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Numerous variational methods have been proposed for solving quantum many-body systems, but they often face exponentially increasing computational complexity as the Hilbert space dimension grows. To address this, we introduce a novel approach using quantum neural networks to simulate the dissipative dynamics of many-body open quantum systems. This method combines neural-network quantum state representation with the time-dependent variational principle, both implemented via quantum algorithms. This results in accurate open quantum dynamics described by the Lindblad quantum master equation, exemplified by the spin-boson and transverse field Ising models. Our approach avoids the computational expense of classical algorithms and demonstrates the potential advantages of quantum computing for many-body simulations. To reduce measurement errors, we introduce a projection reset procedure, which could benefit other quantum simulations. In addition, our approach can be extended to simulate non-Markovian quantum dynamics. [ABSTRACT FROM AUTHOR]

Although selected configuration interaction (SCI) algorithms can tackle much larger Hilbert spaces than the conventional full CI method, the scaling of their computational cost with respect to the system size remains inherently exponential. In addition, inaccuracies in describing the correlation hole at small interelectronic distances lead to the slow convergence of the electronic energy relative to the size of the one-electron basis set. To alleviate these effects, we show that the non-Hermitian, transcorrelated (TC) version of SCI significantly compactifies the determinant space, allowing us to reach a given accuracy with a much smaller number of determinants. Furthermore, we note a significant acceleration in the convergence of the TC-SCI energy as the basis set size increases. The extent of this compression and the energy convergence rate are closely linked to the accuracy of the correlation factor used for the similarity transformation of the Coulombic Hamiltonian. Our systematic investigation of small molecular systems in increasingly large basis sets illustrates the magnitude of these effects. [ABSTRACT FROM AUTHOR]

Two computational approaches for computing the rates of internal conversions in molecular systems where a large set of nuclear degrees of freedom plays a role are discussed and compared. One approach is based on the numerical solution of the time-dependent Schrödinger equation and allows us to include almost the whole set of vibrational coordinates, thanks to the employment of effective procedures for selecting those elements of the Hilbert space which play a significant role in dynamics. The other approach, based on the time-dependent perturbation theory and limited to the use of the harmonic approximation, allows us to include the whole Hilbert space spanned by the vibrational states of the system. The two approaches are applied to the photophysics of azulene, whose anti-Kasha behavior caused by anomalous internal conversion rates is well assessed. The calculated rates for the decays of the first two excited singlet states are in very good agreement with experimental data, indicating the reliability of both methodologies. [ABSTRACT FROM AUTHOR]

In the past decade, the quantum chemical version of the density matrix renormalization group method has established itself as the method of choice for strongly correlated molecular systems. However, despite its favorable scaling, in practice, it is not suitable for computations of dynamic correlation. Several approaches to include that in post-DMRG methods exist; in our group, we focused on the tailored coupled cluster (TCC) approach. This method works well in many situations; however, in exactly degenerate cases (with two or more determinants of equal weight), it exhibits a bias toward the reference determinant representing the Fermi vacuum. Although sometimes it is possible to use a compensation scheme to avoid this bias for energy differences, it is certainly a drawback. In order to overcome this bias of the TCC method, we have developed a Hilbert-space multireference version of tailored CC, which can treat several determinants on an equal footing. We have implemented and compared the performance of three Hilbert-space multireference coupled cluster (MRCC) variants—the state universal one and the Brillouin–Wigner and Mukherjee's state specific ones. We have assessed these approaches on the cyclobutadiene and tetramethyleneethane molecules, which are both diradicals with exactly degenerate determinants at a certain geometry. We have also investigated the sensitivity of the results on the orbital rotation of the highest occupied and lowest unoccupied molecular orbital (HOMO–LUMO) pair, as it is well known that Hilbert-space MRCC methods are not invariant to such transformations. [ABSTRACT FROM AUTHOR]

