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.

翻译：Moore-Penrose伪逆（PInv）是线性系统的基本求解方法。本文提出了一种对PInv的自然推广，将其扩展至一般非线性领域，特别是神经网络。我们引入满射伪可逆神经网络（SPNN），这是一类显式设计以允许可处理的非线性PInv的架构。所提出的非线性PInv及其在SPNN中的实现满足基本几何特性。其中一个特性是零空间投影或“反向投影”$x' = x + A^\dagger(y-Ax)$，它将样本$x$移动到满足$Ax=y$的最接近的一致状态$x'$。我们形式化非线性反向投影（NLBP），该方法通过我们定义的PInv保证非线性映射$f(x)=y$具有相同的一致性约束。我们利用SPNN扩展零样本逆问题的适用范围。基于扩散的零空间投影通过利用闭式反向投影，彻底改变了线性逆问题的零样本求解。我们将该方法推广至非线性退化问题。此处“退化”被广义地理解为包括任何非线性信息损失，涵盖从光学畸变到分类等语义抽象。该方法能够实现复杂退化的零样本逆推，并允许在无需重新训练扩散先验的情况下对生成输出进行精确的语义控制。