Theory

Some references relevant to the algorithms in \(\mathtt{deep\_tensor}\) are collected below.

DIRT Algorithm

  • DIRT algorithm: Cui and Dolgov (2022)
  • TT-IRT (single-layer DIRT) algorithm: Dolgov et al. (2020)

Functional Tensor Trains

  • Functional tensor trains: Bigoni, Engsig-Karup, and Marzouk (2016), Gorodetsky, Karaman, and Marzouk (2019)
  • Extended functional tensor trains: Strössner, Sun, and Kressner (2024)
  • Tensor train decomposition: Oseledets (2011)
  • TT cross-approximation algorithm: Oseledets and Tyrtyshnikov (2010)
  • Maxvol algorithm: Goreinov et al. (2010)

Applications and Extensions of the DIRT Algorithm

  • State and parameter estimation: Zhao and Cui (2024)
  • Optimal experimental design: Koval, Herzog, and Scheichl (2024)
  • Rare event estimation: Cui, Dolgov, and Scheichl (2024)
  • Conditional simulation (amortised inference): Cui, Dolgov, and Zahm (2023)

References

Bigoni, Daniele, Allan P Engsig-Karup, and Youssef M Marzouk. 2016. “Spectral Tensor-Train Decomposition.” SIAM Journal on Scientific Computing 38 (4): A2405–39. https://doi.org/10.1137/15M1036919.
Cui, Tiangang, and Sergey Dolgov. 2022. “Deep Composition of Tensor-Trains Using Squared Inverse Rosenblatt Transports.” Foundations of Computational Mathematics 22 (6): 1863–1922. https://doi.org/10.1007/s10208-021-09537-5.
Cui, Tiangang, Sergey Dolgov, and Robert Scheichl. 2024. “Deep Importance Sampling Using Tensor Trains with Application to a Priori and a Posteriori Rare Events.” SIAM Journal on Scientific Computing 46 (1): C1–29. https://doi.org/10.1137/23M1546981.
Cui, Tiangang, Sergey Dolgov, and Olivier Zahm. 2023. “Scalable Conditional Deep Inverse Rosenblatt Transports Using Tensor Trains and Gradient-Based Dimension Reduction.” Journal of Computational Physics 485: 112103. https://doi.org/10.1016/j.jcp.2023.112103.
Dolgov, Sergey, Karim Anaya-Izquierdo, Colin Fox, and Robert Scheichl. 2020. “Approximation and Sampling of Multivariate Probability Distributions in the Tensor Train Decomposition.” Statistics and Computing 30: 603–25. https://doi.org/10.1007/s11222-019-09910-z.
Goreinov, Sergei A, Ivan V Oseledets, Dimitry V Savostyanov, Eugene E Tyrtyshnikov, and Nikolay L Zamarashkin. 2010. “How to Find a Good Submatrix.” In Matrix Methods: Theory, Algorithms and Applications: Dedicated to the Memory of Gene Golub, 247–56. World Scientific. https://doi.org/10.1142/9789812836021_0015.
Gorodetsky, Alex, Sertac Karaman, and Youssef Marzouk. 2019. “A Continuous Analogue of the Tensor-Train Decomposition.” Computer Methods in Applied Mechanics and Engineering 347: 59–84. https://doi.org/10.1016/j.cma.2018.12.015.
Koval, Karina, Roland Herzog, and Robert Scheichl. 2024. “Tractable Optimal Experimental Design Using Transport Maps.” Inverse Problems 40 (12): 125002. https://doi.org/10.1088/1361-6420/ad8260.
Oseledets, Ivan. 2011. “Tensor-Train Decomposition.” SIAM Journal on Scientific Computing 33 (5): 2295–2317. https://doi.org/10.1137/090752286.
Oseledets, Ivan, and Eugene Tyrtyshnikov. 2010. “TT-Cross Approximation for Multidimensional Arrays.” Linear Algebra and Its Applications 432 (1): 70–88. https://doi.org/10.1016/j.laa.2009.07.024.
Strössner, Christoph, Bonan Sun, and Daniel Kressner. 2024. “Approximation in the Extended Functional Tensor Train Format.” Advances in Computational Mathematics 50 (3): 54. https://doi.org/10.1007/s10444-024-10140-9.
Zhao, Yiran, and Tiangang Cui. 2024. “Tensor-Train Methods for Sequential State and Parameter Learning in State-Space Models.” Journal of Machine Learning Research 25 (244): 1–51. http://jmlr.org/papers/v25/23-0743.html.