Infinite-dimensional Hilbert Space Research Articles

Brownian motion in the Hilbert space of quantum states and the stochastically emergent Lorentz symmetry: A fractal geometric approach from Wiener process to formulating Feynman’s path-integral measure for relativistic quantum fields

This paper aims to provide a consistent, finite-valued, and mathematically well-defined reformulation of Feynman’s path-integral measure for quantum fields obtained by studying the Wiener stochastic process in the infinite-dimensional Hilbert space of quantum states. This reformulation will undoubtedly have a crucial role in formulating quantum gravity within a mathematically well-defined framework. In fact, this study is fundamentally different from any previous research on the relationship between Feynman’s path-integral and the Wiener stochastic process. In this research, we focus on the fact that the classic Wiener measure is no longer applicable in infinite-dimensional Hilbert spaces due to fundamental differences between displacements in low and extremely high dimensions. Thus, an analytic norm motivated by the role of the fractal functions in the Wilsonian renormalization approach is worked out to properly characterize Brownian motion in the Hilbert space of quantum states on a compact flat manifold. This norm, the so-called fractal norm, pushes the rougher functions (physically the quantum states with higher energies) to the farther points of the Hilbert space until the fractal functions as the roughest ones are moved to infinity. Implementing the Wiener stochastic process with the fractal norm, results in a modified form of the Wiener measure called the Wiener fractal measure, which is a well-defined measure for Feynman’s path-integral formulation of quantum fields. Wiener fractal measure has a complicated formula of non-local terms but produces the Klein–Gordon action at the first order of approximation. Using complex integrals to compensate for the removal of non-local terms appearing in higher orders of approximation, the Wiener fractal measure turns into a complex measure and generates Feynman’s path-integral formulation of scalar quantum fields. This brings us to the main objective of this study. Finally, some various significant aspects of quantum field theory (such as renormalizability, RG flow, Wick rotation, regularization, etc.) are revisited by means of the analytical aspects of the Wiener fractal measure.

Fuzzy gauge theory for quantum computers

Continuous gauge theories, because of their bosonic degrees of freedom, have an infinite-dimensional local Hilbert space. Encoding these degrees of freedom on qubit-based hardware demands some sort of “qubitization” scheme, where one approximates the behavior of a theory while using only finitely many degrees of freedom. We propose a novel qubitization strategy for gauge theories, called “fuzzy gauge theory,” building on the success of the fuzzy σ-model in earlier work. We provide arguments that the fuzzy gauge theory lies in the same universality class as regular gauge theory, in which case its use would obviate the need of any further limit besides the usual spatial continuum limit. Furthermore, we demonstrate that these models are relatively resource-efficient for quantum simulations. Published by the American Physical Society 2024