We introduce a generalization of the σ-SCF method to approximate noncollinear spin ground and excited single-reference electronic states by minimizing the Hamiltonian variance. The new method is based on the σ-SCF method, originally proposed by Ye et al. [J. Chem. Phys. 147, 214104 (2017)], and provides a prescription to determine ground and excited noncollinear spin states on an equal footing. Our implementation was carried out utilizing an initial simulated annealing stage followed by a mean-field iterative self-consistent approach to simplify the cumbersome search introduced by generalizing the spin degrees of freedom. The simulated annealing stage ensures a broad exploration of the Hilbert space spanned by the generalized spin single-reference states with random complex element-wise rotations of the generalized density matrix elements in the simulated annealing stage. The mean-field iterative self-consistent stage employs an effective Fockian derived from the variance, which is utilized to converge tightly to the solutions. This process helps us to easily find complex spin structures, avoiding manipulating the initial guess. As proof-of-concept tests, we present results for Hn (n = 3–7) planar rings and polyhedral clusters with geometrical spin frustration. We show that most of these systems have noncollinear spin excited states that can be interpreted in terms of geometric spin frustration. These states are not directly targeted by energy minimization methods, which are meant to converge to the ground state. This stresses the capability of the σ-SCF methodology to find approximate noncollinear spin structures as mean-field excited states. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a novel dual inertial Tseng's extragradient method for solving variational inequality problems in real Hilbert spaces, particularly those involving pseudomonotone and Lipschitz continuous operators. Our secondary method incorporates variable step‐size, updated at each iteration based on some previous iterates. A notable advantage of these algorithms is their ability to operate without prior knowledge of Lipschitz‐type constants and without the need for any line‐search procedure. We establish the convergence theorem of the proposed algorithms under mild assumptions. To illustrate the numerical behavior of the algorithms and to make comparisons with other methods, we conduct several numerical experiments. The results of these evaluations are showcased and thoroughly examined to exemplify the practical significance and effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

In this article, we have derived some sufficient conditions for existence of optimal control for semilinear control system of fractional order (1,2] in Hilbert space. We have discussed existence and uniqueness of mild solution with the help of Banach fixed point theorem for proposed problem. Under certain conditions stated, Lagrange's problem admits at least one optimal control pair. Finally, one example is given to understand theoretical results in better manner. [ABSTRACT FROM AUTHOR]

It is remarkable that Heisenberg's position-momentum uncertainty relation leads to the existence of a maximal acceleration for a physical particle in the context of a geometric reformulation of quantum mechanics. It is also known that the maximal acceleration of a quantum particle is related to the magnitude of the speed of transportation in projective Hilbert space. In this paper, inspired by the study of geometric aspects of quantum evolution by means of the notions of curvature and torsion, we derive an upper bound for the rate of change of the speed of transportation in an arbitrary finite-dimensional projective Hilbert space. The evolution of the physical system being in a pure quantum state is assumed to be governed by an arbitrary time-varying Hermitian Hamiltonian operator. Our derivation, in analogy to the inequalities obtained by L. D. Landau in the theory of fluctuations by means of general commutation relations of quantum-mechanical origin, relies upon a generalization of Heisenberg's uncertainty relation. We show that the acceleration squared of a quantum evolution in projective space is upper bounded by the variance of the temporal rate of change of the Hamiltonian operator. Moreover, focusing for illustrative purposes on the lower-dimensional case of a single spin qubit immersed in an arbitrarily time-varying magnetic field, we discuss the optimal geometric configuration of the magnetic field that yields maximal acceleration along with vanishing curvature and unit geodesic efficiency in projective Hilbert space. Finally, we comment on the consequences that our upper bound imposes on the limit at which one can perform fast manipulations of quantum systems to mitigate dissipative effects and/or obtain a target state in a shorter time. [ABSTRACT FROM AUTHOR]

Let A be a positive bounded operator on a Hilbert space H and let ‖ T ‖ A , w A (T) , and m A (T) denote the A -operator seminorm, the A -numerical radius, and the A -minimum modulus of an operator T in the semi-Hilbertian space (H , ‖ ⋅ ‖ A) , respectively. In this paper, we present new improvements of certain A -Cauchy–Schwarz type inequalities and as applications of our results, we provide refinements of some A -numerical radius inequalities for semi-Hilbertian space operators. It is shown, among other inequalities, that w A (T) ≤ (1 − 1 2 inf λ ∈ C ⁡ m A 2 (I − λ T)) ‖ T ‖ A , where I is the identity operator on H. A refinement of the triangle inequality for semi-Hilbertian space operators is also given. [ABSTRACT FROM AUTHOR]

AbstractConvex PCA, which was introduced in Bigot et al. (2017), modifies Euclidean PCA by restricting the data and the principal components to lie in a given convex subset of a Hilbert space. This setting arises naturally in many applications, including distributional data in the Wasserstein space of an interval, and ranked compositional data under the Aitchison geometry. Our contribution in this paper is threefold. First, we present several new theoretical results including consistency as well as continuity and differentiability of the objective function in the finite dimensional case. Second, we develop a numerical implementation of finite dimensional convex PCA when the convex set is polyhedral, and show that this provides a natural approximation of Wasserstein GPCA. Third, we illustrate our results with two financial applications, namely distributions of stock returns ranked by size and the capital distribution curve, both of which are of independent interest in stochastic portfolio theory. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

