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Open research questions in Tensor decomposition and applications

51 unresolved questions extracted from the limitations and future-work sections of 178 Tensor decomposition and applications papers in our library. Each links back to the study that raised it.

What the literature leaves open

  • Exponential growth of Lebesgue constants for uniform interpolation nodes. Limited accuracy of previous methods. Technical challenges in implementing the new method.

    Accurate bootstrap bounds from optimal interpolation · 2026 · DOI

  • The issue of softening to zero. The need for a rigorous variational-to-geometric mechanism. The requirement for a positive definite metric-type tensor.

    Viscous Time Theory as a Variational Source Geometry · 2026 · DOI

  • Sensitivity to noise - Limited generalization to out-of-training-distribution data - Complexity of diffusion tensor imaging (DTI)

    Reliable deep diffusion tensor estimation: Rethinking the power of data-driven optimization routine · 2026 · DOI

  • The paper identifies a gap in prior work on block folding. The gap is the lack of a unified account of spectral calculus, distortion, experiments, and discussion.

    Corrected Theory of 8196D→32D Block Folding: A Unified Compression-Theory Account — Spectral Calculus, Distortion, Experiments, and Discussion · 2026 · DOI

  • The nonconvexity of the optimization problem. The high dimensionality of the problem. The need to control the noise in the Langevin dynamics.

    Langevin dynamics for high-dimensional optimization: the case of multi-spiked tensor PCA · 2026 · DOI

  • Future research should apply the corrected theory to real-world data. Future research should explore the limitations of the theory.

    Corrected Theory of 8196D→32D Block Folding: A Unified Compression-Theory Account — Spectral Calculus, Distortion, Experiments, and Discussion · 2026 · DOI

  • We also evaluate the robustness of existing single-pass methods on real-world data tensors, including images and videos, a topic that has not been thoroughly examined before.

    Efficient Techniques for Low-Rank Tensor Approximation and Applications in Robust Object Detection · 2026 · DOI

  • This removes the Frobenius-to-operator loss responsible for the previous extra factor $d_{\max}$ and resolves the explicit open problem posed in the earlier work.

    Tensor-normal maximum likelihood estimation at the operator-norm sample threshold · 2026

  • Meanwhile, the Laplacian Kernel, based on the L1 distance, offers greater robustness to outliers and sparse data compared to the L2-based RBF kernel. [58]; Fang and Hu [48], The theoretical guarantee for the convergence of ADMM with more than two variables is lacking.

    Self-weighted anchor representation with tensor rotation for enhanced multi-view clustering · 2026 · DOI

  • The method introduces an additional O(D^6) overhead when differentiating the SVD. 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 · DOI

  • To apply the proposed method to various tensor renormalization group algorithms. To study the application of the method to classical and quantum many-body systems. To 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 · DOI

  • To 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.

    Robust Completion for Rank-1 Tensors with Noises · 2026 · DOI

  • The 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.

    Robust Completion for Rank-1 Tensors with Noises · 2026 · DOI

  • Further development of the method for other types of bootstrap computations. Investigation of the exact optimal interpolation nodes. Application of the method to other areas of physics.

    Accurate bootstrap bounds from optimal interpolation · 2026 · DOI

  • 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.

    Sum of Squares Decompositions for Structured Biquadratic Forms · 2026 · DOI

  • 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.

    Sum of Squares Decompositions for Structured Biquadratic Forms · 2026 · DOI

  • 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.

    A feasible conjugate gradient method for calculating B-eigenpairs of symmetric tensors · 2026 · DOI

  • The computation of extreme M-eigenvalues of fourth order hierarchically symmetric tensors is a challenging problem. There is a need for efficient methods to solve this problem.

    An efficient memory gradient method for extreme M-eigenvalues of elastic type tensors · 2026 · DOI

  • To develop a general method for finding optimal tagging matrices. To explore the connection between tagging matrices and projective varieties in algebraic geometry further.

    Randomized Block Low-Rank Matrix Compression by Tagging · 2026 · DOI

  • 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.

    Randomized Block Low-Rank Matrix Compression by Tagging · 2026 · DOI

  • 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.

    Stochastic Gradient Descent for Incomplete Tensor Linear Systems · 2026 · DOI

  • The question of whether a metric-type structure can appear as a response of an underlying variational system. The need for a rigorous variational-to-geometric mechanism for conditional source formation.

    Viscous Time Theory as a Variational Source Geometry · 2026 · DOI

  • The paper will design a polynomial-time and robust algorithm for separable order-2 nTD in part II of this paper [44].

    Identifiability of Nonnegative Tucker Decompositions—Part I: Theory · 2026 · DOI

  • 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 · DOI

  • Future 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.

    Characterizing n-chain-locality of correlation tensors based on linear networks · 2026 · DOI

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51 open questions have been extracted from the limitations and future-work passages of 178 Tensor decomposition and applications papers in our library. Each one below links back to the study that raised it, so you can read the original claim in context.

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