Who Said Neural Networks Aren't Linear? (assafshocher.github.io)

The authors introduce Linearizer, a construction that makes many conventionally nonlinear neural mappings linear by changing the underlying vector spaces. Concretely, they parameterize models as f(x) = g_y^{-1}(A g_x(x)), where g_x and g_y are invertible neural networks and A is a linear operator. By redefining vector-space addition and scaling via the invertible maps, f becomes linear with respect to those induced spaces, allowing one to bring the full toolkit of linear algebra to bear on models that appear nonlinear in standard coordinates. This reformulation yields practical and striking consequences. Because the inner map is linear, exact inversion via the pseudoinverse is possible, enabling latent-space interpolation between real images with a single forward pass (one-step flow matching & inversion) and collapsing multi-step diffusion sampling into a single step. Enforcing A^2 = A makes the whole network a global projector, a useful prior for projective generative models, and the modular sandwich structure supports composable style transfer. Technically, the approach requires invertible encoders/decoders and a learnable linear core A, but it opens paths to spectral analysis, exact inversion, and modularity in generative modeling — reframing some nonlinear problems as linear ones in engineered coordinates.