Hilbert Space Setting Research Articles

A New Accelerated Fixed-Point Algorithm for Classification and Convex Minimization Problems in Hilbert Spaces with Directed Graphs

A new accelerated algorithm for approximating the common fixed points of a countable family of G-nonexpansive mappings is proposed, and the weak convergence theorem based on our main results is established in the setting of Hilbert spaces with a symmetric directed graph G. As an application, we apply our results for solving classification and convex minimization problems. We also apply our proposed algorithm to estimate the weight connecting the hidden layer and output layer in a regularized extreme learning machine. For numerical experiments, the proposed algorithm gives a higher performance of accuracy of the testing set than that of FISTA-S, FISTA, and nAGA.

Newton-Type Inertial Algorithms for Solving Monotone Equations Governed by Sums of Potential and Nonpotential Operators

In a Hilbert space setting, this paper is devoted to the study of a class of first-order algorithms which aim to solve structured monotone equations involving the sum of potential and nonpotential operators. Precisely, we are looking for the zeros of an operator $$A= \nabla f +B $$ , where $$\nabla f$$ is the gradient of a differentiable convex function f, and B is a nonpotential monotone and cocoercive operator. This study is based on the inertial autonomous dynamic previously studied by the authors, which involves dampings controlled respectively by the Hessian of f, and by a Newton-type correction term attached to B. These geometric dampings attenuate the oscillations which occur with the inertial methods with viscous damping. Temporal discretization of this dynamic provides fully splitted proximal-gradient algorithms. Their convergence properties are proven using Lyapunov analysis. These results open the door to the design of first-order accelerated algorithms in numerical optimization taking into account the specific properties of potential and nonpotential terms.

A Self-Adaptive Step Size Algorithm for Solving Variational Inequalities with the Split Feasibility Problem with Multiple Output Sets Constraints

In this paper, we investigate the problem of solving strongly monotone variational inequality problems over the solution set of the split feasibility problem with multiple output sets in real Hilbert spaces. The strong convergence of the proposed algorithm is proved without knowing any information of the Lipschitz and strongly monotone constants of the mapping. In addition, the implementation of the algorithm does not require the computation or estimation of the norms of the given bounded linear operators. Special cases are considered. Finally, a numerical experiment has been carried out to illustrate the proposed algorithm.

Entropic Regularization of Wasserstein Distance Between Infinite-Dimensional Gaussian Measures and Gaussian Processes

This work studies the entropic regularization formulation of the 2-Wasserstein distance on an infinite-dimensional Hilbert space, in particular for the Gaussian setting. We first present the minimum mutual information property, namely, the joint measures of two Gaussian measures on Hilbert space with the smallest mutual information are joint Gaussian measures. This is the infinite-dimensional generalization of the maximum entropy property of Gaussian densities on Euclidean space. We then give closed-form formulas for the optimal entropic transport plan, entropic 2-Wasserstein distance, and Sinkhorn divergence between two Gaussian measures on a Hilbert space, along with the fixed point equations for the barycenter of a set of Gaussian measures. Our formulations fully exploit the regularization aspect of the entropic formulation and are valid both in singular and in nonsingular settings. In the infinite-dimensional setting, both the entropic 2-Wasserstein distance and Sinkhorn divergence are Fréchet differentiable, in contrast to the exact 2-Wasserstein distance, which is not differentiable. Our Sinkhorn barycenter equation is new and always has a unique solution. In contrast, the finite-dimensional barycenter equation for the entropic 2-Wasserstein distance fails to generalize to the Hilbert space setting. In the setting of reproducing kernel Hilbert spaces, our distance formulas are given explicitly in terms of the corresponding kernel Gram matrices, providing an interpolation between the kernel maximum mean discrepancy and the kernel 2-Wasserstein distance.

Multi-step inertial forward-backward-half forward algorithm for solving monotone inclusion

In this article, we study a multi-step inertial forward-backward-half forward splitting algorithm to solve the solution of the sum of three operators with two of them being respectively, cocoercive and monotone-Lipschitz continuous. Notably, the convergence of the multi-step inertial forward-backward-half forward algorithm is established under the summability condition formulated on the basis of the iterative sequence in a Hilbert space setting. We also display two applications including problems that are: a composite monotone inclusion problem involving mixtures of linearly composed with parallel-sum type operators, and a composite convex optimization problem. As far as the resolvent of the composite operator is concerned, due to difficulty in calculation, then, the composite monotone inclusion problem is addressed by the proposed multi-step inertial algorithm together with the primal-dual idea. Experiments conducted on an image deblurring problem indicate the superiority and efficiency of the proposed method.

