CIRM - Videos & books Library - High-dimentional classification with deep neural networks: decision boundaries, noise, and margin

Multi angle

Auteurs : Petersen, Philipp (Auteur de la Conférence)
CIRM (Editeur )

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Résumé : we discuss classification problems in high dimension. We study classification problems using three classical notions: complexity of decision boundary, noise, and margin. We demonstrate that under suitable conditions on the decision boundary, classification problems can be very efficiently approximated, even in high dimensions. If a margin condition is assumed, then arbitrary fast approximation rates can be achieved, despite the problem being high-dimensional and discontinuous. We extend the approximation results ta learning results and show close ta optimal learning rates for empirical risk minimization in high dimensional classification.

Keywords : curse of dimensionality; deep neural networks; classification

Codes MSC :
41A25 - Rate of convergence, degree of approximation
41A46 - Approximation by arbitrary nonlinear expressions; widths and entropy
62C20 - Minimax procedures
68T05 - Learning and adaptive systems

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Recanzone, Luca

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https://videos.cirm-math.fr/2024-10-29_Petersen

Informations sur la Rencontre

Nom de la rencontre : SIGMA (Signal, Image, Geometry, Modeling, Approximation) / SIGMA (Signal, Image, Géométrie, Modélisation, Approximation)
Organisateurs de la rencontre : Cohen, Albert ; Digne, Julie ; Fadili, Jalal ; Mula, Olga ; Nouy, Anthony
Dates : 28/10/2024 - 01/11/2024
Année de la rencontre : 2024
URL Congrès : https://conferences.cirm-math.fr/3066.html

Données de citation

DOI : 10.24350/CIRM.V.20257603
Citer cette vidéo: Petersen, Philipp (2024). High-dimentional classification with deep neural networks: decision boundaries, noise, and margin. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.20257603
URI : http://dx.doi.org/10.24350/CIRM.V.20257603

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Bibliographie

PETERSEN, Philipp et VOIGTLAENDER, Felix. Optimal approximation of piecewise smooth functions using deep ReLU neural networks. Neural Networks, 2018, vol. 108, p. 296-330. - https://doi.org/10.1016/j.neunet.2018.08.019

CARAGEA, Andrei, PETERSEN, Philipp, et VOIGTLAENDER, Felix. Neural network approximation and estimation of classifiers with classification boundary in a Barron class. The Annals of Applied Probability, 2023, vol. 33, no 4, p. 3039-3079. - https://doi.org/10.1214/22-AAP1884

KIM, Yongdai, OHN, Ilsang, et KIM, Dongha. Fast convergence rates of deep neural networks for classification. Neural Networks, 2021, vol. 138, p. 179-197. - https://doi.org/10.1016/j.neunet.2021.02.012

IMAIZUMI, Masaaki et FUKUMIZU, Kenji. Deep neural networks learn non-smooth functions effectively. In : The 22nd international conference on artificial intelligence and statistics. PMLR, 2019. p. 869-878. - https://proceedings.mlr.press/v89/imaizumi19a.html

LERMA-PINEDA, Andres Felipe, PETERSEN, Philipp, FRIEDER, Simon, et al. Dimension-independent learning rates for high-dimensional classification problems. arXiv preprint arXiv:2409.17991, 2024. - https://doi.org/10.48550/arXiv.2409.17991

GARCIA J., PETERSEN P., Classification problem with Barron regular boundaries and margin condition, ta appear 2024 -

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