Theoretical Foundations for Pseudo-Inversion of Nonlinear Operators

The Moore-Penrose inverse is widely used in physics, statistics, and various fields of engineering. It captures well the notion of inversion of linear operators in the case of overcomplete data. In data science, nonlinear operators are extensively used. In this paper we characterize the fundamental properties of a pseudo-inverse (PI) for nonlinear operators. The concept is defined broadly. First for general sets, and then a refinement for normed spaces. The PI for normed spaces yields the Moore-Penrose inverse when the operator is a matrix. We present conditions for existence and uniqueness of a PI 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 PI of some well-known, non-invertible, nonlinear operators, such as hard- or soft-thresholding and ReLU. Finally, we analyze a neural layer and discuss relations to wavelet thresholding.