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.
Overview
Integrity signals
Registry
Cites
Statistics
Verification
Claims the paper makes
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
In the citing paper
“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
Kolda, T. G., & Bader, B. W. (2009) · SIAM Review · doi:10.1137/07070111x · Cited by 10,624
Carroll, J. D., & Chang, J.-J. (1970) · Psychometrika · doi:10.1007/bf02310791 · Cited by 4,761
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
Beylkin, G., & Mohlenkamp, M. J. (2005) · SIAM Journal on Scientific Computing · doi:10.1137/040604959 · Cited by 357
Incomplete Cross Approximation in the Mosaic-Skeleton Method
Tyrtyshnikov, E. E. (2000) · Computing · doi:10.1007/s006070070031 · Cited by 253
Lathauwer, L. D., Moor, B. D., & Vandewalle, J. P. L. (2004) · SIAM Journal on Matrix Analysis and Applications · doi:10.1137/s089547980139786x · Cited by 229
Grasedyck, L. (2004) · Computing · doi:10.1007/s00607-003-0037-z · Cited by 155
Hackbusch, W., & Khoromskij, B. N. (2005) · Computing · doi:10.1007/s00607-005-0144-0 · Cited by 125
Hierarchical Kronecker tensor-product approximations
Hackbusch, W., Khoromskij, B. N., & Tyrtyshnikov, E. E. (2005) · Journal of Numerical Mathematics · doi:10.1515/1569395054012767 · Cited by 118
A bibliography on semiseparable matrices*
Vandebril, R., Barel, M. V., Golub, G. H., & Mastronardi, N. (2005) · CALCOLO · doi:10.1007/s10092-005-0107-z · Cited by 111
Multigrid Accelerated Tensor Approximation of Function Related Multidimensional Arrays
Khoromskij, B. N., & Khoromskaia, V. (2009) · SIAM Journal on Scientific Computing · doi:10.1137/080730408 · Cited by 107
Approximate iterations for structured matrices
Hackbusch, W., Khoromskij, B. N., & Tyrtyshnikov, E. E. (2008) · Numerische Mathematik · doi:10.1007/s00211-008-0143-0 · Cited by 86
Tensor approximations of matrices generated by asymptotically smooth functions
Tyrtyshnikov, E. E. (2003) · Sbornik Mathematics · doi:10.1070/sm2003v194n06abeh000747 · Cited by 86
Optimal Kronecker Product Approximation of Block Toeplitz Matrices
Kamm, J., & Nagy, J. G. (2000) · SIAM Journal on Matrix Analysis and Applications · doi:10.1137/s0895479898345540 · Cited by 65
Structured matrices and Newton's iteration: unified approach
Pan, V. Y., Rami, Y., & Wang, X. (2002) · Linear Algebra and its Applications · doi:10.1016/s0024-3795(01)00336-6 · Cited by 44
Tensor properties of multilevel Toeplitz and related matrices
Olshevsky, V., Oseledets, I. V., & Tyrtyshnikov, E. E. (2005) · Linear Algebra and its Applications · doi:10.1016/j.laa.2005.03.040 · Cited by 41
Self-citationHackbusch, W., & Khoromskij, B. N. (2005) · Computing · doi:10.1007/s00607-005-0145-z · Cited by 40
Verification of the cross 3D algorithm on quantum chemistry data
Flad, H., Khoromskij, B. N., Savostyanov, D., & Tyrtyshnikov, E. E. (2008) · Russian Journal of Numerical Analysis and Mathematical Modelling · doi:10.1515/rjnamm.2008.020 · Cited by 34
Research works printed but not linked: 10
- M. Espig, Effiziente Bestapproximation mittels Summen von Elementartensoren in hohen Dimensionen , Ph.D. thesis, Universität Leipzig, 2008.
- G. Golub and C. Van Loan, Matrix Computations , 3rd ed., John Hopkins Univ. Press, Baltimore, MD, 1996.
- R. M. Gray, Toeplitz and Circulant Matrices: A Review , Now Publishers, Hanover, MA, 2006. · DOI
- A. Greenbaum, Iterative Methods for Solving Linear Systems , SIAM, Philadelphia, 1997. · DOI
- Harshman R. A., 1970, UCLA Working Papers in Phonetics, 16, 1
- V. Olshevsky, I. V. Oseledets, and E. E. Tyrtyshnikov, Superfast Inversion of Two-level Toeplitz Matrices using Newton Iteration and Tensor-displacement Structure , Oper. Theory Adv. Appl., 179, 2008, Birkhäuser, Basel, pp. 229–240. · DOI
- Oseledets I. V., 2005, Comput. Math. Math. Phys., 45, 302
- I. V. Oseledets, Compact Matrix Form of the d-dimensional Tensor Decomposition , Preprint 2009-01, Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, 2009.
- S. Pissanetzky, Sparse Matrix Technology , Academic Press, New York, 1984. · DOI
- C. F. Van Loan and N. Pitsianis, Approximation with Kronecker products , in Linear Algebra for Large Scale and Real-time Applications NATO Adv. Sci. Inst. Ser. E Appl. Sci. 232, Kluwer Academic Publishers, Dordrecht, 1993, pp. 293–314. · DOI
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
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
