The Moore-Penrose Pseudo-inverse (PInv) serves as the fundamental solution for linear systems. In this paper, we propose a natural generalization of PInv to the nonlinear regime in general and to neural networks in particular. We introduce Surjective Pseudo-invertible Neural Networks (SPNN), a class of architectures explicitly designed to admit a tractable non-linear PInv. The proposed non-linear PInv and its implementation in SPNN satisfy fundamental geometric properties. One such property is null-space projection or "Back-Projection", $x' = x + A^\dagger(y-Ax)$, which moves a sample $x$ to its closest consistent state $x'$ satisfying $Ax=y$. We formalize Non-Linear Back-Projection (NLBP), a method that guarantees the same consistency constraint for non-linear mappings $f(x)=y$ via our defined PInv. We leverage SPNNs to expand the scope of zero-shot inverse problems. Diffusion-based null-space projection has revolutionized zero-shot solving for linear inverse problems by exploiting closed-form back-projection. We extend this method to non-linear degradations. Here, "degradation" is broadly generalized to include any non-linear loss of information, spanning from optical distortions to semantic abstractions like classification. This approach enables zero-shot inversion of complex degradations and allows precise semantic control over generative outputs without retraining the diffusion prior.

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