Recent no‐go theorems have ruled out four‐dimensional classical de Sitter vacua in heterotic string theory. On the other hand, the absence of a well‐defined Wilsonian effective action and other related phenomena also appear to rule out such time‐dependent vacua with de Sitter isometries, even in the presence of quantum corrections. In this note, the authors argued that a four‐dimensional de Sitter space can still exist in SO(32)$SO(32)$ heterotic string theory as a Glauber–Sudarshan state, i.e., as an excited state, over a supersymmetric Minkowski background, albeit within a finite temporal domain. Borel resummation and resurgence play a crucial role in constructing such a state in the Hilbert space of heterotic theory governed entirely by the IR degrees of freedom. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a frame diagonalization of matrices in $ M_n({\mathbb {C}}) $ Mn(C), called QR-frame diagonalization, by using QR-factorization. Furthermore, we show that every matrix A in $ M_n({\mathbb {C}}) $ Mn(C) is QR-frame diagonalizable. Also we introduce another frame diagonalization via Hadamard matrices in $ M_n({\mathbb {R}}) $ Mn(R), called Hadamard frame diagonalization, by using Hadamard matrices for n = 2, 4 or multiple of 4. We show that every matrix A in $ M_n({\mathbb {R}}) $ Mn(R) which n = 2, 4 or multiple of 4 is Hadamard frame diagonalizable. In this frame diagonalization, we can find entries on main diagonal Δ by using inner product. Moreover we introduce frame diagonalization of matrices on left quaternionic Hilbert spaces $ M_n({\mathbb {H}}) $ Mn(H). [ABSTRACT FROM AUTHOR]

Let Ω be a nonempty closed convex subset of a real Hilbert space H . Let ℑ be a nonspreading mapping from Ω into itself. Define two sequences { ψ n } n = 1 ∞ and { ϕ n } n = 1 ∞ as follows: { ψ n + 1 = π n ψ n + (1 − π n) ℑ ψ n , ϕ n = 1 n ∑ n t = 1 ψ t , for n ∈ N , where 0 ≤ π n ≤ 1 , and π n → 0 . In 2010, Kurokawa and Takahashi established weak and strong convergence theorems of the sequences developed from the above Baillion-type iteration method (Nonlinear Anal. 73:1562–1568, 2010). In this paper, we prove weak and strong convergence theorems for a new class of (η , β) -enriched strictly pseudononspreading ((η , β) -ESPN) maps, more general than that studied by Kurokawa and W. Takahashi in the setup of real Hilbert spaces. Further, by means of a robust auxiliary map incorporated in our theorems, the strong convergence of the sequence generated by Halpern-type iterative algorithm is proved thereby resolving in the affirmative the open problem raised by Kurokawa and Takahashi in their concluding remark for the case in which the map ℑ is averaged. Some nontrivial examples are given, and the results obtained extend, improve, and generalize several well-known results in the current literature. [ABSTRACT FROM AUTHOR]

The inertial forward–backward splitting algorithm can be considered as a modified form of the forward–backward algorithm for variational inequality problems with monotone and Lipschitz continuous cost mappings. By using parallel and inertial techniques and the forward–backward splitting algorithm, in this paper, we propose a new parallel inertial forward–backward splitting algorithm for solving variational inequality problems, where the constraints are the intersection of common solution sets of a finite family of variational inclusion problems. Then, strong convergence of proposed iteration sequences is showed under standard assumptions imposed on cost mappings in a real Hilbert space. Finally, some numerical experiments demonstrate the reliability and benefits of the proposed algorithm. [ABSTRACT FROM AUTHOR]

AbstractWe, in this article, focus on functional partially linear quantile regression, where the observations are missing at random, which allows the response or covariates or response and covariates simultaneously missing. Estimation of the unknown function is done based on reproducing kernel method. Under suitable assumptions, we discuss consistency with rates of the estimators, and establish asymptotic normality of the estimator for the parameter. At the same time, we study hypothesis test of the parameter, and prove asymptotic distributions of restricted estimators of the parameter and test statistic under null hypothesis and local alternative hypothesis, respectively. Also, we study variable selection of the linear part of the model. By simulation and real data, finite sample performance of the proposed methods is analyzed. [ABSTRACT FROM AUTHOR]