Some new convergence and stability results for Jungck generalized pseudo-contractive and Lipschitzian type operators using hybrid iterative techniques in the Hilbert space

The convergence of generalized pseudo-contractive operators were studied with regard to a unique fixed point in the article (Verma in Proc Roy Irish Acad Soct A, 97(1):83–86, 1997). In this present article, we introduce Jungck generalized pseudo-contractive and Lipschitzian type operators in the prehilbertian space and in the Hilbert space settings, establish the existence and the uniqueness of a common fixed point for the operator S and a sequence of operators {T_i:Drightarrow D}_{i=1}^k using a new Jungck–Kirk–Mann type fixed point iterative algorithm as well as the general Kirk–Mann type iterative algorithm. The strong convergence of these fixed point iterative algorithms to a unique common fixed point is also investigated for the same set of operators considered. Moreover, some stability results are established for the Jungck–Kirk–Mann type iterative algorithm as well as the general Kirk–Mann type iterative algorithm. In addition, some examples are given to support our arguments. The results are new, original as well as generalizing some existing ones in the literature.

Regularizing linear inverse problems under unknown non-Gaussian white noise allowing repeated measurements

Abstract We deal with the solution of a generic linear inverse problem in the Hilbert space setting. The exact right-hand side is unknown and only accessible through discretized measurements corrupted by white noise with unknown arbitrary distribution. The measuring process can be repeated, which allows to reduce and estimate the measurement error through averaging. We show convergence against the true solution of the infinite-dimensional problem for a priori and a posteriori regularization schemes as the number of measurements and the dimension of the discretization tend to infinity under natural and easily verifiable conditions for the discretization.

Fractional Hardy-type and trace theorems for nonlocal function spaces with heterogeneous localization

This work aims to prove a Hardy-type inequality and a trace theorem for a class of function spaces on smooth domains with a nonlocal character. Functions in these spaces are allowed to be as rough as an [Formula: see text]-function inside the domain of definition but as smooth as a [Formula: see text]-function near the boundary. This feature is captured by a norm that is characterized by a nonlocal interaction kernel defined heterogeneously with a special localization feature on the boundary. Thus, the trace theorem we obtain here can be viewed as an improvement and refinement of the classical trace theorem for fractional Sobolev spaces [Formula: see text]. Similarly, the Hardy-type inequalities we establish for functions that vanish on the boundary show that functions in this generalized space have the same decay rate to the boundary as functions in the smaller space [Formula: see text]. The results we prove extend existing results shown in the Hilbert space setting with p = 2. A Poincaré-type inequality we establish for the function space under consideration together with the new trace theorem allows formulating and proving well-posedness of a nonlinear nonlocal variational problem with conventional local boundary condition.

Characterizations of manifolds modeled on absorbing sets in non-separable Hilbert spaces and the discrete cells property

We characterize infinite-dimensional manifolds modeled on absorbing sets in non-separable Hilbert spaces by using the discrete cells property, which is a general position property. Moreover, we study the discrete (locally finite) approximation property, w

An extension of the generalized admissible S-algorithm for multivalued type contractions in graphic Banach spaces with application to image recovery problem

The purpose of this paper is to introduce a new extension of the generalized admissible S-algorithm for approximating common fixed point of three multivalued mappings satisfying two general classes of contraction conditions in a uniformly convex Banach space endowed with a graph. As an application of our result we establish the solution of image recovery problem in Hilbert space setting.

Approximation of Fixed Points for Enriched Suzuki Nonexpansive Operators with an Application in Hilbert Spaces

In this article, we introduce the class of enriched Suzuki nonexpansive (ESN) mappings. We show that this new class of mappings properly contains the class of Suzuki nonexpansive as well as the class of enriched nonexpansive mappings. We establish existence of fixed point and convergence of fixed point in a Hilbert space setting under the Krasnoselskii iteration process. One of the our main results is applied to solve a split feasibility problem (SFP) in this new setting of mappings. Our main results are a significant improvement of the corresponding results of the literature.

