Learning representations on Lp hyperspheres: The equivalence of loss functions in a MAP approach

Nicolas Michel, Jean-François BERCHER, Toshihiko Yamasaki
Proceedings of the 4th (2025) and 3rd (2024) NeurIPS Workshops on Symmetry and Geometry in Neural Representations, PMLR 282:392-414, 2026.

Abstract

A common practice when training Deep Neural Networks is to force the learned representations to lie on the standard unit hypersphere, with respect to the $L_2$ norms. Such practice has been shown to improve both the stability and final performances of DNNs in many applications. In this paper, we derive a unified theoretical framework for learning representation on any $L_p$ hyperspheres for classification tasks, based on Maximum A Posteriori (MAP) modeling. Specifically, we give an expression of the probability distribution of multivariate Gaussians projected on any $L_p$ hypersphere and derive the general associated loss function. Additionally, we show that this framework demonstrates the theoretical equivalence of all projections on $L_p$ hyperspheres through the MAP modeling. It also provides a new interpretation of traditional Softmax Cross Entropy with temperature (SCE-$\tau$) loss functions. Experiments on standard computer vision datasets give an empirical validation of the equivalence of projections on $L_p$ unit hyperspheres when using adequate objectives. It also shows that the SCE-$\tau$ on projected representations, with optimally chosen temperature, shows comparable performances.

Cite this Paper

BibTeX
@InProceedings{pmlr-v282-michel26a, title = {Learning representations on Lp hyperspheres: The equivalence of loss functions in a MAP approach}, author = {Michel, Nicolas and BERCHER, Jean-Fran\c{c}ois and Yamasaki, Toshihiko}, booktitle = {Proceedings of the 4th (2025) and 3rd (2024) NeurIPS Workshops on Symmetry and Geometry in Neural Representations}, pages = {392--414}, year = {2026}, editor = {Acosta, Francisco and Azeglio, Simone and Tolooshams, Bahareh and van de Geijn, Chase and Shewmake, Christian and Sanborn, Sophia and Miolane, Nina}, volume = {282}, series = {Proceedings of Machine Learning Research}, month = {14 Dec 2024--07 Dec 2025}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v282/main/assets/michel26a/michel26a.pdf}, url = {https://proceedings.mlr.press/v282/michel26a.html}, abstract = {A common practice when training Deep Neural Networks is to force the learned representations to lie on the standard unit hypersphere, with respect to the $L_2$ norms. Such practice has been shown to improve both the stability and final performances of DNNs in many applications. In this paper, we derive a unified theoretical framework for learning representation on any $L_p$ hyperspheres for classification tasks, based on Maximum A Posteriori (MAP) modeling. Specifically, we give an expression of the probability distribution of multivariate Gaussians projected on any $L_p$ hypersphere and derive the general associated loss function. Additionally, we show that this framework demonstrates the theoretical equivalence of all projections on $L_p$ hyperspheres through the MAP modeling. It also provides a new interpretation of traditional Softmax Cross Entropy with temperature (SCE-$\tau$) loss functions. Experiments on standard computer vision datasets give an empirical validation of the equivalence of projections on $L_p$ unit hyperspheres when using adequate objectives. It also shows that the SCE-$\tau$ on projected representations, with optimally chosen temperature, shows comparable performances.} }
Endnote
%0 Conference Paper %T Learning representations on Lp hyperspheres: The equivalence of loss functions in a MAP approach %A Nicolas Michel %A Jean-François BERCHER %A Toshihiko Yamasaki %B Proceedings of the 4th (2025) and 3rd (2024) NeurIPS Workshops on Symmetry and Geometry in Neural Representations %C Proceedings of Machine Learning Research %D 2026 %E Francisco Acosta %E Simone Azeglio %E Bahareh Tolooshams %E Chase van de Geijn %E Christian Shewmake %E Sophia Sanborn %E Nina Miolane %F pmlr-v282-michel26a %I PMLR %P 392--414 %U https://proceedings.mlr.press/v282/michel26a.html %V 282 %X A common practice when training Deep Neural Networks is to force the learned representations to lie on the standard unit hypersphere, with respect to the $L_2$ norms. Such practice has been shown to improve both the stability and final performances of DNNs in many applications. In this paper, we derive a unified theoretical framework for learning representation on any $L_p$ hyperspheres for classification tasks, based on Maximum A Posteriori (MAP) modeling. Specifically, we give an expression of the probability distribution of multivariate Gaussians projected on any $L_p$ hypersphere and derive the general associated loss function. Additionally, we show that this framework demonstrates the theoretical equivalence of all projections on $L_p$ hyperspheres through the MAP modeling. It also provides a new interpretation of traditional Softmax Cross Entropy with temperature (SCE-$\tau$) loss functions. Experiments on standard computer vision datasets give an empirical validation of the equivalence of projections on $L_p$ unit hyperspheres when using adequate objectives. It also shows that the SCE-$\tau$ on projected representations, with optimally chosen temperature, shows comparable performances.
APA
Michel, N., BERCHER, J. & Yamasaki, T.. (2026). Learning representations on Lp hyperspheres: The equivalence of loss functions in a MAP approach. Proceedings of the 4th (2025) and 3rd (2024) NeurIPS Workshops on Symmetry and Geometry in Neural Representations, in Proceedings of Machine Learning Research 282:392-414 Available from https://proceedings.mlr.press/v282/michel26a.html.

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