English
Approximation and learning with tree tensor networks - lecture 1
Multi angle
Auteurs : Nouy, Anthony (Auteur de la Conférence)
CIRM (Editeur )
15A69 - Multilinear algebra, tensor products
Ressources complémentaires :
http://smai.emath.fr/cemracs/cemracs21/data/presentation-speakers/nouy.pdf
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...
Keywords : tensors; tensor networks; high dimension; approximation; learning; algorithms
Codes MSC : 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...
Keywords : tensors; tensor networks; high dimension; approximation; learning; algorithms
15A69 - Multilinear algebra, tensor products
Ressources complémentaires :
http://smai.emath.fr/cemracs/cemracs21/data/presentation-speakers/nouy.pdf
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Informations sur la Vidéo
Réalisateur : Hennenfent, GuillaumeLangue : Anglais Date de publication : 16/08/2021 Date de captation : 20/07/2021 Sous collection : Research School arXiv category : Numerical Analysis Domaine : Numerical Analysis & Scientific Computing Format : MP4 (.mp4) - HD Durée : 02:04:45 Audience : Researchers Download : https://videos.cirm-math.fr/2021-07-20_Nouy_1.mp4 
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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 dimensionOrganisateurs 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 Congrès : https://conferences.cirm-math.fr/2412.html
Données de citation
DOI : 10.24350/CIRM.V.19780003Citer 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
- [Multi angle] Bayesian data assimilation and filtering - lecture 2 / Auteur de la Conférence Schillings, Claudia.
- [Multi angle] How to compute transition times? / Auteur de la Conférence Lelièvre, Tony.
- [Multi angle] Coupling models instead of coupling codes: two toy examples / Auteur de la Conférence Lafitte, Olivier.
- [Virtualconference] Statistical theory for deep neural networks - lecture 2 / Auteur de la Conférence Schmidt-Hieber, Johannes.
- [Virtualconference] Statistical theory for deep neural networks - lecture 1 / Auteur de la Conférence Schmidt-Hieber, Johannes.
- [Multi angle] Approximation and learning with tree tensor networks - lecture 2 / Auteur de la Conférence Nouy, Anthony.
- [Virtualconference] Bayesian methods for inverse problems - lecture 2 / Auteur de la Conférence Dashti, Masoumeh.
- [Multi angle] Theory of approximation of high-dimensional functions - lecture 2 / Auteur de la Conférence Cohen, Albert.
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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