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High-dimentional classification with deep neural networks: decision boundaries, noise, and margin
Multi angle
Authors : Petersen, Philipp (Author of the conference)
CIRM (Publisher )
41A25 - Rate of convergence, degree of approximation
41A46 - Approximation by arbitrary nonlinear expressions; widths and entropy
62C20 - Minimax procedures
68T05 - Learning and adaptive systems
CIRM (Publisher )
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Abstract :
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
MSC Codes : Keywords : curse of dimensionality; deep neural networks; classification
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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Information on the Video
Film maker : Recanzone, LucaLanguage : English Available date : 22/11/2024 Conference Date : 29/10/2024 Subseries : Research talks arXiv category : Numerical Analysis ; Machine Learning Mathematical Area(s) : Analysis and its Applications ; Computer Science ; Probability & Statistics Format : MP4 (.mp4) - HD Video Time : 00:44:49 Targeted Audience : Researchers ; Graduate Students ; Doctoral Students, Post-Doctoral Students Download : https://videos.cirm-math.fr/2024-10-29_Petersen 
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Information on the Event
Event Title : SIGMA (Signal, Image, Geometry, Modeling, Approximation) / SIGMA (Signal, Image, Géométrie, Modélisation, Approximation)Event Organizers : Cohen, Albert ; Digne, Julie ; Fadili, Jalal ; Mula, Olga ; Nouy, Anthony Dates : 28/10/2024 - 01/11/2024 Event Year : 2024 Event URL : https://conferences.cirm-math.fr/3066.html
Citation Data
DOI : 10.24350/CIRM.V.20257603Cite this video as: 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 |
See Also
- [Multi angle] Hierarchical matrices for 3D Helmholtz problems in multi-patch IgA-BEM setting / Author of the conference Sampoli, Maria-Lucia.
- [Multi angle] Navigating, restructuring and reshaping learned latent spaces / Author of the conference Solomon, Justin.
- [Multi angle] New dynamical low-complexity approximations for the Schrödinger equation / Author of the conference Ehrlacher, Virginie.
- [Multi angle] Barycenters for transport costs / Author of the conference Delon, Julie.
Bibliography
- 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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