Design of Quantum Gates Using Quantum Scattering Theory

Given two Hamiltonians \(H_0,H_1=H_0+V\) corresponding respectively to a free particle Hamiltonian and the free particle Hamiltonian plus its interaction energy with a scattering centre, we can compute the S-matrix, i.e., the matrix that determines the quantum mechanical amplitude of the process involving a particle coming from a free particle state at an infinite distance from the scattering centre at time \(t=-\infty\), entering into an initial scattered state, interacting with the scattering centre, entering into a final scattered state, and then escaping away to infinity at time \(t\rightarrow\infty\) to a final free particle state. Such amplitudes were first studied by Lippmann and Schwinger (Steven Weinberg, The Quantum Theory of Fields, Vol. 1, W.O. Amrein, Hilbert Space Methods in Quantum Mechanics) and then analyzed in a mathematically precise way by [T. Kato], [W. Amrein], and [K.B. Sinha]. In this paper, we first derive the Lippmann-Schwinger equation for the input and output scattered states in terms of the input and output free particle states using intuitive arguments originally due to Lippmann and Schwinger, and then, using rigorous operator theoretic arguments revolving around the spectral theorem for unbounded self-adjoint operators in a Hilbert space, we obtain an elegant formula for the scattering matrix elements in terms of the interaction potential and the free particle Hamiltonian between two free particle states corresponding to the same measure. We then discuss a computationally more efficient method based on the Dyson series expansion (Steven Weinberg, The Quantum Theory of Fields, Vol. 1), which works even in the case when the interaction Hamiltonian is time varying, in contrast to the Lippmann-Schwinger method, for calculating the S-matrix. We assume that the scattering potential, which can be time varying, can be controlled by incorporating control parameters and explain how the resulting unitary S-matrix can be made to approximate a given unitary gate in infinite-dimensional Hilbert space by optimizing the Hilbert-Schmidt/Frobenius norm distance between the given gate and the S-matrix gate, in which the latter is approximated by a truncated Dyson series. Of course, the S-matrix given by the Dyson series is a Taylor series functional expansion in the interaction potential and hence is a highly nonlinear function of the control parameters, and hence we propose numerical methods for such optimization. We then observe that if the control parameters are taken as random variables with joint moments, then the TPCP map involving statistical averaging of the adjoint action of the S-matrix acting on an initial mixed state w.r.t. the probability distribution of the control parameters will result in a final mixed state that is linear in the statistical moments of the control parameters, and hence optimization w.r.t. these moments can be easily carried out. Our method of TPCP map design is to optimize the sum of Frobenius distance squares between a sequence of desired output mixed states and the S-matrix based formula for the TPCP map acting on the corresponding input mixed states w.r.t. the statistical moments of the control parameters. We then generalize this method to include the case of TPCP maps obtained from the Hudson-Parthasarathy quantum noisy Schrödinger equation. In summary, we simulate the S-matrix gate for independent realizations of the control parameters having a probability distribution defined by their joint moments obtained by the above optimization procedure and apply the adjoint action of these independent S-matrices on a given input state, and take their ensemble average to obtain a good approximation to the action of the desired TPCP map on any given input state. This suggests that TPCP maps of arbitrarily large size can be realized using scattering experiments in the laboratory.

Strong convergence of split equality variational inequality, variational inclusion, and multiple sets fixed point problems in Hilbert spaces with application

This paper introduces an innovative inertial simultaneous cyclic iterative algorithm designed to address a range of mathematical problems within the realm of split equality variational inequalities. Specifically, the algorithm accommodates finite families of split equality variational inequality problems, infinite families of split equality variational inclusion problems, and multiple-sets split equality fixed point problems involving demicontractive operators in infinite-dimensional Hilbert spaces. The algorithm integrates well-established methods, including the cyclic method, the inertial method, the viscosity approximation method, and the projection method. We establish the strong convergence of this proposed algorithm, demonstrating its applicability in various scenarios and unifying disparate findings from existing literature. Additionally, a numerical example is presented to validate the primary convergence theorem.

Synchronization of delayed stochastic reaction–diffusion Hopfield neural networks via sliding mode control

Synchronization of stochastic reaction–diffusion Hopfield neural networks with s-delays via sliding mode control is investigated in this article. To begin with, we choose suitable functional space for state variables, then the system is transformed into a functional differential equation in an infinite-dimensional Hilbert space by using appropriate functional analysis technique. Based on above preliminary preparation, sliding mode control (SMC) is constructed to drive the error trajectory into the designed switching surface. Specifically, the switching surface is constructed as linear combination of state variables, which is related to control gains. Then novel SMC law is designed which involving delay, reaction diffusion term, and reaching law. Furthermore, the criterion of mean-square exponential synchronization for stochastic delayed reaction–diffusion Hopfield neural networks with s-delays is given in the form of matrix form. This criterion is less restrictive and easy to check in computer. Meanwhile, a different novel Lyapunov–Krasovskii functional (LKF) mixed with Itô’s formula, Young inequality, Hanalay inequality is employed in this proof procedure. At last, a numerical example is presented to validate the availability of theoretical result. The simulation is based on the finite difference method, and numerical result coincides with the theoretical result proposed.

