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
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
Items 101–173 of 173
Using the Tensor-Train Approach to Solve the Ground-State Eigenproblem for Hydrogen Molecules
Veit, A. D., & Scott, R. (2017) · SIAM Journal on Scientific Computing · doi:10.1137/15m102808x · Cited by 7
A low-rank approach to the computation of path integrals
Litsarev, M. S., & Oseledets, I. V. (2015) · Journal of Computational Physics · doi:10.1016/j.jcp.2015.11.009 · Cited by 6
Self-citationTensor rank bounds and explicit QTT representations for the inverses of circulant matrices
Vysotsky, L. I., & Rakhuba, M. (2022) · Numerical Linear Algebra with Applications · doi:10.1002/nla.2461 · Cited by 6
TT-ViT: Vision Transformer Compression Using Tensor-Train Decomposition
Minh, H. P., Xuan, N. N., & Thai, S. T. (2022) · Lecture notes in computer science · doi:10.1007/978-3-031-16014-1_59 · Cited by 7
Superfast solution of linear convolutional Volterra equations using QTT approximation
Roberts, J. A., Savostyanov, D., & Tyrtyshnikov, E. E. (2013) · Journal of Computational and Applied Mathematics · doi:10.1016/j.cam.2013.10.025 · Cited by 6
Khoromskaia, V., & Khoromskij, B. N. (2021) · Computational Mathematics and Mathematical Physics · doi:10.1134/s0965542521050110 · Cited by 6
The preconditioned inverse iteration for hierarchical matrices
Benner, P., & Mach, T. (2012) · Numerical Linear Algebra with Applications · doi:10.1002/nla.1830 · Cited by 6
Approximation of Singularities by Quantized‐Tensor FEM
Kazeev, V. A., & Schwab, C. (2015) · PAMM · doi:10.1002/pamm.201510353 · Cited by 6
Enforcing symmetry in tensor network MIMO Volterra identification
Batselier, K. (2021) · IFAC-PapersOnLine · doi:10.1016/j.ifacol.2021.08.404 · Cited by 5
A Novel Tensor Ring Sparsity Measurement for Image Completion
Zeng, J., Qiu, Y., Ma, Y., Wang, A., & Zhao, Q. (2024) · Entropy · doi:10.3390/e26020105 · Cited by 5
Numerical Study of Low Rank Approximation Methods for Mechanics Data and Its Analysis
Lestandi, L. (2021) · Journal of Scientific Computing · doi:10.1007/s10915-021-01421-2 · Cited by 5
Tensor rank bounds for point singularities in ℝ3
Marcati, C., Rakhuba, M., & Schwab, C. (2022) · Advances in Computational Mathematics · doi:10.1007/s10444-022-09925-7 · Cited by 5
Low-Rank Tree Tensor Network Operators for Long-Range Pairwise Interactions
Ceruti, G., Kreßner, D., & Sulz, D. (2025) · SIAM Journal on Scientific Computing · doi:10.1137/24m1661996 · Cited by 4
Tensor decomposition of Toeplitz Jacket matrices for big data processing
Li, J., Yan, Y., Duan, W., Song, S., & Lee, M. H. (2015) · International Conference on Big Data and Smart Computing · doi:10.1109/35021bigcomp.2015.7072840 · Cited by 4
Dolgov, S., Kazeev, V. A., & Khoromskij, B. N. (2017) · Mathematics and Computers in Simulation · doi:10.1016/j.matcom.2017.10.009 · Cited by 4
Zhao, Y., Yang, L. T., Zhang, Y., Sun, J., Wang, X., Zhang, C., & Zhang, G. (2021) · IEEE Transactions on Network Science and Engineering · doi:10.1109/tnse.2021.3119324 · Cited by 4
TT Ranks of Approximate Tensorizations of Some Smooth Functions
Vysotsky, L. I. (2021) · Computational Mathematics and Mathematical Physics · doi:10.1134/s096554252105016x · Cited by 4
