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Approximation and learning with tree tensor networks - lecture 1

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Auteurs : Nouy, Anthony (Auteur de la conférence)
CIRM (Editeur )

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Résumé : 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...

Mots-Clés : tensors; tensor networks; high dimension; approximation; learning; algorithms

Codes MSC :
15A69 - Multilinear algebra, tensor products

Ressources complémentaires :
http://smai.emath.fr/cemracs/cemracs21/data/presentation-speakers/nouy.pdf

    Informations sur la Vidéo

    Réalisateur : Hennenfent, Guillaume
    Langue : Anglais
    Date de Publication : 16/08/2021
    Date de Captation : 20/07/2021
    Sous Collection : Research School
    Catégorie arXiv : Numerical Analysis
    Domaine(s) : Analyse Numérique & Calcul Formel
    Format : MP4 (.mp4) - HD
    Durée : 02:04:45
    Audience : Chercheurs
    Download : https://videos.cirm-math.fr/2021-07-20_Nouy_1.mp4

Informations sur la Rencontre

Nom de la Rencontre : 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
Organisateurs de la Rencontre : Ehrlacher, Virginie ; Lombardi, Damiano ; Mula Hernandez, Olga ; Nobile, Fabio ; Taddei, Tommaso
Dates : 19/07/2021 - 23/07/2021
Année de la rencontre : 2021
URL de la Rencontre : https://conferences.cirm-math.fr/2412.html

Données de citation

DOI : 10.24350/CIRM.V.19780003
Citer cette vidéo: Nouy, Anthony (2021). Approximation and learning with tree tensor networks - lecture 1. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19780003
URI : http://dx.doi.org/10.24350/CIRM.V.19780003

Voir Aussi

Bibliographie

  • 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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