A structured L-BFGS method and its application to inverse problems

Many inverse problems are phrased as optimization problems in which the objective function is the sum of a data-fidelity term and a regularization. Often, the Hessian of the fidelity term is computationally unavailable while the Hessian of the regularizer allows for cheap matrix-vector products. In this paper, we study an L-BFGS method that takes advantage of this structure. We show that the method converges globally without convexity assumptions and that the convergence is linear under a Kurdyka–Łojasiewicz-type inequality. In addition, we prove linear convergence to cluster points near which the objective function is strongly convex. To the best of our knowledge, this is the first time that linear convergence of an L-BFGS method is established in a non-convex setting. The convergence analysis is carried out in infinite dimensional Hilbert space, which is appropriate for inverse problems but has not been done before. Numerical results show that the new method outperforms other structured L-BFGS methods and classical L-BFGS on non-convex real-life problems from medical image registration. It also compares favorably with classical L-BFGS on ill-conditioned quadratic model problems. An implementation of the method is freely available.

Polynomial convergence of iterations of certain random operators in Hilbert space

Abstract We study the convergence of a random iterative sequence of a family of operators on infinite-dimensional Hilbert spaces, inspired by the stochastic gradient descent (SGD) algorithm in the case of the noiseless regression. We identify conditions that are strictly broader than previously known for polynomial convergence rate in various norms, and characterize the roles the randomness plays in determining the best multiplicative constants. Additionally, we prove almost sure convergence of the sequence.

Lipschitz continuity of the dilation of Bloch functions on the unit ball of a Hilbert space and applications

Let BE\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_E$$\\end{document} be the open unit ball of a complex finite- or infinite-dimensional Hilbert space. If f belongs to the space B(BE)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {B}(B_E)$$\\end{document} of Bloch functions on BE\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_E$$\\end{document}, we prove that the dilation map given by x↦(1-‖x‖2)Rf(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x \\mapsto (1-\\Vert x\\Vert ^2) \\mathcal {R}f(x)$$\\end{document} for x∈BE\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x \\in B_E$$\\end{document}, where Rf\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {R}f$$\\end{document} denotes the radial derivative of f, is Lipschitz continuous with respect to the pseudohyperbolic distance ρE\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho _E$$\\end{document} in BE\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_E$$\\end{document}, which extends to the finite- and infinite-dimensional setting the result given for the classical Bloch space B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {B}$$\\end{document}. To provide this result, we will need to prove that ρE(zx,zy)≤|z|ρE(x,y)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho _E(zx,zy) \\le |z| \\rho _E(x,y)$$\\end{document} for x,y∈BE\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x,y \\in B_E$$\\end{document} under some conditions on z∈C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z \\in \\mathbb {C}$$\\end{document}. Lipschitz continuity of x↦(1-‖x‖2)Rf(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x \\mapsto (1-\\Vert x\\Vert ^2) \\mathcal {R}f(x)$$\\end{document} will yield some applications on interpolating sequences for B(BE)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {B}(B_E)$$\\end{document} which also extends classical results from B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {B}$$\\end{document} to B(BE)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {B}(B_E)$$\\end{document}. Indeed, we show that it is necessary for a sequence in BE\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_E$$\\end{document} to be separated to be interpolating for B(BE)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {B}(B_E)$$\\end{document} and we also prove that any interpolating sequence for B(BE)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {B}(B_E)$$\\end{document} can be slightly perturbed and it remains interpolating.

Chow's theorem for Hilbert Grassmannians as a Wigner-type theorem

Let H be an infinite-dimensional complex Hilbert space. Denote by G ∞ ( H ) the Grassmannian formed by closed subspaces of H whose dimension and codimension both are infinite. We say that X , Y ∈ G ∞ ( H ) are ortho-adjacent if they are compatible and X ∩ Y is a hyperplane in both X, Y. A subset C ⊂ G ∞ ( H ) is called an A-component if for any X , Y ∈ C the intersection X ∩ Y is of the same finite codimension in both X, Y and C is maximal with respect to this property. Let f be a bijective transformation of G ∞ ( H ) preserving the ortho-adjacency relation in both directions. We show that the restriction of f to every A-component is induced by a unitary or anti-unitary operator or it is the composition of the orthocomplementary map and a map induced by a unitary or anti-unitary operator. Note that the restrictions of f to distinct A-components can be related to different operators.