We propose a novel method for testing serial independence of object-valued time series in metric spaces, which are more general than Euclidean or Hilbert spaces. The proposed method is fully nonparametric, free of tuning parameters and can capture all nonlinear pairwise dependence. The key concept used in this paper is the distance covariance in metric spaces, which is extended to the autodistance covariance for object-valued time series. Furthermore, we propose a generalized spectral density function to account for pairwise dependence at all lags and construct a Cramér–von Mises-type test statistic. New theoretical arguments are developed to establish the asymptotic behaviour of the test statistic. A wild bootstrap is also introduced to obtain the critical values of the nonpivotal limiting null distribution. Extensive numerical simulations and two real data applications on cumulative intraday returns and human mortality data are conducted to illustrate the effectiveness and versatility of our proposed test. [ABSTRACT FROM AUTHOR]

Kernel two-sample tests have been widely used for multivariate data to test equality of distributions. However, existing tests based on mapping distributions into a reproducing kernel Hilbert space mainly target specific alternatives and do not work well for some scenarios when the dimension of the data is moderate to high due to the curse of dimensionality. We propose a new test statistic that makes use of a common pattern under moderate and high dimensions and achieves substantial power improvements over existing kernel two-sample tests for a wide range of alternatives. We also propose alternative testing procedures that maintain high power with low computational cost, offering easy off-the-shelf tools for large datasets. The new approaches are compared to other state-of-the-art tests under various settings and show good performance. We showcase the new approaches through two applications: the comparison of musks and nonmusks using the shape of molecules, and the comparison of taxi trips starting from John F. Kennedy airport in consecutive months. All proposed methods are implemented in an R package kerTests. [ABSTRACT FROM AUTHOR]

Semi-supervised classification has gained widespread popularity because of their superior ability to handle unlabeled samples in practical problems. This paper has presented a novel estimation error-ranked LSSVM method with double Mahalanobis-kernel which is used for semi-supervised classification. The main point is to construct two Mahalanobis distances in Hilbert space to form double Mahalanobis-kernel by considering the relationship between the characteristics of two sorts of samples, so as to reduce the influence of non-informational dimensions. Furthermore, the implementation of the proposed method is required to solve the label security problem of unlabeled samples. The unlabeled sample with the minimum evaluated error is selected for labeling, which effectively ensures the accuracy of the unlabeled sample labeling. This method not only considers the similarity of sample features, but also focuses on the security of unlabeled samples. And based on the experimental results of four artificial data sets and several UCI data sets, it verifies the effectiveness of the semi-supervised method with double Mahalanobis-kernel. Especially considering the experimental results of five disease diagnosis data sets, it demonstrates the potential of the proposed semi-supervised classification method in medical diagnosis. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a modified Tseng's extragradient method with a new step-length rule to solve pseudo-monotone variational inequalities in real Hilbert spaces. Under suitable conditions, the sequence generated by this algorithm strongly converges to the common elements of the solution set of pseudo-monotone variational inequality problems and the fixed point set of k-demicontractive mappings. Finally, we give some numerical experiments to illustrate the effectiveness of the proposed algorithm. The main results of this paper generalize and improve some known results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we explore various transforms associated with a bounded linear operator T on a Hilbert space. These transforms include the Aluthge, λ -Aluthge, Duggal, generalized mean, and λ -mean transforms. Our aim is to investigate the connections between T and these transforms, focusing on aspects such as norm inequalities and numerical ranges, while also highlighting certain essential properties. Furthermore, we aim to determine the conditions under which an operator T coincides in norm with its transformed counterparts through these transformations. Several characterizations and properties are also derived. [ABSTRACT FROM AUTHOR]

In the present paper, we propose a new approach to solving a class of generalized split problems. This approach will open some new directions for research to solve the other split problems, for instance, the split common zero point problem and the split common fixed point problem. More precisely, we study the split common solution problem for monotone operator equations in real Hilbert spaces. To find a solution to this problem, we propose and establish the strong convergence of the two new iterative methods by using the Tikhonov regularization method. Meantime, we also study the stability of the iterative methods. Finally, two numerical examples are also given to illustrate the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

Within the framework of quantum information theory, the performance of conventional 2-D techniques is often unsatisfactory for the study of Bell states that hold a unique status as maximally entangled states. Therefore, 3-D approaches are increasingly employed to achieve more accurate and detailed analysis, offering improved performance and insights in complex scenarios. Among the diverse background of mixed two-qubit states, certain configurations exhibit unique quantum correlations that harness advanced quantum information processing tasks such as Bell diagonal states. Although, these states may appear superficially simple and exhibit a rich spectrum of correlations. The present research employs a methodology that involves a convex combination of distinct Bell states to generate the entire class of Bell diagonal states. This work explores the time evolution of Bell diagonal states, when exposed to various quantum channels and investigates the dynamics of quantum correlations such as entanglement, discord, and state of separability. Finally, the behaviour of Bell diagonal states is analysed and results are compared between theory and practice. A threedimensional visual approach is used to illustrate a deeper understanding of various quantum features and dynamic behaviour of the Bell diagonal states. [ABSTRACT FROM AUTHOR]

