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.
“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]
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 Model, matrices that are well approximated as a sum of n-fold tensor products, and more generally any k-sparse Hamiltonian without evident tensor product structure.
“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]
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Computing Inner Eigenvalues of Matrices in Tensor Train Matrix Format
Mach, T. (2012) · Springer eBooks · doi:10.1007/978-3-642-33134-3_82 · Cited by 12 · see citing works
Kühn, S., & Richter, M. (2020) · Physical review. B./Physical review. B · doi:10.1103/physrevb.101.075302 · Cited by 11 · see citing works
QTT-isogeometric solver in two dimensions
Markeeva, L., Tsybulin, I. V., & Oseledets, I. V. (2020) · Journal of Computational Physics · doi:10.1016/j.jcp.2020.109835 · Cited by 11 · see citing works
Self-citationMulti-Branch Tensor Network Structure for Tensor-Train Discriminant Analysis
Sofuoglu, S. E., & Aviyente, S. (2021) · IEEE Transactions on Image Processing · doi:10.1109/tip.2021.3120871 · Cited by 11 · see citing works
Baranov, V., & Oseledets, I. V. (2015) · The Journal of Chemical Physics · doi:10.1063/1.4935017 · Cited by 11 · see citing works
Self-citationNip, M., Hespanha, J. P., & Khammash, M. (2013) · IEEE Conference on Decision and Control (CDC) · doi:10.1109/cdc.2013.6760167 · Cited by 11 · see citing works
Rakhuba, M. (2021) · SIAM Journal on Scientific Computing · doi:10.1137/19m1280156 · Cited by 10 · see citing works
Quantized Tensor FEM for Multiscale Problems: Diffusion Problems in Two and Three Dimensions
Kazeev, V. A., Oseledets, I. V., Rakhuba, M., & Schwab, C. (2022) · Multiscale Modeling and Simulation · doi:10.1137/20m1341659 · Cited by 10 · see citing works
Self-citationMultigrid Methods for Tensor Structured Markov Chains with Low Rank Approximation
Bolten, M., Kahl, K., & Sokolović, S. (2016) · SIAM Journal on Scientific Computing · doi:10.1137/140994447 · Cited by 10 · see citing works
How Competitive is the ADI for Tensor Structured Equations?
Mach, T., & Saak, J. (2012) · PAMM · doi:10.1002/pamm.201210306 · Cited by 9 · see citing works
Tensor Train Accelerated Solvers for Nonsmooth Rigid Body Dynamics
Corona, E., Gorsich, D., Jayakumar, P., & Veerapaneni, S. (2019) · Applied Mechanics Reviews · doi:10.1115/1.4043324 · Cited by 9 · see citing works
Tensor train based isogeometric analysis for PDE approximation on parameter dependent geometries
Ion, I. G., Loukrezis, D., & Gersem, H. D. (2022) · Computer Methods in Applied Mechanics and Engineering · doi:10.1016/j.cma.2022.115593 · Cited by 8 · see citing works
Zhang, J., Wen, Z., & Zhang, Y. (2016) · Journal of Scientific Computing · doi:10.1007/s10915-016-0255-0 · Cited by 7 · see citing works
- 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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