Surjective Pseudo-Invertible Neural Networks

The Moore-Penrose Pseudo-inverse (PInv) is the fundamental tool for inverting linear operators. We propose a natural generalization to the non-linear regime and introduce {Surjective Pseudo-invertible Neural Networks (SPNN)}: architectures that admit a tractable non-linear PInv by construction and satisfy the corresponding geometric properties. Building on this, we formalize {Non-Linear Back-Projection (NLBP)}, the non-linear analogue of $x' = x + A^\dagger(y-Ax)$: an update that projects any sample to its closest state consistent with $f(x)=y$. Diffusion-based null-space projection revolutionized zero-shot solving of linear inverse problems via closed-form back-projection; NLBP extends this paradigm to non-linear learned ``degradations'' in the broad sense, spanning semantic mappings such as classification and object detection. The result is zero-shot inversion of complex degradations and precise semantic control over generative outputs without retraining the prior.