The DOI matches its Crossref record; 23 of the 24 references it cites were found in a registry or an index.
Approximation of $2^d\times2^d$ Matrices Using Tensor Decomposition
Abstract
A new method for structured representation of matrices and vectors is presented. The method is based on the representation of a matrix as a d-dimensional tensor and applying the TT-decomposition proposed recently. It turned out that for many important cases the number of parameters to represent an $n\times n$ matrix falls down to $\mathcal{O}(\log^{\alpha}n)$, giving a logarithmic storage. It is shown that this format can be used not only for storage reduction, but also for linear algebra operations. Possible applications include differential and integral equations, and data and image compression.
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“We consider Hamiltonians that can be written as a polynomial number of terms, with respect to the system size.”
From A variational eigenvalue solver on a photonic quantum processor (2014) · cites this paper as [22]
In the citing paper
“This class of Hamiltonians encompasses a wide range of physical systems, including the electronic structure Hamiltonian of quantum chemistry, the quantum Ising Model, the Heisenberg Model2021, matrices that are well approximated as a sum of n-fold tensor products2223, and more generally any k-sparse Hamiltonian without evident tensor product structure”
From A variational eigenvalue solver on a photonic quantum processor (2014) · cites this paper as [22]
Citations
Cites
Tensor Decompositions and Applications
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A Multilinear Singular Value Decomposition
Lathauwer, L. D., Moor, B. D., & Vandewalle, J. P. L. (2000) · SIAM Journal on Matrix Analysis and Applications · doi:10.1137/s0895479896305696 · Cited by 4,222
On the Best Rank-1 and Rank-( R 1 , R 2 ,. . ., R N ) Approximation of Higher-Order Tensors
Lathauwer, L. D., Moor, B. D., & Vandewalle, J. P. L. (2000) · SIAM Journal on Matrix Analysis and Applications · doi:10.1137/s0895479898346995 · Cited by 1,622
Tensor Rank and the Ill-Posedness of the Best Low-Rank Approximation Problem
de Silva, V., & Lim, L. (2008) · SIAM Journal on Matrix Analysis and Applications · doi:10.1137/06066518x · Cited by 971
Approximation of boundary element matrices
Bebendorf, M. (2000) · Numerische Mathematik · doi:10.1007/pl00005410 · Cited by 868
TT-cross approximation for multidimensional arrays
Oseledets, I. V., & Tyrtyshnikov, E. E. (2009) · Linear Algebra and its Applications · doi:10.1016/j.laa.2009.07.024 · Cited by 573
Self-citationBreaking the Curse of Dimensionality, Or How to Use SVD in Many Dimensions
Oseledets, I. V., & Tyrtyshnikov, E. E. (2009) · SIAM Journal on Scientific Computing · doi:10.1137/090748330 · Cited by 468
Self-citationAlgorithms for Numerical Analysis in High Dimensions
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Tyrtyshnikov, E. E. (2000) · Computing · doi:10.1007/s006070070031 · Cited by 253
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Cited by
Sorted by citations
A variational eigenvalue solver on a photonic quantum processor
Peruzzo, A., McClean, J., Shadbolt, P., Yung, M.-H., Zhou, X.-Q., Love, P. J., Aspuru-Guzik, A., & O’Brien, J. L. (2014) · Nature Communications · doi:10.1038/ncomms5213 · Cited by 4,552
Model Compression and Hardware Acceleration for Neural Networks: A Comprehensive Survey
Deng, B. L., Li, G., Han, S., Shi, L., & Xie, Y. (2020) · Proceedings of the IEEE · doi:10.1109/jproc.2020.2976475 · Cited by 996
A literature survey of low‐rank tensor approximation techniques
Grasedyck, L., Kreßner, D., & Tobler, C. (2013) · GAMM-Mitteilungen · doi:10.1002/gamm.201310004 · Cited by 627
Variational quantum algorithms for nonlinear problems
Lubasch, M., Joo, J., Moinier, P., Kiffner, M., & Jaksch, D. (2020) · Physical Review A · doi:10.1103/physreva.101.010301 · Cited by 332
Tensor Ring DecompositionPreprint
Zhao, Q., Zhou, G., Xie, S., Zhang, L., & Cichocki, A. S. (2016) · arXiv (Cornell University) · doi:10.48550/arxiv.1606.05535 · Cited by 573
Metadata
- Authors
- Oseledets, I. V.
- Corresponding author
- Oseledets, I. V.
- ORCID
- Oseledets, I. V.
- Venue
- SIAM Journal on Matrix Analysis and Applications
- Published
- 2010-01
- Volume
- 31
- Issue
- 4
- Pages
- 2130-2145
- ISSN
- 0895-4798 (ISSN-L)
- 1095-7162
- OpenAlex
- W2025267249
- Cited by
- 174
- References
- 24
- Citations by year
- 2026: 8 · 2025: 11 · 2024: 8 · 2023: 10 · 2022: 15 · 2021: 13 · 2020: 17 · 2019: 9 · 2018: 10 · 2017: 13 · 2016: 7 · 2015: 14 · 2014: 8 · 2013: 10 · 2012: 10
- FWCI
- 5.16
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- SIAM Journal on Matrix Analysis and Applications (publishedVersion)
- Topics
- Physical Sciences › Mathematics › Computational Mathematics › Tensor decomposition and applications
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Matrix Theory and Algorithms
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Keywords
- tensor train decomposition, low-rank tensor approximation, storage reduction, linear algebra operations
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