Nonexpansive Operators Research Articles

Fixed points of nonexpansive operators and normal structure concerning s-Orlicz convex sets

The concept of normal structure for s-Orlicz convex sets (0<s≤1) is introduced and a class of nonexpansive operators in p-normed spaces (0<p≤1) are considered. By developing the ideas and methods related to convex sets, some new fixed point theorems for nonexpansive operators concerning s-Orlicz convex sets are proved: the weak compactness condition of sets is replaced by the coverage condition of operators.

Designing stable neural networks using convex analysis and ODEs

Motivated by classical work on the numerical integration of ordinary differential equations we present a ResNet-styled neural network architecture that encodes non-expansive (1-Lipschitz) operators, as long as the spectral norms of the weights are appropriately constrained. This is to be contrasted with the ordinary ResNet architecture which, even if the spectral norms of the weights are constrained, has a Lipschitz constant that, in the worst case, grows exponentially with the depth of the network. Further analysis of the proposed architecture shows that the spectral norms of the weights can be further constrained to ensure that the network is an averaged operator, making it a natural candidate for a learned denoiser in Plug-and-Play algorithms. Using a novel adaptive way of enforcing the spectral norm constraints, we show that, even with these constraints, it is possible to train performant networks. The proposed architecture is applied to the problem of adversarially robust image classification, to image denoising, and finally to the inverse problem of deblurring.

Generating operators between Banach spaces

We introduce and study the notion of generating operators as those norm-one operators G:X⟶Y\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G:X\\longrightarrow Y$$\\end{document} such that for every 0<δ<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<\\delta <1$$\\end{document}, the set {x∈X:‖x‖⩽1,‖Gx‖>1-δ}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{x\\in X:\\Vert x\\Vert \\leqslant 1,\\ \\Vert Gx\\Vert >1-\\delta \\}$$\\end{document} generates the unit ball of X by closed convex hull. This class of operators includes isometric embeddings, spear operators (actually, operators with the alternative Daugavet property), and other examples like the natural inclusions of ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document} into c0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c_0$$\\end{document} and of L∞[0,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_\\infty [0,1]$$\\end{document} into L1[0,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1[0,1]$$\\end{document}. We first present a characterization in terms of the adjoint operator, make a discussion on the behaviour of diagonal generating operators on c0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c_0$$\\end{document}-, ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-, and ℓ∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _\\infty $$\\end{document}-sums, and present examples in some classical Banach spaces. Even though rank-one generating operators always attain their norm, there are generating operators, even of rank-two, which do not attain their norm. We discuss when a Banach space can be the domain of a generating operator which does not attain its norm in terms of the behaviour of some spear sets of the dual space. Finally, we study when the set of all generating operators between two Banach spaces X and Y generates all non-expansive operators by closed convex hull. We show that this is the case when X=L1(μ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X=L_1(\\mu )$$\\end{document} and Y has the Radon-Nikodým property with respect to μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}. Therefore, when X=ℓ1(Γ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X=\\ell _1(\\Gamma )$$\\end{document}, this is the case for every target space Y. Conversely, we also show that a real finite-dimensional space X satisfies that generating operators from X to Y generate all non-expansive operators by closed convex hull only in the case that X is an ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-space.

A New Four-Step Iterative Approximation Scheme for Reich-Suzuki-Type Nonexpansive Operators in Banach Spaces

A New Four-Step Iterative Approximation Scheme for Reich-Suzuki-Type Nonexpansive Operators in Banach Spaces

The Aleksandrov problem and the Tingley problem for expansive and nonexpansive operators in p-normed spaces

The Aleksandrov problem and the Tingley problem for expansive and nonexpansive operators in p-normed spaces

Generalized Permutants and Graph GENEOs

This paper is part of a line of research devoted to developing a compositional and geometric theory of Group Equivariant Non-Expansive Operators (GENEOs) for Geometric Deep Learning. It has two objectives. The first objective is to generalize the notions of permutants and permutant measures, originally defined for the identity of a single “perception pair”, to a map between two such pairs. The second and main objective is to extend the application domain of the whole theory, which arose in the set-theoretical and topological environments, to graphs. This is performed using classical methods of mathematical definitions and arguments. The theoretical outcome is that, both in the case of vertex-weighted and edge-weighted graphs, a coherent theory is developed. Several simple examples show what may be hoped from GENEOs and permutants in graph theory and how they can be built. Rather than being a competitor to other methods in Geometric Deep Learning, this theory is proposed as an approach that can be integrated with such methods.

