Natural unique non-linear pseudo-inverse beyond surjectivity

Determine a principled definition that yields a unique "natural" non-linear pseudo-inverse for an operator g: X -> Y when a measurement y lies outside the range of g, in the framework where a bijective completion G(x) = [g(x) | q(x)] augments g with auxiliary coordinates q(x); specifically, resolve how to uniquely select among the additional degrees of freedom introduced by G so that a canonical pseudo-inverse is obtained.

Background

The paper defines a non-linear pseudo-inverse for surjective operators using a bijective completion G that augments the output g(x) with auxiliary coordinates q(x), and shows that this construction satisfies the reflexive Penrose identities. The Surjective Pseudo-invertible Neural Network (SPNN) architecture realizes this pseudo-inverse by learning the missing components via an auxiliary network.

However, the approach assumes surjectivity of g. When y lies outside Range(g), the authors note that while their definition remains formally applicable, the bijective completion G has additional degrees of freedom, preventing a unique selection of the pre-image. Establishing a canonical, "natural" choice in this general setting remains unresolved.