Halpern-type relaxed algorithms with alternated and multi-step inertia for split feasibility problems with applications in classification problems

In this work, we construct four efficient multi-step inertial relaxed algorithms based on the monotonic step-length criterion which does not require any information about the norm of the underlying operator or the use of a line search procedure for split feasibility problems in infinite-dimensional Hilbert spaces. The first and the third are the general multi-step inertial-type methods, which unify two steps of the multi-step inertial terms with the golden ratio-based and an alternating golden ratio-based extrapolation steps, respectively, to improve the speed of convergence of their sequences of iterates to a solution of the problem, while the second and the fourth are the three-term conjugate gradient-like and multi-step inertial-type methods, which integrate both the three-term conjugate gradient-like direction and a multi-step inertial term with the golden ratio-based and an alternating golden ratio-based extrapolation steps, respectively, to accelerate their sequences of iterates toward a solution of the problem. Under some simple and weaker assumptions, we formulate and prove some strong convergence theorems for each of these algorithms based on the convergence theorem of a golden ratio-based relaxed algorithm with perturbations and the alternating golden ratio-based relaxed algorithm with perturbations in infinite-dimensional real Hilbert spaces. Moreover, we analyze their applications in classification problems for an interesting real-world dataset based on the extreme learning machine (ELM) with the $\ell_{1}-\ell_{2}$ hybrid regularization approach and in solving constrained minimization problems in infinite-dimensional Hilbert spaces. In all the experiments, our proposed algorithms, which generalizes several algorithms in the literature, comparatively achieve better performance than some related algorithms.

In this article, we construct an accelerated relaxed algorithm with an alternating inertial extrapolation step. The proposed algorithm uses a three-term conjugate gradient-like direction, which helps to fasten the sequence of its iterates to a point in a solution set. The algorithm employs a self-adaptive step-length criterion that does not require any information related to the norm of the operator or the use of a line-search procedure. Moreover, we formulate and prove a strong convergence theorem for the algorithm to a minimum-norm solution of a split feasibility problem in infinite-dimensional real Hilbert spaces. Furthermore, we investigate its applications in breast cancer detection by solving classification problems for an interesting real-world breast cancer dataset, based on the extreme learning machine (ELM) with the \(\ell_{1}\)-regularization approach (i.e., the Lasso model) and the \(\ell_{1}\)-\(\ell_{2}\) hybrid regularization technique. The performance results of the experiments demonstrate that the proposed algorithm is robust, efficient, and achieves better generalization performance and stability than some existing algorithms in the literature.

We studied the approximate split equality problem (ASEP) in the framework of infinite-dimensional Hilbert spaces. Let , , and   be infinite-dimensional real Hilbert spaces, let and   be two nonempty closed convex sets, and let and   be two bounded linear operators. The ASEP in infinite-dimensional Hilbert spaces is to minimize the function over and . Recently, Moudafi and Byrne had proposed several algorithms for solving the split equality problem and proved their convergence. Note that their algorithms have only weak convergence in infinite-dimensional Hilbert spaces. In this paper, we used the regularization method to establish a single-step iterative for solving the ASEP in infinite-dimensional Hilbert spaces and showed that the sequence generated by such algorithm strongly converges to the minimum-norm solution of the ASEP. Note that, by taking in the ASEP, we recover the approximate split feasibility problem (ASFP).

The δ Equation on a Hilbert Space and Some Applications to Complex Analysis on Infinite Dimensional Vector Spaces

In this paper, some existence results for a nonlinear complementarity problem involving a pseudo-monotone mapping over an arbitrary closed convex cone in a real Hilbert space are established. In particular, some known existence results for a nonlinear complementarity problem in a finite-dimensional Hilbert space are generalized to an infinite-dimensional real Hilbert space. Applications to a class of nonlinear complementarity problems and the study of the post-critical equilibrium state of a thin elastic plate subjected to unilateral conditions are given.

The sandwiched Rényi divergences of two finite-dimensional density operators quantify their asymptotic distinguishability in the strong converse domain. This establishes the sandwiched Rényi divergences as the operationally relevant ones among the infinitely many quantum extensions of the classical Rényi divergences for Rényi parameter alpha >1. The known proof of this goes by showing that the sandwiched Rényi divergence coincides with the regularized measured Rényi divergence, which in turn is proved by asymptotic pinching, a fundamentally finite-dimensional technique. Thus, while the notion of the sandwiched Rényi divergences was extended recently to density operators on an infinite-dimensional Hilbert space (in fact, even for states of an arbitrary von Neumann algebra), these quantities were so far lacking an operational interpretation similar to the finite-dimensional case, and it has also been open whether they coincide with the regularized measured Rényi divergences. In this paper we fill this gap by answering both questions in the positive for density operators on an infinite-dimensional Hilbert space, using a simple finite-dimensional approximation technique. We also initiate the study of the sandwiched Rényi divergences, and the related problem of the strong converse exponent, for pairs of positive semi-definite operators that are not necessarily trace-class (this corresponds to considering weights in a general von Neumann algebra setting). This is motivated by the need to define conditional Rényi entropies in the infinite-dimensional setting, while it might also be interesting from the purely mathematical point of view of extending the concept of Rényi (and other) divergences to settings beyond the standard one of positive trace-class operators (positive normal functionals in the von Neumann algebra setting). In this spirit, we also discuss the definition and some properties of the more general family of Rényi (alpha ,z)-divergences of positive semi-definite operators on an infinite-dimensional separable Hilbert space.

