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CEMRACS 2021: Data Assimilation and Model Reduction in High Dimensional Problems / CEMRACS 2021: Assimilation de données et réduction de modèle pour des problêmes en grande dimension

Collection CEMRACS 2021: Data Assimilation and Model Reduction in High Dimensional Problems / CEMRACS 2021: Assimilation de données et réduction de modèle pour des problêmes en grande dimension

Organizer(s) Ehrlacher, Virginie ; Lombardi, Damiano ; Mula Hernandez, Olga ; Nobile, Fabio ; Taddei, Tommaso
Date(s) 19/07/2021 - 23/07/2021
linked URL https://conferences.cirm-math.fr/2412.html
00:00:00 / 00:00:00
3 9

Approximation and learning with tree tensor networks - lecture 1

By Anthony Nouy

Many problems in computational and data science require the approximation of high-dimensional functions. Examples of such problems can be found in physics, stochastic analysis, statistics, machine learning or uncertainty quantification. The approximation of high-dimensional functions requires the introduction of approximation tools that capture specific features of these functions. In this lecture, we will give an introduction to tree tensor networks (TNs), or tree-based tensor formats. In part I, we will present some general notions about tensors, tensor ranks, tensor formats and tensorization of vectors and functions. Then in part II, we will introduce approximation tools based on TNs, present results on the approximation power (or expressivity) of TNs and discuss the role of tensorization and architecture of TNs. Finally in part III, we will present algorithms for computing with TNs. This includes algorithms for tensor truncation, for the solution of optimization problems, for learning functions from samples...

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Citation data

  • DOI 10.24350/CIRM.V.19780003
  • Cite this video Nouy, Anthony (20/07/2021). Approximation and learning with tree tensor networks - lecture 1. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19780003
  • URL https://dx.doi.org/10.24350/CIRM.V.19780003

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Bibliography

  • HACKBUSCH, Wolfgang. Tensor spaces and numerical tensor calculus. Berlin : Springer, 2012. - https://doi.org/10.1007/978-3-030-35554-8
  • NOUY, Anthony. Low-rank methods for high-dimensional approximation and model order reduction. Model reduction and approximation, P. Benner, A. Cohen, M. Ohlberger, and K. Willcox, eds., SIAM, Philadelphia, PA, 2017, p. 171-226.
  • ALI, Mazen et NOUY, Anthony. Approximation with tensor networks. Part i: Approximation spaces. arXiv preprint arXiv:2007.00118, 2020. - https://arxiv.org/abs/2007.00118
  • ALI, Mazen et NOUY, Anthony. Approximation with tensor networks. Part ii: Approximation rates for smoothness classes. arXiv preprint arXiv:2007.00128, 2020. - https://arxiv.org/abs/2007.00128
  • ALI, Mazen et NOUY, Anthony. Approximation with Tensor Networks. Part III: Multivariate Approximation. arXiv preprint arXiv:2101.11932, 2021. - https://arxiv.org/abs/2101.11932
  • MICHEL, Bertrand et NOUY, Anthony. Learning with tree tensor networks: complexity estimates and model selection. arXiv preprint arXiv:2007.01165, 2020. - https://arxiv.org/abs/2007.01165
  • NOUY, Anthony. Higher-order principal component analysis for the approximation of tensors in tree-based low-rank formats. Numerische Mathematik, 2019, vol. 141, no 3, p. 743-789. - https://doi.org/10.1007/s00211-018-1017-8
  • HABERSTICH, Cécile, NOUY, Anthony, et PERRIN, Guillaume. Active learning of tree tensor networks using optimal least-squares. arXiv preprint arXiv:2104.13436, 2021. - https://arxiv.org/abs/2104.13436

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