Pseudo-Invertible Neural Networks

TL;DR

This paper introduces Surjective Pseudo-invertible Neural Networks (SPNN), a novel architecture that generalizes the Moore-Penrose Pseudo-inverse to nonlinear neural networks, enabling zero-shot inverse problem solving and semantic control in complex degradations.

Contribution

The paper proposes a new class of neural networks, SPNN, that admits a tractable nonlinear pseudo-inverse, extending linear inverse techniques to nonlinear settings for the first time.

Findings

SPNNs satisfy geometric properties like null-space projection.

Non-Linear Back-Projection guarantees consistency in nonlinear mappings.

Extended zero-shot inverse problem solving to nonlinear degradations.

Abstract

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.