A Unifying Framework for Economic Equilibrium and Optimal Transport in Infinite-Dimensional Hilbert Spaces

A Unifying Framework for Economic Equilibrium and Optimal Transport in Infinite-Dimensional Hilbert Spaces

Quantum detection problem for fusion frames

A quantum injective frame is a frame whose measurements for density operators can be used as a distinguishing feature in a quantum system, and the frame quantum detection problem demands a characterization of all such frames. Very recently, the quantum detection problem for continuous as well as discrete frames in both finite and infinite dimensional Hilbert spaces received significant attention. The quantum detection problem pertaining to the characterization of informationally complete positive operator-valued measures (POVM) can be split into two cases: The quantum injectivity or state separability problem and the rang analysis or quantum state estimation problem. Building upon this notion, this note is aimed at the quantum detection problem for fusion frames. The injectivity of a family of vectors and a family of closed subspaces is characterized in terms of some operator equations in Hilbert–Schmidt and trace classes.

Estimating truncation effects of quantum bosonic systems using sampling algorithms

To simulate bosons on a qubit- or qudit-based quantum computer, one has to regularize the theory by truncating infinite-dimensional local Hilbert spaces to finite dimensions. In the search for practical quantum applications, it is important to know how big the truncation errors can be. In general, it is not easy to estimate errors unless we have a good quantum computer. In this paper, we show that traditional sampling methods on classical devices, specifically Markov Chain Monte Carlo, can address this issue for a rather generic class of bosonic systems with a reasonable amount of computational resources available today. As a demonstration, we apply this idea to the scalar field theory on a two-dimensional lattice, with a size that goes beyond what is achievable using exact diagonalization methods. This method can be used to estimate the resources needed for realistic quantum simulations of bosonic theories, and also, to check the validity of the results of the corresponding quantum simulations.

Quantum and Classical Spin-Network Algorithms for q-Deformed Kogut-Susskind Gauge Theories.

Treating the infinite-dimensional Hilbert space of non-Abelian gauge theories is an outstanding challenge for classical and quantum simulations. Here, we employ q-deformed Kogut-Susskind lattice gauge theories, obtained by deforming the defining symmetry algebra to a quantum group. In contrast to other formulations, this approach simultaneously provides a controlled regularization of the infinite-dimensional local Hilbert space while preserving essential symmetry-related properties. This enables the development of both quantum as well as quantum-inspired classical spin-network algorithms for q-deformed gauge theories. To be explicit, we focus on SU(2)_{k} gauge theories with k∈N that are controlled by the deformation parameter q=e^{2πi/(k+2)}, a root of unity, and converge to the standard SU(2) Kogut-Susskind model as k→∞. In particular, we demonstrate that this formulation is well suited for efficient tensor network representations by variational ground-state simulations in 2D, providing first evidence that the continuum limit can be reached with k=O(10). Finally, we develop a scalable quantum algorithm for Trotterized real-time evolution by analytically diagonalizing the SU(2)_{k} plaquette interactions. Our work gives a new perspective for the application of tensor network methods to high-energy physics and paves the way for quantum simulations of non-Abelian gauge theories far from equilibrium where no other methods are currently available.

Simulating gauge theories with variational quantum eigensolvers in superconducting microwave cavities

Quantum-enhanced computing methods are promising candidates to solve currently intractable problems. We consider here a variational quantum eigensolver (VQE), that delegates costly state preparations and measurements to quantum hardware, while classical optimization techniques guide the quantum hardware to create a desired target state. In this work, we propose a bosonic VQE using superconducting microwave cavities, overcoming the typical restriction of a small Hilbert space when the VQE is qubit based. The considered platform allows for strong nonlinearities between photon modes, which are highly customisable and can be tuned in situ, i.e. during running experiments. Our proposal hence allows for the realization of a wide range of bosonic ansatz states, and is therefore especially useful when simulating models involving degrees of freedom that cannot be simply mapped to qubits, such as gauge theories, that include components which require infinite-dimensional Hilbert spaces. We thus propose to experimentally apply this bosonic VQE to the U(1) Higgs model including a topological term, which in general introduces a sign problem in the model, making it intractable with conventional Monte Carlo methods.

