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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]
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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 ›
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