A weak law of large numbers for realised covariation in a Hilbert space setting

This article generalises the concept of realised covariation to Hilbert-space-valued stochastic processes. More precisely, based on high-frequency functional data, we construct an estimator of the trace-class operator-valued integrated volatility process arising in general mild solutions of Hilbert space-valued stochastic evolution equations in the sense of Da Prato and Zabczyk (2014). We prove a weak law of large numbers for this estimator, where the convergence is uniform on compacts in probability with respect to the Hilbert–Schmidt norm. In addition, we determine convergence rates for common stochastic volatility models in Hilbert spaces.

Optimal Convergence of the Discrepancy Principle for Polynomially and Exponentially Ill-Posed Operators under White Noise

We consider a linear ill-posed equation in the Hilbert space setting under white noise. Known convergence results for the discrepancy principle are either restricted to Hilbert-Schmidt operators (and they require a self-similarity condition for the unknown solution additional to a classical source condition) or to polynomially ill-posed operators (excluding exponentially ill-posed problems). In this work, we show optimal convergence for a modified discrepancy principle for both polynomially and exponentially ill-posed operators (without further restrictions) solely under either Hölder-type or logarithmic source conditions. In particular, the method includes only a single simple hyper parameter, which does not need to be adapted to the type of ill-posedness.

Convergence of iterates for first-order optimization algorithms with inertia and Hessian driven damping

In a Hilbert space setting, for convex optimization, we show the convergence of the iterates to optimal solutions for a class of accelerated first-order algorithms. They can be interpreted as discrete temporal versions of an inertial dynamic involving both viscous damping and Hessian-driven damping. The asymptotically vanishing viscous damping is linked to the accelerated gradient method of Nesterov while the Hessian driven damping makes it possible to significantly attenuate the oscillations. By treating the Hessian-driven damping as the time derivative of the gradient term, this gives, in discretized form, first-order algorithms. These results complement the previous work of the authors where it was shown the fast convergence of the values, and the fast convergence towards zero of the gradients.

Inertial accelerated algorithms for solving split feasibility with multiple output sets in Hilbert spaces

Abstract In this paper, we propose two inertial accelerated algorithms which do not require prior knowledge of operator norm for solving split feasibility problem with multiple output sets in real Hilbert spaces. We prove weak and strong convergence results for approximating the solution of the considered problem under certain mild conditions. We also give some numerical examples to demonstrate the performance and efficiency of our proposed algorithms over some existing related algorithms in the literature.

Two strongly convergent self-adaptive iterative schemes for solving pseudo-monotone equilibrium problems with applications

Abstract The aim of this paper is to propose two new modified extragradient methods to solve the pseudo-monotone equilibrium problem in a real Hilbert space with the Lipschitz-type condition. The iterative schemes use a new step size rule that is updated on each iteration based on the value of previous iterations. By using mild conditions on a bi-function, two strong convergence theorems are established. The applications of proposed results are studied to solve variational inequalities and fixed point problems in the setting of real Hilbert spaces. Many numerical experiments have been provided in order to show the algorithmic performance of the proposed methods and compare them with the existing ones.

A modified discrepancy principle to attain optimal convergence rates under unknown noise

We consider a linear ill-posed equation in the Hilbert space setting. Multiple independent unbiased measurements of the right-hand side are available. A natural approach is to take the average of the measurements as an approximation of the right-hand side and to estimate the data error as the inverse of the square root of the number of measurements. We calculate the optimal convergence rate (as the number of measurements tends to infinity) under classical source conditions and introduce a modified discrepancy principle, which asymptotically attains this rate.

Inertial Extragradient Methods for Solving Split Equilibrium Problems

This paper presents two inertial extragradient algorithms for finding a solution of split pseudomonotone equilibrium problems in the setting of real Hilbert spaces. The weak and strong convergence theorems of the introduced algorithms are presented under some constraint qualifications of the scalar sequences. The discussions on the numerical experiments are also provided to demonstrate the effectiveness of the proposed algorithms.

The difference vectors for convex sets and a resolution of the geometry conjecture

The geometry conjecture, which was posed nearly a quarter of a century ago, states that the fixed point set of the composition of projectors onto nonempty closed convex sets in Hilbert space is actually equal to the intersection of certain translations of the underlying sets.

Direct and inverse problems for time-fractional pseudo-parabolic equations

The purpose of this paper is to establish the solvability results to direct and inverse problems for time-fractional pseudo-parabolic equations with the self-adjoint operators. We are especially interested in proving existence and uniqueness of the solutions in the abstract setting of Hilbert spaces.