Enforcing Idempotency in Neural Networks

Nikolaj Banke Jensen, Jamie Vicary
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:27070-27090, 2025.

Abstract

In this work, we propose a new architecture-agnostic method for training idempotent neural networks. An idempotent operator satisfies $f(x) = f(f(x))$, meaning it can be applied iteratively with no effect beyond the first application. Some neural networks used in data transformation tasks, such as image generation and augmentation, can represent non-linear idempotent projections. Using methods from perturbation theory we derive the recurrence relation ${\mathbf{K}’ \leftarrow 3\mathbf{K}^2 - 2\mathbf{K}^3}$ for iteratively projecting a real-valued matrix $\mathbf{K}$ onto the manifold of idempotent matrices. Our analysis shows that for linear, single-layer MLP networks this projection 1) has idempotent fixed points, and 2) is attracting only around idempotent points. We give an extension to non-linear networks by considering our approach as a substitution of the gradient for the canonical loss function, achieving an architecture-agnostic training scheme. We provide experimental results for MLP- and CNN-based architectures with significant improvement in idempotent error over the canonical gradient-based approach. Finally, we demonstrate practical applications of the method as we train a generative network successfully using only a simple reconstruction loss paired with our method.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-jensen25a, title = {Enforcing Idempotency in Neural Networks}, author = {Jensen, Nikolaj Banke and Vicary, Jamie}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {27070--27090}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/jensen25a/jensen25a.pdf}, url = {https://proceedings.mlr.press/v267/jensen25a.html}, abstract = {In this work, we propose a new architecture-agnostic method for training idempotent neural networks. An idempotent operator satisfies $f(x) = f(f(x))$, meaning it can be applied iteratively with no effect beyond the first application. Some neural networks used in data transformation tasks, such as image generation and augmentation, can represent non-linear idempotent projections. Using methods from perturbation theory we derive the recurrence relation ${\mathbf{K}’ \leftarrow 3\mathbf{K}^2 - 2\mathbf{K}^3}$ for iteratively projecting a real-valued matrix $\mathbf{K}$ onto the manifold of idempotent matrices. Our analysis shows that for linear, single-layer MLP networks this projection 1) has idempotent fixed points, and 2) is attracting only around idempotent points. We give an extension to non-linear networks by considering our approach as a substitution of the gradient for the canonical loss function, achieving an architecture-agnostic training scheme. We provide experimental results for MLP- and CNN-based architectures with significant improvement in idempotent error over the canonical gradient-based approach. Finally, we demonstrate practical applications of the method as we train a generative network successfully using only a simple reconstruction loss paired with our method.} }
Endnote
%0 Conference Paper %T Enforcing Idempotency in Neural Networks %A Nikolaj Banke Jensen %A Jamie Vicary %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-jensen25a %I PMLR %P 27070--27090 %U https://proceedings.mlr.press/v267/jensen25a.html %V 267 %X In this work, we propose a new architecture-agnostic method for training idempotent neural networks. An idempotent operator satisfies $f(x) = f(f(x))$, meaning it can be applied iteratively with no effect beyond the first application. Some neural networks used in data transformation tasks, such as image generation and augmentation, can represent non-linear idempotent projections. Using methods from perturbation theory we derive the recurrence relation ${\mathbf{K}’ \leftarrow 3\mathbf{K}^2 - 2\mathbf{K}^3}$ for iteratively projecting a real-valued matrix $\mathbf{K}$ onto the manifold of idempotent matrices. Our analysis shows that for linear, single-layer MLP networks this projection 1) has idempotent fixed points, and 2) is attracting only around idempotent points. We give an extension to non-linear networks by considering our approach as a substitution of the gradient for the canonical loss function, achieving an architecture-agnostic training scheme. We provide experimental results for MLP- and CNN-based architectures with significant improvement in idempotent error over the canonical gradient-based approach. Finally, we demonstrate practical applications of the method as we train a generative network successfully using only a simple reconstruction loss paired with our method.
APA
Jensen, N.B. & Vicary, J.. (2025). Enforcing Idempotency in Neural Networks. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:27070-27090 Available from https://proceedings.mlr.press/v267/jensen25a.html.

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