for a method that can efficiently detect anomalies in hyperspectral images.
Spectral-Spatial Extraction through Layered Tensor Decomposition for Hyperspectral Anomaly Detection · 2026 · DOIThe existing literature on quaternion tensor decompositions is limited. There is a need for a comprehensive framework for the simultaneous decomposition of two sets of five quaternion tensors. The application of quaternion tensor decompositions to color video processing is underexplored.
Simultaneous Decompositions of Two Sets of Five Quaternion Tensors and Applications in Color Videos Processing · 2026 · DOIThe existing methods for the rank-1 tensor completion problem may not be robust to noises in the observed tensor entries. There is a need for a robust method that can handle noises in the observed tensor entries.
Classical DMRG is limited by its sequential minimization approach. The choice of coarse space impacts convergence rates.
An Additive Two-Level Parallel Variant of the DMRG Algorithm with Coarse-Space Correction · 2026 · DOIThe existing analysis of the HOOI and ASI methods has gaps and is only applicable to real tensors. There is a need for a detailed convergence analysis of the two methods for complex tensors.
Convergence Analysis of Two Alternating Iterative Schemes for Tucker Decomposition · 2026The algorithm may not be suitable for very large-scale problems due to computational complexity. The method assumes that the underlying graph structure changes smoothly over time. The experimental evaluation is limited to a few datasets.
Dynamic Elliptical Graph Factor Models via Riemannian Optimization with Geodesic Temporal Regularization · 2026An open question is whether one can use the FHT-W ansatz to efficiently solve the backward Kolmogorov equation and the Hamilton-Jacobi equation on lattice models.
To address the question of whether a symmetric biquadratic M-tensor is an SOS tensor. To develop new techniques to bridge the gap between M-eigenvalues and matrix eigenvalues.
Limited methods that handle data that are both multi-source and multi-way. Need for a technique that can reduce dimension and approximate underlying signals in multi-way data. Lack of an approach that can handle incomplete data with imputation in multi-source and multi-way data analysis.
Future research can focus on testing the FCG method on large-scale tensors. Future research can focus on applying the FCG method to other problems related to symmetric tensors.
Existing algorithms for compressing data-sparse matrices can be improved by invoking rank structure. There is a need for efficient methods for compressing flat rank-structured matrices.
Alternatively, the convergence of iterative methods other than SGD, such as Gauss-Seidel variants, could be studied using a proof technique similar to that of Theorem 2.
The existing methods for completely positive tensor decomposition problems are computationally expensive and do not exploit the sparsity of the tensor. There is a need for a novel framework and algorithm to generate maximal cliques of multi-hypergraphs and reformulate the problem into an ideal-sparse generalized moment problem.
An Ideal-Sparse Generalized Moment Problem Reformulation for Completely Positive Tensor Decomposition Exploiting Maximal Cliques of Multi-hypergraphs · 2026 · DOIFuture research could explore the application of n-chain-locality to quantum information processing protocols. Further work could investigate the relationship between n-chain-locality and other notions of nonlocality.
There is a gap in the understanding of n-chain-locality in linear networks. The paper aims to address this gap by characterizing n-chain-locality using discrete and continuous n-chain-local hidden variable models.
The remaining tensor contraction operations required to evaluate the derivatives are also multiplied by (k + 1)(k + 2)/2.
Forward-mode automatic differentiation for the tensor renormalization group and its relation to the impurity method · 2026 · DOITo investigate the potential of the method for calculating physical quantities with higher accuracy and efficiency.
Forward-mode automatic differentiation for the tensor renormalization group and its relation to the impurity method · 2026 · DOITo investigate the theoretical guarantee of the proposed method. To apply the proposed method to other tensor completion problems. To explore the application of the proposed method in other fields.
To investigate the theoretical properties of the proposed framework.
