Generalized Inversion of Nonlinear Operators (2111.10755v3)

Abstract: 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.

• The paper introduces a robust framework for constructing generalized pseudo-inverses for nonlinear operators by extending the first two Moore-Penrose axioms.
• It leverages normed spaces and best approximate solutions to ensure existence and uniqueness of pseudo-inverses in continuous settings.
• Applications in neural networks and wavelet thresholding highlight the method’s potential for computational efficiency in complex operator inversion.

Summary of "Generalized Inversion of Nonlinear Operators"

The paper "Generalized Inversion of Nonlinear Operators" explores the intricate concept of inverting nonlinear operators. While the Moore-Penrose inverse provides a robust framework for linear operators, the extension of this concept to nonlinear cases remains underexplored yet essential for fields such as data science, particularly in machine learning and image processing.

Key Contributions

1. Generalized Inversion Framework: The authors develop a generalized framework for inverting nonlinear operators by extending the first two Moore-Penrose (MP) axioms. They provide a systematic method to construct $\{1,2\}$-inverses and explore their properties, emphasizing the need for distinct approaches from the linear case due to the non-applicability of adjoint operations.
2. Pseudo-Inverses in Normed Spaces: By leveraging normed spaces, the authors propose a robust definition of pseudo-inverses for nonlinear operators. This definition adheres closely to the linear domain's MP axioms, utilizing best approximate solutions (BAS) criteria and MP2. This approach necessitates a nuanced application of optimization principles.
3. Existence and Uniqueness: Several theoretical results are presented regarding the existence and uniqueness of pseudo-inverses, especially under continuous operator scenarios or when operating over compact sets. Notably, the authors highlight scenarios where pseudo-inverses are expressible using straightforward conditions.
4. Neural Networks and Wavelet Thresholding: The applicability of the theory is demonstrated through practical examples, such as neural network layers and wavelet thresholding, where pseudo-inverses are analytically derived, showcasing the framework's versatility in encoding complex operations into simpler forms.
5. Polynomial Representations: The paper explores expressing generalized inverses through polynomials of forward applications, akin to leveraging the Cayley-Hamilton theorem in nonlinear settings. This aspect points to potential computational efficiencies in inverting complex operators via known forward operations.
6. Drazin and Left-Drazin Inverses: The paper extends to specific scenarios—invertible nonlinear operators—introducing the Drazin and Left-Drazin inverses which aim to respect the operator's algebraic properties and capture a broader class than the more restricted MP inverse. These inverses, defined by polynomial expressions, offer theoretical insights into the operator's structural characteristics.

Implications and Future Directions

This research injects significant insights into the mathematical treatment of inverse problems, offering computational strategies that could enhance nonlinear operator handling across computational fields. By laying the groundwork for robust inverse definitions beyond linear paradigms, this work presents a strategic leap toward more intricate data science applications.

Looking forward, the implications of these developments suggest various future research avenues, including refining the computational algorithms for approximating nonlinear inverses and exploring applications within emerging areas of artificial intelligence. The concepts of vanishing polynomials and the efficient usage of forward operator applications hold potential for new algorithmic designs, particularly in computational learning systems.

Conclusion

The paper presents a comprehensive theoretical contribution to the understanding and calculation of generalized inverses for nonlinear operators. By bridging previous gaps and introducing new perspectives, it opens a pathway for both theoretical exploration and practical applications in diverse scientific domains.