mlsanityData to 7 Oct 2026

Mode-wise Tensor Decompositions: Multi-dimensional Generalizations of CUR Decompositions

arXiv 2103.11037 ↗First posted 19 Mar 2021math.NA cs.IT cs.LGcited by 13

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Abstract (arXiv)

Low rank tensor approximation is a fundamental tool in modern machine learning and data science. In this paper, we study the characterization, perturbation analysis, and an efficient sampling strategy for two primary tensor CUR approximations, namely Chidori and Fiber CUR. We characterize exact tensor CUR decompositions for low multilinear rank tensors. We also present theoretical error bounds of the tensor CUR approximations when (adversarial or Gaussian) noise appears. Moreover, we show that low cost uniform sampling is sufficient for tensor CUR approximations if the tensor has an incoherent structure. Empirical performance evaluations, with both synthetic and real-world datasets, establish the speed advantage of the tensor CUR approximations over other state-of-the-art low multilinear rank tensor approximations.

Key facts. Cited by 13 papers in the map (data through 7 Oct 2026). Builds on 13 of its 73 references that are in the map. First posted on arXiv 19 Mar 2021.

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13 of 73 references.

References · 13

  1. 2014Sketching as a Tool for Numerical Linear Algebracited by 285
  2. 2016Tensor Decomposition for Signal Processing and Machine Learningcited by 223
  3. 2013Revisiting the Nystrom Method for Improved Large-Scale Machine Learningcited by 76
  4. 2017Relative Error Tensor Low Rank Approximationcited by 45
  5. 2014Optimal CUR Matrix Decompositionscited by 29
  6. 2017Accelerated Alternating Projections for Robust Principal Component Analysiscited by 25
  7. 2017Statistically Optimal and Computationally Efficient Low Rank Tensor Completion from Noisy Entriescited by 23
  8. 2013Improving CUR Matrix Decomposition and the Nystr"om Approximation via Adaptive Samplingcited by 21
  9. 2021Robust CUR Decomposition: Theory and Imaging Applicationscited by 20
  10. 2020Rapid Robust Principal Component Analysis: CUR Accelerated Inexact Low Rank Estimationcited by 16
  11. 2018Extension of PCA to Higher Order Data Structures: An Introduction to Tensors, Tensor Decompositions, and Tensor PCAcited by 14
  12. 2014Matrix Coherence and the Nystrom Methodcited by 8
1 more
  1. 2017CUR Decompositions, Similarity Matrices, and Subspace Clusteringcited by 4

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 1; 2022: 2; 2023: 3; 2024: 4; 2025: 2; 2026: 1.

13 papers.

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  1. 2022Matrix Completion with Cross-Concentrated Sampling: Bridging Uniform Sampling and CUR Samplingcited by 12
  2. 2023Robust Tensor CUR Decompositions: Rapid Low-Tucker-Rank Tensor Recovery with Sparse Corruptioncited by 11
  3. 2021Fast Robust Tensor Principal Component Analysis via Fiber CUR Decompositioncited by 8
  4. 2024CURSOR: Scalable Mixed-Order Hypergraph Matching with CUR Decompositioncited by 3
  5. 2024Guaranteed Sampling Flexibility for Low-tubal-rank Tensor Completioncited by 3
  6. 2024S3Attention: Improving Long Sequence Attention with Smoothed Skeleton Sketchingcited by 2
  7. 2022Error Analysis of Tensor-Train Cross Approximationcited by 1
  8. 2023Non-convex approaches for low-rank tensor completion under tubal samplingcited by 1
  9. 2023Maximal Volume Matrix Cross Approximation for Image Compression and Least Squares Solutioncited by 0
  10. 2024Hyperspectral Band Selection based on Generalized 3DTV and Tensor CUR Decompositioncited by 0
  11. 2025Robust Anomaly Detection via Tensor Pseudoskeleton Decompositioncited by 0
  12. 2025Property Inheritance for Subtensors in Tensor Train Decompositionscited by 0
1 more
  1. 2026Robust Low-Tubal-Rank Tensor Completion under Cross-Concentrated Samplingcited by 0