Inversion of operators is a fundamental concept in data processing. Inversion of linear operators is well studied, supported by established theory. When an inverse either does not exist or is not unique, generalized inverses are used. Most notable is the Moore-Penrose inverse, widely used in physics, statistics, and various fields of engineering. This work investigates generalized inversion of nonlinear operators. We first address broadly the desired properties of generalized inverses, guided by the Moore-Penrose axioms. We define the notion for general sets, and then a refinement, termed pseudo-inverse, for normed spaces. We present conditions for existence and uniqueness of a pseudo-inverse and establish theoretical results investigating its properties, such as continuity, its value for operator compositions and projection operators, and others. Analytic expressions are given for the pseudo-inverse of some well-known, non-invertible, nonlinear operators, such as hard- or soft-thresholding and ReLU. We analyze a neural layer and discuss relations to wavelet thresholding. Next, the Drazin inverse, and a relaxation, are investigated for operators with equal domain and range. We present scenarios where inversion is expressible as a linear combination of forward applications of the operator. Such scenarios arise for classes of nonlinear operators with vanishing polynomials, similar to the minimal or characteristic polynomials for matrices. Inversion using forward applications may facilitate the development of new efficient algorithms for approximating generalized inversion of complex nonlinear operators.

翻译：算子求逆是数据处理中的基本概念。线性算子的求逆已有成熟理论支持，而广义逆则用于处理逆不存在或不唯一的情形。其中最著名的是Moore-Penrose逆，广泛应用于物理学、统计学及各类工程领域。本研究探讨非线性算子的广义求逆问题。我们首先以Moore-Penrose公理为指导，系统阐述广义逆的理想性质。我们在一般集合上定义该概念，进而针对赋范空间提出其精细化形式——伪逆。给出了伪逆存在唯一性的条件，并建立关于其性质（如连续性、算子复合与投影算子的取值等）的理论结果。针对硬阈值、软阈值及ReLU等常见不可逆非线性算子，给出了伪逆的解析表达式。我们分析了神经网络的某一层，并讨论了与小波阈值法的关联。随后，研究了定义域与值域相同的算子的Drazin逆及其松弛形式。论文提出了一种场景，其中求逆可表示为算子正向应用的线性组合。此类场景出现于具有零化多项式的非线性算子族，类似于矩阵的最小多项式或特征多项式。利用正向应用来求逆，有助于开发逼近复杂非线性算子广义逆的新高效算法。