Dynamic Elliptical Graph Factor Models via Riemannian Optimization with Geodesic Temporal Regularization · 2026We have argued that to obtain accurate numerical conformal bootstrap bounds, we should approximate the optimal functional αopt(F∆,j) as closely as possible in the regime where it has interesting structure. For functionals built from a finite number of derivatives of the crossing equation, the interesting region is a finite range of ∆ where the functional exhibits “wiggles,” beyond which it decays exponentially.
The question of whether a symmetric biquadratic M-tensor is an SOS tensor is still open. The general theory for the subclass of biquadratic tensors is not directly applicable.
To explore the connection between tagging matrices and projective varieties in algebraic geometry further.
The paper will design a polynomial-time and robust algorithm for separable order-2 nTD in part II of this another study.
To improve the computational efficiency of the proposed method. To evaluate the performance of the proposed method on more datasets. To apply the proposed method to other applications.
Spectral-Spatial Extraction through Layered Tensor Decomposition for Hyperspectral Anomaly Detection · 2026 · DOIFurther study of the properties of Langevin dynamics in nonconvex optimization problems. Development of new methods for tensor PCA. Investigation of the applications of the results to other areas of machine learning and optimization.
Langevin dynamics for high-dimensional optimization: the case of multi-spiked tensor PCA · 2026 · DOI
Most-cited papers in Tensor decomposition and applications
- Algorithms and applications for approximate nonnegative matrix factorization · Computational Statistics & Data Analysis · 2006 · 1,197 citations
- Tensor product variable binding and the representation of symbolic structures in connectionist systems · Artificial Intelligence · 1990 · 539 citations
- Gradient Projection Algorithms and Software for Arbitrary Rotation Criteria in Factor Analysis · Educational and Psychological Measurement · 2005 · 447 citations
- Modern Multidimensional Scaling: Theory and Applications · Journal of Educational Measurement · 2003 · 262 citations
- Nonlocal Low-Rank Tensor Completion for Visual Data · IEEE Transactions on Cybernetics · 2021 · 216 citations
- Guaranteed Tensor Recovery Fused Low-rankness and Smoothness · IEEE Transactions on Pattern Analysis and Machine Intelligence · 2023 · 147 citations
- ADTT: A Highly Efficient Distributed Tensor-Train Decomposition Method for IIoT Big Data · IEEE Transactions on Industrial Informatics · 2021 · 141 citations
- Dynamic factor models with infinite-dimensional factor spaces: One-sided representations · Journal of Econometrics · 2014 · 117 citations
- Uniqueness Proof for a Family of Models Sharing Features of Tucker's Three-Mode Factor Analysis and PARAFAC/Candecomp · Psychometrika · 1996 · 91 citations
- Tensor-based unsupervised feature selection for error-robust handling of unbalanced incomplete multi-view data · Information Fusion · 2024 · 78 citations
Most recent work
- An efficient recursive support vector machine regression method based on tensor train decomposition and Aitken acceleration technique · Journal of Computational and Applied Mathematics · 2026
- Solving the Gross-Pitaevskii equation on multiple different scales using the quantics tensor train representation · Physical Review Research · 2026
- Loop series expansions for tensor networks · Physical Review Research · 2026
- Non-negative matrix factorization algorithms generally improve topic model fits · Statistics and Computing · 2026
- Robust tensor completion via tensor wheel decomposition with difference factors regularization for hyperspectral image restoration · Applied Mathematical Modelling · 2026
- A power method for computing singular value decomposition · Journal of Computational and Applied Mathematics · 2026
- Tensor Wheel Decomposition: Theory and Application to Tensor Completion · IEEE Transactions on Image Processing · 2026
- A practical introduction to tensor network renormalization with TNRKit.jl · SciPost Physics Codebases · 2026
- High-order tensor completion via a learnable t-product-induced Fully-Connected Tensor Network decomposition · Pattern Recognition · 2026
- Fast and Accurate Domain Adaptation for Irregular and Regular Tensor Decomposition · IEEE Transactions on Knowledge and Data Engineering · 2026
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