Who Said Neural Networks Aren’t Linear?

Nimrod Berman, Assaf Hallak, Assaf Shocher
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:7683-7706, 2026.

Abstract

Neural networks are famously nonlinear. However, linearity is defined relative to a pair of vector spaces, $f:\mathcal{X}\to\mathcal{Y}$. Leveraging the algebraic concept of transport of structure, we propose a method to explicitly identify non-standard vector spaces where a neural network acts as a linear operator. When sandwiching a linear operator $A$ between two invertible neural networks, $f(x)=g_y^{-1}(A g_x(x))$, the corresponding vector spaces $\mathcal{X}$ and $\mathcal{Y}$ are induced by newly defined addition and scaling actions derived from $g_x$ and $g_y$. We term this kind of architecture a Linearizer. This framework makes the entire arsenal of linear algebra, including SVD, pseudo-inverse, orthogonal projection and more, applicable to nonlinear mappings. Furthermore, we show that the composition of two Linearizers that share a neural network is also a Linearizer. We leverage this property and demonstrate that training diffusion models using our architecture makes the hundreds of sampling steps collapse into a single step. We further utilize our framework to enforce idempotency (i.e. $f(f(x))=f(x)$) on networks leading to a globally projective generative model and to demonstrate modular style transfer.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-berman26c, title = {Who Said Neural Networks Aren’t Linear?}, author = {Berman, Nimrod and Hallak, Assaf and Shocher, Assaf}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {7683--7706}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/berman26c/berman26c.pdf}, url = {https://proceedings.mlr.press/v306/berman26c.html}, abstract = {Neural networks are famously nonlinear. However, linearity is defined relative to a pair of vector spaces, $f:\mathcal{X}\to\mathcal{Y}$. Leveraging the algebraic concept of transport of structure, we propose a method to explicitly identify non-standard vector spaces where a neural network acts as a linear operator. When sandwiching a linear operator $A$ between two invertible neural networks, $f(x)=g_y^{-1}(A g_x(x))$, the corresponding vector spaces $\mathcal{X}$ and $\mathcal{Y}$ are induced by newly defined addition and scaling actions derived from $g_x$ and $g_y$. We term this kind of architecture a Linearizer. This framework makes the entire arsenal of linear algebra, including SVD, pseudo-inverse, orthogonal projection and more, applicable to nonlinear mappings. Furthermore, we show that the composition of two Linearizers that share a neural network is also a Linearizer. We leverage this property and demonstrate that training diffusion models using our architecture makes the hundreds of sampling steps collapse into a single step. We further utilize our framework to enforce idempotency (i.e. $f(f(x))=f(x)$) on networks leading to a globally projective generative model and to demonstrate modular style transfer.} }
Endnote
%0 Conference Paper %T Who Said Neural Networks Aren’t Linear? %A Nimrod Berman %A Assaf Hallak %A Assaf Shocher %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-berman26c %I PMLR %P 7683--7706 %U https://proceedings.mlr.press/v306/berman26c.html %V 306 %X Neural networks are famously nonlinear. However, linearity is defined relative to a pair of vector spaces, $f:\mathcal{X}\to\mathcal{Y}$. Leveraging the algebraic concept of transport of structure, we propose a method to explicitly identify non-standard vector spaces where a neural network acts as a linear operator. When sandwiching a linear operator $A$ between two invertible neural networks, $f(x)=g_y^{-1}(A g_x(x))$, the corresponding vector spaces $\mathcal{X}$ and $\mathcal{Y}$ are induced by newly defined addition and scaling actions derived from $g_x$ and $g_y$. We term this kind of architecture a Linearizer. This framework makes the entire arsenal of linear algebra, including SVD, pseudo-inverse, orthogonal projection and more, applicable to nonlinear mappings. Furthermore, we show that the composition of two Linearizers that share a neural network is also a Linearizer. We leverage this property and demonstrate that training diffusion models using our architecture makes the hundreds of sampling steps collapse into a single step. We further utilize our framework to enforce idempotency (i.e. $f(f(x))=f(x)$) on networks leading to a globally projective generative model and to demonstrate modular style transfer.
APA
Berman, N., Hallak, A. & Shocher, A.. (2026). Who Said Neural Networks Aren’t Linear?. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:7683-7706 Available from https://proceedings.mlr.press/v306/berman26c.html.

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