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Approximation and learning with tree tensor networks - lecture 2
Multi angle
Authors : Nouy, Anthony (Author of the conference)
CIRM (Publisher )
15A69 - Multilinear algebra, tensor products
Additional resources :
http://smai.emath.fr/cemracs/cemracs21/data/presentation-speakers/nouy.pdf
CIRM (Publisher )
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Abstract :
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
MSC Codes : 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
Additional resources :
http://smai.emath.fr/cemracs/cemracs21/data/presentation-speakers/nouy.pdf
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 16/08/2021 Conference Date : 20/07/2021 Subseries : Research School arXiv category : Numerical Analysis Mathematical Area(s) : Numerical Analysis & Scientific Computing Format : MP4 (.mp4) - HD Video Time : 02:07:48 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2021-07-20_Nouy_2.mp4 
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Information on the Event
Event Title : 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 dimensionEvent Organizers : Ehrlacher, Virginie ; Lombardi, Damiano ; Mula Hernandez, Olga ; Nobile, Fabio ; Taddei, Tommaso Dates : 19/07/2021 - 23/07/2021 Event Year : 2021 Event URL : https://conferences.cirm-math.fr/2412.html
Citation Data
DOI : 10.24350/CIRM.V.19780403Cite this video as: Nouy, Anthony (2021). Approximation and learning with tree tensor networks - lecture 2. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19780403 URI : http://dx.doi.org/10.24350/CIRM.V.19780403 |
See Also
- [Multi angle] The tensor Harish-Chandra-Itzykson-Zuber integral / Author of the conference Gurău, Răzvan.
- [Multi angle] Tensor PCA / Author of the conference Ben Arous, Gérard.
- [Multi angle] Bayesian data assimilation and filtering - lecture 2 / Author of the conference Schillings, Claudia.
- [Multi angle] How to compute transition times? / Author of the conference Lelièvre, Tony.
- [Multi angle] Coupling models instead of coupling codes: two toy examples / Author of the conference Lafitte, Olivier.
- [Virtualconference] Statistical theory for deep neural networks - lecture 2 / Author of the conference Schmidt-Hieber, Johannes.
- [Virtualconference] Statistical theory for deep neural networks - lecture 1 / Author of the conference Schmidt-Hieber, Johannes.
- [Multi angle] Approximation and learning with tree tensor networks - lecture 1 / Author of the conference Nouy, Anthony.
- [Virtualconference] Bayesian methods for inverse problems - lecture 2 / Author of the conference Dashti, Masoumeh.
- [Multi angle] Theory of approximation of high-dimensional functions - lecture 2 / Author of the conference Cohen, Albert.
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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