One of the major problems in the theory of maximal monotone operators is to find a point in the solution set Zer( ), set of zeros of maximal monotone mapping . The problem of finding a zero of a maximal monotone in real Hilbert space has been investigated by many researchers. Rockafellar considered the proximal point algorithm and proved the weak convergence of this algorithm with the maximal monotone operator. Güler gave an example showing that Rockafellar’s proximal point algorithm does not converge strongly in an infinite-dimensional Hilbert space. In this paper, we consider an explicit method that is strong convergence in an infinite-dimensional Hilbert space and a simple variant of the hybrid steepest-descent method, introduced by Yamada. The strong convergence of this method is proved under some mild conditions. Finally, we give an application for the optimization problem and present some numerical experiments to illustrate the effectiveness of the proposed algorithm.

In this paper, we propose a new modification of the Gradient Projection Algorithm and the Forward–Backward Algorithm. Using our proposed algorithms, we establish two strong convergence theorems for solving convex minimization problem, monotone variational inclusion problem and fixed point problem for demicontractive mappings in a real Hilbert space. Furthermore, we apply our results to solve split feasibility and optimal control problems. We also give two numerical examples of our algorithm in real Euclidean space of dimension 4 and in an infinite dimensional Hilbert space, to show the efficiency and advantage of our results.

To be the best of our knowledge, the convergence theorem for the DC program and split DC program are proposed in finite-dimensional real Hilbert spaces or Euclidean spaces. In this paper, to study the split DC program, we give a hybrid proximal linearized algorithm and propose related convergence theorems in the settings of finite- and infinite-dimensional real Hilbert spaces, respectively.

In this work, we prove that any element in the tensor product of separable infinite-dimensional Hilbert spaces can be expressed as a matrix product state (MPS) of possibly infinite bond dimension. The proof is based on the singular value decomposition of compact operators and the connection between tensor products and Hilbert–Schmidt operators via the Schmidt decomposition in infinite-dimensional separable Hilbert spaces. The construction of infinite-dimensional MPS (idMPS) is analogous to the well-known finite-dimensional construction in terms of singular value decompositions of matrices. The infinite matrices in idMPS give rise to operators acting on (possibly infinite-dimensional) auxiliary Hilbert spaces. As an example we explicitly construct an MPS representation for certain eigenstates of a chain of three coupled harmonic oscillators.

Finite dimensional approximation and Newton-based algorithm for stochastic approximation in Hilbert space

Mann iteration is weakly convergent in infinite dimensional spaces. We, in this paper, use the nearest point projection to force the strong convergence of a Mann‐based iteration for nonexpansive and monotone operators. A strong convergence theorem of common elements is obtained in an infinite dimensional Hilbert space. No compact conditions are needed.

Our interest in this paper is to introduce a Halpern-type algorithm with both inertial terms and errors for approximating fixed point of a nonexpansive mapping. We obtain strong convergence of the sequence generated by our proposed method in real Hilbert spaces under some reasonable assumptions on the sequence of parameters. As applications, we present some strong convergence results for monotone inclusion, variational inequality problem, linear inverse problem, and LASSO problem in Compressed Sensing. Our result improves the rate of convergence of existing Halpern method for monotone inclusion, variational inequality problem, linear inverse problem and LASSO problem in compressed sensing as illustrated in our numerical examples both in finite and infinite dimensional Hilbert spaces.

We obtain the admissible sets on the unit circle to be the spectrum of a strict $m$-isometry on an $n$-finite dimensional Hilbert space. This property gives a better picture of the correct spectrum of an $m$-isometry. We determine that the only $m$-isometries on $\mathbb{R}^2$ are $3$-isometries and isometries giving by $\pm I+Q$, where $Q$ is a nilpotent operator. Moreover, on real Hilbert space, we obtain that $m$-isometries preserve volumes. Also we present a way to construct a strict $(m+1)$-isometry with an $m$-isometry given, using ideas of Aleman and Suciu \cite[Proposition 5.2]{AS} on infinite dimensional Hilbert space.

Nonnegative Moore–Penrose inverses of Gram operators