The semi-Fredholmness and property (ω) for 2×2 upper triangular operator matrices

Let be a upper triangular operator matrix acting on , where and are complex infinite dimensional separable Hilbert spaces. In this paper, we study the semi-Fredholmness for for some invertible , and using the conclusions obtained, we characterize the equivalent conditions such that for any invertible .

Master Bellman equation in the Wasserstein space: Uniqueness of viscosity solutions

We study the Bellman equation in the Wasserstein space arising in the study of mean field control problems, namely stochastic optimal control problems for McKean-Vlasov diffusion processes. Using the standard notion of viscosity solution à la Crandall-Lions extended to our Wasserstein setting, we prove a comparison result under general conditions on the drift and reward coefficients, which coupled with the dynamic programming principle, implies that the value function is the unique viscosity solution of the Master Bellman equation. This is the first uniqueness result in such a second-order context. The classical arguments used in the standard cases of equations in finite-dimensional spaces or in infinite-dimensional separable Hilbert spaces do not extend to the present framework, due to the awkward nature of the underlying Wasserstein space. The adopted strategy is based on finite-dimensional approximations of the value function obtained in terms of the related cooperative n n -player game, and on the construction of a smooth gauge-type function, built starting from a regularization of a sharp estimate of the Wasserstein metric; such a gauge-type function is used to generate maxima/minima through a suitable extension of the Borwein-Preiss generalization of Ekeland’s variational principle on the Wasserstein space.

Novel extended Halpern-type convergence algorithms for the split common fixed point problem involving α-demicontractive operators

As is well known, it is quite valuable and challenging to solve the split common fixed point problem (SCFP) associated with two different nonlinear operators. In this study, in order to solve (SCFP) with a more general setting of α-demicontractive operators due to Măruşter and Măruşter in 2011, we develop a kind of novel iterative approach in infinite dimensional Hilbert spaces. Further, using Xu's inequality and Maingé's lemma, strong convergence theorems of the suggested algorithms are established under some mild conditions. Finally, some special cases are considered, and the main results of this work are supported via two numerical comparisons.

Entangled coherent states and violations of Bell-CHSH inequalities

Three classes of entangled coherent states are employed to study the Bell-CHSH inequality. By using pseudospin operators in infinite dimensional Hilbert spaces, four dichotomic operators (A,A′,B,B′) entering the inequality are constructed. For each class of coherent states, we compute the correlator 〈ψ|AB+A′B+AB′−A′B′|ψ〉, analyzing the set of parameters that leads to a Bell-CHSH inequality violation and, particularly, to the saturation of Tsirelson's bound.

Theoretical Aspects in Penalty Hyperparameters Optimization

Learning processes play an important role in enhancing understanding and analyzing real phenomena. Most of these methodologies revolve around solving penalized optimization problems. A significant challenge arises in the choice of the penalty hyperparameter, which is typically user-specified or determined through Grid search approaches. There is a lack of automated tuning procedures for the estimation of these hyperparameters, particularly in unsupervised learning scenarios. In this paper, we focus on the unsupervised context and propose a bi-level strategy to address the issue of tuning the penalty hyperparameter. We establish suitable conditions for the existence of a minimizer in an infinite-dimensional Hilbert space, along with presenting some theoretical considerations. These results can be applied in situations where obtaining an exact minimizer is unfeasible. Working on the estimation of the hyperparameter with the gradient-based method, we also introduce a modified version of Ekeland’s principle as a stopping criterion for these methods. Our approach distinguishes from conventional techniques by reducing reliance on random or black-box strategies, resulting in stronger mathematical generalization.

Bounded below composition operators on the space of Bloch functions on the unit ball of a Hilbert space

Let B_E be the open unit ball of a complex finite or infinite dimensional Hilbert space E and consider the space mathcal {B}(B_E) of Bloch functions on B_E. Using Lipschitz continuity of the dilation map on B_E given by x mapsto (1-Vert xVert ^2) mathcal {R}f(x) for x in B_E, where mathcal {R}f denotes the radial derivative of f in mathcal {B}(B_E), we study when a composition operator on mathcal {B}(B_E) is bounded below.