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About this paper
Eyal Gofer, Guy Gilboa
Gofer and Gilboa develop a comprehensive theoretical framework for generalized inversion of nonlinear operators, extending concepts like Moore-Penrose and Drazin inverses from linear algebra. The work defines and characterizes various types of nonlinear inverses, outlines conditions for their existence and uniqueness, and derives analytic expressions for common nonlinear functions, addressing a theoretical gap in modern data science.
Key insights
- Nonlinear {1,2}-inverses, while existing, are not unique and can be constructed by selecting a unique pre-image for elements in the operator's image and defining the inverse arbitrarily outside.
- The 'best approximate solution' property is crucial for a strong pseudo-inverse in normed spaces, and unlike the linear case, it does not generally imply the second Moore-Penrose axiom (MP2).
- For nonlinear endofunctions, the Drazin inverse exists if the operator restricted to its sufficiently iterated image becomes a bijection, and a Left-Drazin inverse can be expressed as a polynomial in the operator itself if a 'vanishing polynomial' exists.
Summary by alphaXiv.
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