The paper concerns a new iterative method for approximating a solution of an equilibrium problem involving a monotone and Lipschitz-type bifunction in a Hilbert space. Our method combines the proximal mapping with a regularization technique. When compared with known extragradient methods, the proposed method is seen to have a simpler and more elegant structure. A strong convergence theorem is established under some appropriate conditions imposed on the control parameters. Several experiments are performed in order to illustrate the numerical effectiveness of our new algorithm in comparison with that of existing methods. [ABSTRACT FROM AUTHOR]

The aim of this paper is to develop a marginal screening method for variable screening in high-dimensional binary classification based on the Wasserstein distance accounting for the distributional difference. Many existing screening methods, such as the two-sample t-test and Kolmogorov test, have been developed under the parametric/nonparametric modeling assumptions to reduce the dimension of the predictors. However, such modeling specifications or nonparametric approaches are associated with the probability measure induced by the predictor in a Euclidean space. While many machine learning methods have successfully found the nonlinear decision boundary in the transformed space, called the reproducing kernel Hilbert space (RKHS), we consider the Wasserstein filter's capacity to detect the distributional difference between two probability measures induced by the nonlinear function of the predictor in the RKHS. Thereby, we can flexibly filter out the non-informative predictors associated with the binary classification, as well as escape the modeling assumptions required in a Euclidean space. We prove that the Wasserstein filter satisfies the sure screening property under some mild conditions. We also demonstrate the advantages of our proposed approach by comparing the finite sample performance of it with those of the existing choices through simulation studies, as well as through application to lung cancer data. [ABSTRACT FROM AUTHOR]

We are concerned in this paper with the convergence analysis of the primal-dual splitting (PDS) and the split Douglas-Rachford (SDR) algorithms for monotone inclusions by using an operator-oriented approach. We shall show that both PDS and SDR algorithms can be driven by a (firmly) nonexpansive mapping in a product Hilbert space. We are then able to apply the Krasnoselskii-Mann and Halpern fixed point algorithms to PDS and SDR to get weakly and strongly convergent algorithms for finding solutions of the primal and dual monotone inclusions. Moreover, an additional projection technique is used to derive strong convergence of a modified SDR algorithm. [ABSTRACT FROM AUTHOR]

In real Hilbert spaces, given a single-valued Lipschitz continuous and monotone operator, we study generalized split feasibility problem (GSFP) over solution set of monotone variational inclusion problem. An inertia iterative method is proposed to solve this problem, by showing that the sequence generated by the iteration converges strongly to solution of GSFP. As against previous methods, our step size is chosen to be simple and not depending on norm of associated bounded linear map as well as Lipschitz constant of the single-valued operator. The obtained result was applied to study split linear inverse problem, precisely, the LASSO problem. Lastly, with the aid of numerical examples, we exhibited efficiency of our algorithm and its dominance over other existing schemes. [ABSTRACT FROM AUTHOR]

In this paper, we survey iterative algorithms for solving split variational inequalities and fixed point problems in Hilbert spaces. The investigated split problem is involved in two pseudomonotone operators and two pseudocontractive operators. We propose a self-adaptive Tseng-type algorithm for finding a solution of the split problem. Strong convergence of the suggested algorithm is shown under weaker conditions than sequential weak-to-weak continuity imposed on two pseudomonotone operators. [ABSTRACT FROM AUTHOR]

In this research, we study convex bi-level optimization problems for which the inner level consists of the sum of two proper, convex, and lower semi-continuous functions. We propose and analyze a new accelerated forward-backward algorithm using linesearch and inertial techniques for solving a solution of convex bi-level optimization. We then establish a strong convergence theorem of the proposed method under some suitable conditions. As an application, we apply our algorithm to solving data classifications of some non-communicable diseases. We conduct a comparative analysis with existing algorithms to show the effectiveness of our algorithm. Our numerical experiments confirm that our proposed algorithm outperforms other methods in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new method for solving variational inclusion problems in Hilbert spaces. This algorithm uses two inertial terms to speed up the convergence. In order to avoid computing the Lipschitz stepsize, we use an updated stepsize which is not necessary to know the Lipschitz constant of the operator. The weak convergence is established under some mild conditions. We present numerical performance of the proposed algorithm and compare our algorithm with other algorithms in literature. Finally, we deduce our algorithm for solving the convex minimization problem and give an application to the data classification problem of heart failure dataset. [ABSTRACT FROM AUTHOR]