Two Convergence Results for Inexact Orbits of Nonexpansive Operators in Metric Spaces with Graphs

In this work we show that if iterates of a nonexpansive self-mapping of a complete metric with a graph converge uniformly on a subset of the space, then this convergence is stable under the presence of small computational errors.

Frugal and decentralised resolvent splittings defined by nonexpansive operators

Frugal resolvent splittings are a class of fixed point algorithms for finding a zero in the sum of the sum of finitely many set-valued monotone operators, where the fixed point operator uses only vector addition, scalar multiplication and the resolvent of each monotone operator once per iteration. In the literature, the convergence analyses of these schemes are performed in an inefficient, algorithm-by-algorithm basis. In this work, we address this by developing a general framework for frugal resolvent splitting which simultaneously covers and extends several important schemes in the literature. The framework also yields a new resolvent splitting algorithm which is suitable for decentralised implementation on regular networks.

Modified Inertial-Type Subgradient Extragradient Methods for Variational Inequalities and Fixed Points of Finite Bregman Relatively Nonexpansive and Demicontractive Mappings

In this paper, we design two inertial-type subgradient extragradient algorithms with the linear-search process for resolving the two pseudomonotone variational inequality problems (VIPs) of and the common fixed point problem (CFPP) of finite Bregman relatively nonexpansive operators and Bregman relatively demicontractive operators in Banach spaces of both p-uniform convexity and uniform smoothness, which are more general than Hilbert ones. By the aid of suitable restrictions, it is shown that the sequences fabricated by the suggested schemes converge weakly and strongly to a solution of a pair of VIPs with a CFPP constraint, respectively. Additionally, the illustrative instance is furnished to back up the practicability and implementability of the suggested methods. This paper reveals the competitive advantage of the proposed algorithms over the existing algorithms; that is, the existing hybrid projection method for a single VIP with an FPP constraint is extended to develop the modified inertial-type subgradient extragradient method for a pair of VIPs with an CFPP constraint.

A Probabilistic Result on Impulsive Noise Reduction in Topological Data Analysis through Group Equivariant Non-Expansive Operators.

In recent years, group equivariant non-expansive operators (GENEOs) have started to find applications in the fields of Topological Data Analysis and Machine Learning. In this paper we show how these operators can be of use also for the removal of impulsive noise and to increase the stability of TDA in the presence of noisy data. In particular, we prove that GENEOs can control the expected value of the perturbation of persistence diagrams caused by uniformly distributed impulsive noise, when data are represented by L-Lipschitz functions from R to R.

Tikhonov regularized iterative methods for nonlinear problems

ABSTRACT We consider the monotone inclusion problems in real Hilbert spaces. Proximal splitting algorithms are very popular technique to solve it and generally achieve weak convergence under mild assumptions. Researchers assume the strong conditions like strong convexity or strong monotonicity on the considered operators to prove strong convergence of the algorithms. Mann iteration method and normal S-iteration method are popular methods to solve fixed point problems. We propose a new common fixed point algorithm based on normal S-iteration method using Tikhonov regularization to find common fixed point of non-expansive operators and prove strong convergence of the generated sequence to the set of common fixed points without assuming strong convexity and strong monotonicity. Based on proposed fixed point algorithm, we propose a forward–backward-type algorithm and a Douglas–Rachford algorithm in connection with Tikhonov regularization to find the solution of monotone inclusion problems. Further, we consider the complexly structured monotone inclusion problems which are very popular these days. We also propose a strongly convergent forward–backward-type primal–dual algorithm and a Douglas–Rachford-type primal–dual algorithm to solve the monotone inclusion problems. Finally, we conduct a numerical experiment to solve image deblurring problems.

Existence results for nonexpansive multi-valued operators and nonlinear integral inclusions

Existence results for nonexpansive multi-valued operators and nonlinear integral inclusions

On an efficient iterative scheme for a class of generalized nonexpansive operators in Banach spaces

The main purpose of this work is to connect the recently introduced iterative approximation scheme known as [Formula: see text]-iterative scheme with the class of generalized [Formula: see text]-nonexpansive mappings. We consider the setting of a Banach space to prove our weak and strong convergence results. Once again, we prove that the class of [Formula: see text]-nonexpansive mapping includes properly the classes of Suzuki nonexpansive, generalized [Formula: see text]-nonexpansive and as well Reich–Suzuki type nonexpansive mappings by providing a numerical example. Eventually, we prove that [Formula: see text]-iterative scheme of this example is essentially more efficient than the other well-known iterative scheme of the literature. The presented work is new and extends some other well-known results in the iterations theory.

