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Introduction to Tensor Decompositions and their Applications in Machine Learning

arXiv 1711.10781 ↗First posted 29 Nov 2017stat.ML cs.LGcited by 46

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Abstract (arXiv)

Tensors are multidimensional arrays of numerical values and therefore generalize matrices to multiple dimensions. While tensors first emerged in the psychometrics community in the 20th century, they have since then spread to numerous other disciplines, including machine learning. Tensors and their decompositions are especially beneficial in unsupervised learning settings, but are gaining popularity in other sub-disciplines like temporal and multi-relational data analysis, too. The scope of this paper is to give a broad overview of tensors, their decompositions, and how they are used in machine learning. As part of this, we are going to introduce basic tensor concepts, discuss why tensors can be considered more rigid than matrices with respect to the uniqueness of their decomposition, explain the most important factorization algorithms and their properties, provide concrete examples of tensor decomposition applications in machine learning, conduct a case study on tensor-based estimation of mixture models, talk about the current state of research, and provide references to available software libraries.

Key facts. Cited by 46 papers in the map (data through 14 Oct 2026). First posted on arXiv 29 Nov 2017.

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Citing papers per year: 2021: 12; 2022: 12; 2023: 11; 2024: 5; 2025: 4; 2026: 2.

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  1. 2021FedPara: Low-Rank Hadamard Product for Communication-Efficient Federated Learningcited by 141
  2. 2021Learning Frequency Domain Approximation for Binary Neural Networkscited by 38
  3. 2023Estimator Meets Equilibrium Perspective: A Rectified Straight Through Estimator for Binary Neural Networks Trainingcited by 19
  4. 2023Factor Graph Neural Networkscited by 15
  5. 2022CoordX: Accelerating Implicit Neural Representation with a Split MLP Architecturecited by 13
  6. 2023Robust Tensor CUR Decompositions: Rapid Low-Tucker-Rank Tensor Recovery with Sparse Corruptioncited by 11
  7. 2021When Random Tensors meet Random Matricescited by 10
  8. 2022Exploiting Low-Rank Tensor-Train Deep Neural Networks Based on Riemannian Gradient Descent With Illustrations of Speech Processingcited by 10
  9. 2022Dynamic Programming in Rank Space: Scaling Structured Inference with Low-Rank HMMs and PCFGscited by 10
  10. 2023Faster Robust Tensor Power Method for Arbitrary Ordercited by 8
  11. 2025Tensor Logic: The Language of AIcited by 8
  12. 2021STNN-DDI: A Substructure-aware Tensor Neural Network to Predict Drug-Drug Interactionscited by 7
34 more
  1. 2022Unsupervised Discontinuous Constituency Parsing with Mildly Context-Sensitive Grammarscited by 7
  2. 2024Unlocking Data-free Low-bit Quantization with Matrix Decomposition for KV Cache Compressioncited by 7
  3. 2021Support Recovery of Sparse Signals from a Mixture of Linear Measurementscited by 6
  4. 2022Subquadratic Kronecker Regression with Applications to Tensor Decompositioncited by 6
  5. 2022Average-Case Complexity of Tensor Decomposition for Low-Degree Polynomialscited by 6
  6. 2021Tensor decomposition for learning Gaussian mixtures from momentscited by 5
  7. 2021Fast Low-Rank Tensor Decomposition by Ridge Leverage Score Samplingcited by 5
  8. 2022Fast algorithm for overcomplete order-3 tensor decompositioncited by 5
  9. 2023Optimizing Orthogonalized Tensor Deflation via Random Tensor Theorycited by 4
  10. 2021SetConv: A New Approach for Learning from Imbalanced Datacited by 3
  11. 2023Singular Value Representation: A New Graph Perspective On Neural Networkscited by 3
  12. 2024A Random Matrix Approach to Low-Multilinear-Rank Tensor Approximationcited by 3
  13. 2024Tensor learning with orthogonal, Lorentz, and symplectic symmetriescited by 3
  14. 2022Tensor Train for Global Optimization Problems in Roboticscited by 2
  15. 2023Bayesian longitudinal tensor response regression for modeling neuroplasticitycited by 2
  16. 2025Towards Quantum Tensor Decomposition in Biomedical Applicationscited by 2
  17. 2021Self-Reflective Terrain-Aware Robot Adaptation for Consistent Off-Road Ground Navigationcited by 1
  18. 2022Modeling Label Correlations for Second-Order Semantic Dependency Parsing with Mean-Field Inferencecited by 1
  19. 2023Improving Grammar-based Sequence-to-Sequence Modeling with Decomposition and Constraintscited by 1
  20. 2023Parameter-efficient Multi-Task and Multi-Domain Learning using Factorized Tensor Networkscited by 1
  21. 2023Simple but Effective Unsupervised Classification for Specified Domain Images: A Case Study on Fungi Imagescited by 1
  22. 2025Multiresolution Analysis and Statistical Thresholding on Dynamic Networkscited by 1
  23. 2021An Iterative Contextualization Algorithm with Second-Order Attentioncited by 0
  24. 2021DynACPD Embedding Algorithm for Prediction Tasks in Dynamic Networkscited by 0
  25. 2021An Accelerated Stochastic Gradient for Canonical Polyadic Decompositioncited by 0
  26. 2022A knowledge graph representation learning approach to predict novel kinase-substrate interactionscited by 0
  27. 2022NumS: Scalable Array Programming for the Cloudcited by 0
  28. 2022Linear Multidimensional Regression with Interactive Fixed-Effectscited by 0
  29. 2023Uniform tensor clustering by jointly exploring sample affinities of various orderscited by 0
  30. 2024Improving LSH via Tensorized Random Projectioncited by 0
  31. 2024Overcomplete Tensor Decomposition via Koszul-Young Flatteningscited by 0
  32. 2025A Tight Lower Bound for the Approximation Guarantee of Higher-Order Singular Value Decompositioncited by 0
  33. 2026Methods for Knowledge Graph Construction from Text Collections: Development and Applicationscited by 0
  34. 2026On Probabilistic Inference Through Parametric Tensor Decomposition in Base Tensor Networkscited by 0