Risthaus, L., & Schneider, M. (2022) · Applied Numerical Mathematics · doi:10.1016/j.apnum.2022.04.002 · Cited by 4
Gavrilyuk, I. P., & Khoromskij, B. N. (2018) · Computational Methods in Applied Mathematics · doi:10.1515/cmam-2018-0021 · Cited by 4
Alternating Linear Scheme in a Bayesian Framework for Low-Rank Tensor Approximation
Menzen, C., Kok, M., & Batselier, K. (2022) · SIAM Journal on Scientific Computing · doi:10.1137/20m1386414 · Cited by 7
Tensor approximation of cooperative games and their semivalues
Ballester‐Ripoll, R. (2021) · International Journal of Approximate Reasoning · doi:10.1016/j.ijar.2021.11.007 · Cited by 4
Khoromskaia, V., & Khoromskij, B. N. (2017) · Computational Methods in Applied Mathematics · doi:10.1515/cmam-2017-0004 · Cited by 4
Tensor network square root Kalman filter for online Gaussian process regression
Menzen, C., Kok, M., & Batselier, K. (2025) · Automatica · doi:10.1016/j.automatica.2025.112694 · Cited by 3
Tensor-programmable quantum circuits for solving differential equations
Siegl, P., Reese, G. S., Hashizume, T., van Hülst, N.-L., & Jaksch, D. (2025) · Physical Review Research · doi:10.1103/2qzh-yf49 · Cited by 5
Ayvaz, M., & Demіralp, M. (2011) · Cited by 3
Tensorisation in the Solution of Smoluchowski Type Equations
Timokhin, I. V. (2020) · Lecture notes in computer science · doi:10.1007/978-3-030-41032-2_20 · Cited by 3
On Tensor-Train Ranks of Tensorized Polynomials
Vysotsky, L. I. (2020) · Lecture notes in computer science · doi:10.1007/978-3-030-41032-2_21 · Cited by 3
Nonnegatively Constrained Tensor Network for Classification Problems
Zdunek, R., & Fonał, K. (2019) · International Conference on Ubiquitous and Future Networks · doi:10.1109/icufn.2019.8806176 · Cited by 3
Low-rank tensor approximation of singularly perturbed boundary value problems in one dimension
Marcati, C., Rakhuba, M., & Ulander, J. E. M. (2021) · CALCOLO · doi:10.1007/s10092-021-00439-0 · Cited by 3
Multilinear state space system identification with matrix product operators
Batselier, K., Ko, C.-Y., Phan, A. H., Cichocki, A. S., & Wong, N. (2018) · IFAC-PapersOnLine · doi:10.1016/j.ifacol.2018.09.219 · Cited by 3
Extended Kalman Filtering with Low-Rank Tensor Networks for MIMO Volterra System Identification
Batselier, K., Ko, C.-Y., & Wong, N. (2019) · IEEE Conference on Decision and Control (CDC) · doi:10.1109/cdc40024.2019.9028895 · Cited by 3
Solving Nonlinear Filtering Problems Using a Tensor Train Decomposition Method
Li, S., Wang, Z., Yau, S. S. ‐., & Zhang, Z. (2022) · IEEE Transactions on Automatic Control · doi:10.1109/tac.2022.3223319 · Cited by 3
Numerical Approaches for High-Dimensional PDEs for Quantum Chemistry
Schneider, R., Rohwedder, T., & Legeza, Ö. (2015) · Springer eBooks · doi:10.1007/978-3-540-70529-1_245 · Cited by 3
A Quantum-Inspired Algorithm for Wave Simulation Using Tensor Networks
Lively, K., Pagni, V., & Camacho, G. (2025) · IEEE International Conference on Quantum Computing and Engineering (QCE) · doi:10.1109/qce65121.2025.00210 · Cited by 5
Simulating quantum turbulence with matrix-product states
Gómez-Lozada, F., Perico-García, N., Gourianov, N., Salman, H., & Mendoza-Arenas, J. J. (2026) · Physical Review Applied · doi:10.1103/x5xg-whh1 · Cited by 6
Array flattening based univariate high dimensional model representation (AFBUHDMR)