A new inertial algorithm for equilibrium problem and a family of nonexpansive operators in Hilbert space

In this article, we suggest and study a new inertial algorithm combining inertial extrapolation, S‐iteration process and viscosity approximation method for computing a common solution of an equilibrium problem and a family of nonexpansive operators in real Hilbert space. We provide a strong convergence result of the suggested algorithm under some simple assumptions on control sequences. Further, we compare our suggested algorithm with some existing well‐known algorithms by an analytical example. Some prominent recent findings in this field have been improved, generalized, and extended as an outcome of this paper.

Mann Hybrid Deepest-Descent Extragradient Method with Line-Search Process for Hierarchical Variational Inequalities for Countable Nonexpansive Mappings

In real Hilbert spaces, let the CFPP indicate a common fixed-point problem of asymptotically nonexpansive operator and countably many nonexpansive operators, and suppose that the HVI and VIP represent a hierarchical variational inequality and a variational inequality problem, respectively. We put forward Mann hybrid deepest-descent extragradient approach for solving the HVI with the CFPP and VIP constraints. The proposed algorithms are on the basis of Mann’s iterative technique, viscosity approximation method, subgradient extragradient rule with linear-search process, and hybrid deepest-descent rule. Under suitable restrictions, it is shown that the sequences constructed by the algorithms converge strongly to a solution of the HVI with the CFPP and VIP constraints.

Gurariĭ operators are generic

Answering a question of Garbulińska-Wȩgrzyn and Kubiś, we prove that Gurariĭ operators form a dense G_delta -set in the space mathcal B(mathbb G) of all nonexpansive operators on the Gurariĭ space mathbb G, endowed with the strong operator topology. This implies that universal operators on mathbb G form a residual set in mathcal B(mathbb G).

A new paradigm for artificial intelligence based on group equivariant non-expansive operators

The recent frantic surge of machine learning and, more broadly, of artificial intelligence (AI) brings to light old and new open issues, and among them, the so-called eXplainable artificial intelligence (XAI) – AI that humans can understand – as opposed to black-box learning systems where even their designers cannot explain AI decisions. One of the major XAI questions is how to design transparent learning systems that incorporate prior knowledge. These topics are becoming more relevant and pervasive as AI systems become more unfathomable and entangled with human factors. Recently a new paradigm for XAI has been introduced in literature, based on group equivariant non-expansive operators (GENEOs), which are able to inject prior knowledge in a learning system. Hence, the use of GENEOs dramatically reduces the number of unknown parameters to be identified and the size of the related training set, providing both computational advantages and an increased degree of interpretability of the results. Here we will illustrate the main characteristics of GENEOs and the encouraging results already obtained on a couple of industrial case studies.

Random fixed point results for generalized asymptotically nonexpansive random operators

In this paper, we define an implicit random iterative process with errors for three finite families of generalized asymptotically nonexpansive random operators. We also prove some convergence theorems using this iteration method in separable Banach spaces.

On the finite representation of linear group equivariant operators via permutant measures

Recent advances in machine learning have highlighted the importance of using group equivariant non-expansive operators for building neural networks in a more transparent and interpretable way. An operator is called equivariant with respect to a group if the action of the group commutes with the operator. Group equivariant non-expansive operators can be seen as multi-level components that can be joined and connected in order to form neural networks by applying the operations of chaining, convex combination and direct product. In this paper we prove that each linear G-equivariant non-expansive operator (GENEO) can be produced by a weighted summation associated with a suitable permutant measure, provided that the group G transitively acts on a finite signal domain. This result is based on the Birkhoff–von Neumann decomposition of doubly stochastic matrices and some well known facts in group theory. Our theorem makes available a new method to build all linear GENEOs with respect to a transitively acting group in the finite setting. This work is part of the research devoted to develop a good mathematical theory of GENEOs, seen as relevant components in machine learning.

Reaching Takahashi-type nonexpansive operators via modular structures with application to best proximity point

Reaching Takahashi-type nonexpansive operators via modular structures with application to best proximity point