Generalized Inversion of Nonlinear Operators

IF 1.3 4区 数学 Q4 COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE
Eyal Gofer, Guy Gilboa
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引用次数: 0

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.

Abstract Image

非线性算子的广义反演
算子反演是数据处理中的一个基本概念。线性算子的逆运算已得到深入研究,并有成熟的理论支持。当逆运算不存在或不唯一时,就会使用广义逆运算。最著名的是 Moore-Penrose 反演,它被广泛应用于物理学、统计学和工程学的各个领域。这项工作研究非线性算子的广义反演。首先,我们在摩尔-彭罗斯公理的指导下,大致探讨了广义反演所需的性质。我们定义了一般集合的概念,然后定义了规范空间的细化概念,称为伪逆。我们提出了伪逆存在性和唯一性的条件,并建立了研究其性质的理论结果,如连续性、其对算子组合和投影算子的价值等。我们给出了一些著名的非可逆非线性算子(如硬或软阈值和 ReLU)的伪逆分析表达式。我们分析了一个神经层,并讨论了与小波阈值的关系。接下来,我们研究了具有相等域和范围的算子的 Drazin 逆和松弛。我们介绍了反演可表示为算子正向应用的线性组合的情况。这种情况出现在具有消失多项式的非线性算子类中,类似于矩阵的最小多项式或特征多项式。使用前向应用进行反演可能有助于开发新的高效算法,以近似对复杂非线性算子进行广义反演。
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来源期刊
Journal of Mathematical Imaging and Vision
Journal of Mathematical Imaging and Vision 工程技术-计算机:人工智能
CiteScore
4.30
自引率
5.00%
发文量
70
审稿时长
3.3 months
期刊介绍: The Journal of Mathematical Imaging and Vision is a technical journal publishing important new developments in mathematical imaging. The journal publishes research articles, invited papers, and expository articles. Current developments in new image processing hardware, the advent of multisensor data fusion, and rapid advances in vision research have led to an explosive growth in the interdisciplinary field of imaging science. This growth has resulted in the development of highly sophisticated mathematical models and theories. The journal emphasizes the role of mathematics as a rigorous basis for imaging science. This provides a sound alternative to present journals in this area. Contributions are judged on the basis of mathematical content. Articles may be physically speculative but need to be mathematically sound. Emphasis is placed on innovative or established mathematical techniques applied to vision and imaging problems in a novel way, as well as new developments and problems in mathematics arising from these applications. The scope of the journal includes: computational models of vision; imaging algebra and mathematical morphology mathematical methods in reconstruction, compactification, and coding filter theory probabilistic, statistical, geometric, topological, and fractal techniques and models in imaging science inverse optics wave theory. Specific application areas of interest include, but are not limited to: all aspects of image formation and representation medical, biological, industrial, geophysical, astronomical and military imaging image analysis and image understanding parallel and distributed computing computer vision architecture design.
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