We introduce a novel class of asymptotically α-hemicontractive mappings and demonstrate its relationship with the existing related families of mappings. We establish certain interesting properties of the fixed point set of the new class of mappings. Furthermore, we propose and investigate a new iterative algorithm for solving split common fixed point problem for the new class of mappings. In particular, weak and strong convergence theorems for solving split common fixed point problem for our new class of mappings in Hilbert spaces are proved. Moreover, using our method, we require no prior knowledge of norm of the transfer operator. The results presented in the paper extend and improve the results of Censor and Segal [Censor, Y.; Segal, A. The split common fixed point problem for directed operators. J. Convex Anal. 16 (2009), no. 2, 587-600.], Moudafi [Moudafi, A. The split common fixed-point problem for demicontractive mappings. Inverse Problems 26 (2010), no. 5:055007.; Moudafi, A. A note on the split common fixed-point problem for quasi-nonexpansive operators. Nonlinear Anal. 74 (2011), no. 12, 4083-4087.], Chima and Osilike [Chima, E. E.; Osilike, M. O. Split common fixed point problem for class of asymptotically hemicontractive mappings. J. Nigerian Math. Soc. 38 (2019), no. 3, 363-390.], Fan et al [Fan, Q.; Peng, J.; He, H. Weak and strong convergence theorems for the split common fixed point problem with demicontractive operators. Optimization 70 (2021), no. 5-6, 1409-1423.] and host of other related results in literature. [ABSTRACT FROM AUTHOR]

Interacting particle or agent systems that exhibit diverse swarming behaviors are prevalent in science and engineering. Developing effective differential equation models to understand the connection between individual interaction rules and swarming is a fundamental and challenging goal. In this paper, we study the data-driven discovery of a second-order particle swarming model that describes the evolution of N particles in \mathbb {R}^d under radial interactions. We propose a learning approach that models the latent radial interaction function as Gaussian processes, which can simultaneously fulfill two inference goals: one is the nonparametric inference of the interaction function with pointwise uncertainty quantification, and the other is the inference of unknown scalar parameters in the noncollective friction forces of the system. We formulate the learning problem as a statistical inverse learning problem and introduce an operator-theoretic framework that provides a detailed analysis of recoverability conditions, establishing that a coercivity condition is sufficient for recoverability. Given data collected from M i.i.d trajectories with independent Gaussian observational noise, we provide a finite-sample analysis, showing that our posterior mean estimator converges in a Reproducing Kernel Hilbert Space norm, at an optimal rate in M equal to the one in the classical 1-dimensional Kernel Ridge regression. As a byproduct, we show we can obtain a parametric learning rate in M for the posterior marginal variance using L^{\infty } norm and that the rate could also involve N and L (the number of observation time instances for each trajectory) depending on the condition number of the inverse problem. We provide numerical results on systems exhibiting different swarming behaviors, highlighting the effectiveness of our approach in the scarce, noisy trajectory data regime. [ABSTRACT FROM AUTHOR]

A Riesz basis for a separable Hilbert space H is the image of an orthonormal basis under a bounded, linear and bijective operator acting on H. Equivalently, it is an exact frame that shares the properties of a basis for H. Let G be a σ -compact and metrizable locally compact abelian group, and s and r be positive integers. It is illustrated that the image of a matrix-valued orthonormal basis under a bijective, bounded and linear operator acting on the matrix-valued signal space L 2 (G , ℂ s × r) may not be a frame, hence not a basis of the space L 2 (G , ℂ s × r). We introduce a notion of matrix-valued Riesz basis in the space L 2 (G , ℂ s × r) , where the adjointability of a bounded linear operator in the definition of Riesz basis with respect to the matrix-valued inner product plays a crucial role. We establish the existence of matrix-valued Riesz bases of the space L 2 (G , ℂ s × r). Extending results for standard Riesz bases of separable Hilbert spaces, we give necessary and sufficient conditions, and a characterization of matrix-valued Riesz bases of the space L 2 (G , ℂ s × r). [ABSTRACT FROM AUTHOR]