Demiralp, E., & Demіralp, M. (2011) · Cited by 2
Adak, D., Truong, D. P., Vuchkov, R. G., De, S., DeSantis, D., Roberts, N. V., Rasmussen, K. Ø., & Alexandrov, B. S. (2025) · Mathematics · doi:10.3390/math13142277 · Cited by 2
Muravleva, E. A., & Oseledets, I. V. (2018) · Computational Methods in Applied Mathematics · doi:10.1515/cmam-2018-0026 · Cited by 2
Self-citationUsvyatsov, M., Makarova, A., Ballester‐Ripoll, R., Rakhuba, M., Krause, A., & Schindler, K. (2021) · IEEE/CVF International Conference on Computer Vision (ICCV) · doi:10.1109/iccv48922.2021.01123 · Cited by 2
Vico-Greengard-Ferrando quadratures in the tensor solver for integral equations
Khrulkov, V., Rakhuba, M., & Oseledets, I. V. (2017) · Photonics & Electromagnetics Research Symposium - Spring (PIERS-Spring) · doi:10.1109/piers.2017.8262142 · Cited by 1
Self-citationA low-rank QTT-based finite element method for elasticity problems
Benvenuti, E., Manzini, G., Nale, M., & Pizzolato, S. (2025) · Computer Methods in Applied Mechanics and Engineering · doi:10.1016/j.cma.2025.118276 · Cited by 1
Element-wise Multiplication of Tensor Trains
Michailidis, A. A., Fenton, C., & Kiffner, M. (2025) · SIAM Journal on Scientific Computing · doi:10.1137/24m1714149 · Cited by 1
Structured Matrix Approximations via Tensor Decompositions
Kilmer, M. E., & Saibaba, A. K. (2022) · SIAM Journal on Matrix Analysis and Applications · doi:10.1137/21m1418290 · Cited by 1
Denoising convolution algorithms and applications to SAR signal processing
Chertock, A. E., Leonard, C., Tsynkov, S., & Utyuzhnikov, S. V. (2023) · Communications on Analysis and Computation · doi:10.3934/cac.2023008 · Cited by 1
Enabling Large-Scale Probabilistic Seizure Detection with a Tensor-Network Kalman Filter for LS-SVM
de Rooij, S. J. S., Batselier, K., & Hunyadi, B. (2023) · IEEE International Conference on Acoustics, Speech, and Signal Processing Workshops (ICASSPW) · doi:10.1109/icasspw59220.2023.10193615 · Cited by 2
A Khatri-Rao product tensor network for efficient symmetric MIMO Volterra identification
Batselier, K. (2023) · IFAC-PapersOnLine · doi:10.1016/j.ifacol.2023.10.339 · Cited by 1
Bayesian Framework for a MIMO Volterra Tensor Network
Memmel, E., Menzen, C., & Batselier, K. (2023) · IFAC-PapersOnLine · doi:10.1016/j.ifacol.2023.10.341 · Cited by 1
Tensor Network Kalman Filter for LTI Systems
Gedon, D., Piscaer, P., Batselier, K., Smith, C., & Verhaegen, M. (2019) · European Signal Processing Conference (EUSIPCO) · doi:10.23919/eusipco.2019.8902976 · Cited by 1
Local convergence of alternating low‐rank optimization methods with overrelaxation
Oseledets, I. V., Rakhuba, M., & Uschmajew, A. (2022) · Numerical Linear Algebra with Applications · doi:10.1002/nla.2459 · Cited by 1
Self-citationViability of Tensor Train Methods for Geophysical Fluid DynamicsPreprint
Lilly, J. R., DeSantis, D., & Petersen, M. R. (2026) · SSRN Electronic Journal · doi:10.2139/ssrn.6212054 · Cited by 1
Numerical Approximation of Poisson Problems in Long Domains
Chipot, M. M., Hackbusch, W., Sauter, S. A., & Veit, A. D. (2021) · Vietnam Journal of Mathematics · doi:10.1007/s10013-021-00512-9
Quantized CP approximation and sparse tensor interpolation of function‐generated data
Khoromskij, B. N., Kumar, N. K., & Schneider, J. (2019) · Numerical Linear Algebra with Applications · doi:10.1002/nla.2262