We consider bounded linear operators from the view of functional reproducing property. For some bounded linear operators associated with orthogonal polynomials we define an inner product space associated with a kernel constructed with orthogonal polynomials, show that it is a functional reproducing kernel Hilbert space (FRKHS) associated with these bounded linear operators and give decay rate for the best FRKHS approximation with a K -functional associated with the FRKHS. On this basis, we provide a learning rate for kernel regularized regression whose hypothesis space is the defined FRKHS. As applications, we define some concrete FRKHSs associated with polynomial operators such as the Bernstein–Durrmeyer operators, the de la Vallée Poussin operators on both the unit sphere S d − 1 and the unit ball B d . We show that these polynomial operators have reproducing property with respect to the corresponding concrete FRKHSs and show the learning rate for the kernel regularized regression. In short, we provide a way of constructing FRKHS operators with Fourier multipliers and show a learning framework from the view of operator approximation. [ABSTRACT FROM AUTHOR]

We introduce a framework for performing vector-valued regression in finite-dimensional Hilbert spaces. Using Lipschitz smoothness as our regularizer, we leverage Kirszbraun's extension theorem for off-data prediction. We analyze the statistical and computational aspects of this method—to our knowledge, its first application to supervised learning. We decompose this task into two stages: training (which corresponds operationally to smoothing/regularization) and prediction (which is achieved via Kirszbraun extension). Both are solved algorithmically via a novel multiplicative weight updates (MWU) scheme, which, for our problem formulation, achieves significant runtime speedups over generic interior point methods. Our empirical results indicate a dramatic advantage over standard off-the-shelf solvers in our regression setting. [ABSTRACT FROM AUTHOR]

This manuscript proposes a generalization of weak convergence and study of fixed point in a real Hilbert space for quasi-nonexpanding maps. In this work, we introduce a class of fixed points theorems for nonexpansive mapping and generalized form of nonexpansive mapping under the Hilbert space. In addition, we obtained under quasi nonexpansive mapping weak convergence with respect to Hilbert space by Mann’s Type. [ABSTRACT FROM AUTHOR]

We present two limit theorems for the zero-range process with nonlocal diffusion on inhomogeneous networks. The deterministic model is governed by the reaction–diffusion equation with an integral term in space instead of a Laplacian. By constructing the reproducing kernel Hilbert space to consider the inhomogeneities of the network structure, we prove that the law of large numbers and the central limit theorem hold for our models. Furthermore, under a special case of the kernel, we can show that the fluctuation limit obtained by the central limit theorem has a continuous sample path. [ABSTRACT FROM AUTHOR]

We fully characterize noncommutative symmetric spaces E(\mathcal {M},\tau) affiliated with a semifinite von Neumann algebra \mathcal {M} equipped with a faithful normal semifinite trace \tau on a (not necessarily separable) Hilbert space having the Gelfand–Phillips property and the WCG-property. The complete list of their relations with other classical structural properties (such as the Dunford–Pettis property, the Schur property and their variations) is given in the general setting of noncommutative symmetric spaces. [ABSTRACT FROM AUTHOR]

We study the positive-definite completion problem for kernels on a variety of domains and prove results concerning the existence, uniqueness, and characterization of solutions. In particular, we study a special solution called the canonical completion which is the reproducing kernel analogue of the determinant-maximizing completion known to exist for matrices. We establish several results concerning its existence and uniqueness, which include algebraic and variational characterizations. Notably, we prove the existence of a canonical completion for domains which are equivalent to the band containing the diagonal. This corresponds to the existence of a canonical extension in the context of the classical extension problem of positive-definite functions, which can be understood as the solution to an abstract Cauchy problem in a certain reproducing kernel Hilbert space. [ABSTRACT FROM AUTHOR]

We propose a new functional additive hazards model to investigate the potential effects of functional and scalar predictors on mortality risks, and develop a penalized least squares estimation method for model parameters based on a pseudoscore estimating equation. A reproducing kernel Hilbert space approach is used to establish the consistency, convergence rate, and joint asymptotic distribution of the resulting estimators for finite‐dimensional and infinite‐dimensional parameters. Our simulation studies demonstrate that the proposed estimation procedure performs well. For illustration, we apply the proposed method to the Medical Information Mart for Intensive Care III dataset. [ABSTRACT FROM AUTHOR]

The usual theory of negative type (and p-negative type) is heavily dependent on an embedding result of Schoenberg, which states that a metric space isometrically embeds in some Hilbert space if and only if it has 2-negative type. A generalisation of this embedding result to the setting of bi-lipschitz embeddings was given by Linial, London and Rabinovich. In this article we use this newer embedding result to define the concept of distorted p-negative type and extend much of the known theory of p-negative type to the setting of bi-lipschitz embeddings. In particular we show that a metric space (X , d X) has p-negative type with distortion C ( 0 ≤ p < ∞ , 1 ≤ C < ∞ ) if and only if (X , d X p / 2) admits a bi-lipschitz embedding into some Hilbert space with distortion at most C. Analogues of strict p-negative type and polygonal equalities in this new setting are given and systematically studied. Finally, we provide explicit examples of these concepts in the bi-lipschitz setting for the bipartite graphs K m , n . [ABSTRACT FROM AUTHOR]

In the framework of a second‐order semilinear integro‐differential control system in Hilbert spaces, the paper provides sufficient conditions for proving the existence of optimal control. The Banach fixed point theorem is used to investigate the existence and uniqueness of mild solutions for the proposed problem. Additionally, it is shown that, under specific assumptions, there exists at least one optimal control pair for the Lagrange's problem as presented in the article. An example for validation is included in the paper to further support the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study a class of non-Lipschitz quasi-monotone variational inequalities. We introduce a new inertial algorithm with self-adaptive step sizes for finding the minimum-norm solutions of the variational inequalities. Unlike the existing results on non-Lipschitz variational inequalities, our algorithm does not require any linesearch technique. We prove that the sequence generated by our proposed algorithm converges strongly to the minimum-norm solutions of the variational inequalities in Hilbert spaces. In addition, we also consider a class of non-Lipschitz variational inequalities without monotonicity. Finally, we present several numerical examples to illustrate our algorithm by comparing it with some existing methods. [ABSTRACT FROM AUTHOR]

This paper investigates numerical solutions of pseudomontone variational inequality problems in real Hilbert spaces. Two new algorithms based on the celebrated extragradient method are introduced. Under suitable conditions, it is proved that the two algorithms achieve weak convergence and linear convergence rates. The results extend and improve upon related works in the literature on the associated extragradient algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the problem of solving strongly monotone variational inequalities over the solution sets of split feasibility problems with multiple output sets in real Hilbert spaces. We present two new iterative algorithms when the involved subsets are given as the level sets of convex functions. In our algorithms, the projection to the half-space is replaced by the one to the intersection of two half-spaces. The algorithms are accelerated using the inertial technique and eliminate the need for calculating or estimating the norms of linear operators by employing self-adaptive step size criteria. We give convergence of the sequence generated by our algorithms under some suitable assumptions. Some applications to monotone variational inequalities over the solution set of the split feasibility problem are also reported. Finally, we present some numerical examples to illustrate the efficiency and implementation of our algorithms in comparison with existing algorithms in the literature. [ABSTRACT FROM AUTHOR]

This paper presents two new algorithms for solving multivalued variational inequality problems in a real Hilbert space. By combining the nonexpansiveness of proximal operators associated with the proper lower semicontinuous convex function of the problems and inertial techniques, we demonstrate the weak convergence of the iteration sequences generated by our first algorithm under monotone and Lipschitz continuous assumptions of the cost mappings. Next, we use Mann iteration technique to obtain the second algorithm and show its strong convergence. Finally, we give some numerical results for two proposed algorithms and comparison with some other known algorithms. [ABSTRACT FROM AUTHOR]

The purpose of our paper is to present a few convergence results for the Krasnoselskii–Mann iteration and also the modified Krasnoselskii–Mann iteration used for approximating the fixed point of k-strictly pseudo contractive map in the Banach space, which can be derived from the corresponding convergence theorems in the class of non-expansive maps. Also, using the enrichment techniques for contractive type map T, i.e., the averaged operator T λ (u) = (1 - λ) u + λ T u , where λ ∈ (0 , 1 ] , we identified that pseudo-contractive and demi-contractive maps are an unsaturated class of mappings in the Banach space. Illustrative examples and numerical calculations are given to support the obtained results, which are in fact the generalizations of approximations results in the Banach space from the Hilbert space. [ABSTRACT FROM AUTHOR]

In this work, we propose an inertial extragradient method for solving strongly pseudomonotone equilibrium problems utilizing a novel self-adaptive stepsize approach. We establish the R-linear convergence rate of the proposed method without prior knowledge of the Lipschitz-type constants associated with the bifunction. We also discuss the application of the obtained results to variational inequality problems involving strongly pseudomonotone and Lipschitz continuous mapping. Numerical examples are presented to illustrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

We introduce a new optimization approach to solving systems of split equality problems in real Hilbert spaces. We use the inertial method in order to improve the convergence rate of the proposed algorithms. Our algorithms do not depend on the norms of the bounded linear operators which appear in each split equality problem of the system under consideration. This is also a strong point of our algorithms because it is known that it is difficult to compute or estimate the norm of a linear operator in the general case. [ABSTRACT FROM AUTHOR]