Numerical Tensor Techniques for Multidimensional Convolution Products
Hackbusch, W. (2018) · Vietnam Journal of Mathematics · doi:10.1007/s10013-018-0300-4
Fast Low-Rank Solution of the Multidimensional Hyperbolic Problems
Zhong, Z., Wang, S., & Wang, K. (2018) · Computational Mathematics and Modeling · doi:10.1007/s10598-018-9414-5
Miculescu, D., & Karaman, S. (2018) · Proceedings of the ... American Control Conference/Proceedings of the American Control Conference · doi:10.23919/acc.2018.8431657
Tensor-Train-based Algorithms for Aggregate State Estimation of Swarms with Interacting Agents
Miculescu, D., & Karaman, S. (2020) · Proceedings of the ... American Control Conference/Proceedings of the American Control Conference · doi:10.23919/acc45564.2020.9147339
Zhang, L., Zhang, G., & Liang, Z. (2022) · Computers & Mathematics with Applications · doi:10.1016/j.camwa.2022.01.003
Baray, M., Levadoux, D., & Poirier, J.-R. (2022) · IEEE Transactions on Antennas and Propagation · doi:10.1109/tap.2022.3164182
Solving phase‐field fracture problems in the tensor train format
Risthaus, L., & Schneider, M. (2023) · PAMM · doi:10.1002/pamm.202200098
Tindall, J., & Fishman, M. (2023) · SciPost Physics · doi:10.21468/scipost.report.7931
Tindall, J., & Fishman, M. (2023) · SciPost Physics · doi:10.21468/scipost.report.7878
Tensor Train Optimization of Parameterized Quantum Circuits
Paradezhenko, G. V., Pervishko, A. A., & Yudin, D. (2023) · Journal of Experimental and Theoretical Physics Letters · doi:10.1134/s0021364023603056
Einstein Product-Based Dynamical Systems
Chen, C. (2024) · Synthesis lectures on mathematics and statistics · doi:10.1007/978-3-031-54505-4_3
Chen, C. (2024) · Synthesis lectures on mathematics and statistics · doi:10.1007/978-3-031-54505-4_1
Tensor Train Quantum State Tomography Using Compressed Sensing
Sofi, S. S., Vermeylen, C., & Lathauwer, L. D. (2025) · European Signal Processing Conference (EUSIPCO) · doi:10.23919/eusipco63237.2025.11226412
Optimization on the extended tensor-train manifold with shared factors
Molozhavenko, A., & Rakhuba, M. (2026) · Computational and Applied Mathematics · doi:10.1007/s40314-025-03605-0
Provable low-rank tensor-train approximations in the inverse of large-scale structured matrices
Xiao, C., Tang, K., & Zhu, Z. (2026) · Mathematics of Computation · doi:10.1090/mcom/4184
Koopman and Transfer Operator Techniques from the Perspective of Quantum Theory
Giannakis, D., & Montgomery, M. (2025) · Operator Theory · doi:10.1007/978-3-0348-0692-3_125-1
Stable full-field simulation of a multiscale elliptic equation by means of Quantized Tensor TrainsPreprint
Josien, M., Hachimi, A. E., & Ramière, I. (2026) · SSRN Electronic Journal · doi:10.2139/ssrn.6816125
Anonymous, & Richter, M. (2026) · Physical review. B./Physical review. B · doi:10.1103/mmwh-bjpn
Koopman and Transfer Operator Techniques from the Perspective of Quantum Theory
Giannakis, D., & Montgomery, M. (2026) · Operator Theory · doi:10.1007/978-3-032-16356-1_125
Viability of tensor train methods for geophysical fluid dynamics
Lilly, J. R., DeSantis, D., & Petersen, M. (2026) · Journal of Computational Physics · doi:10.1016/j.jcp.2026.115465
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
