We present a variety of projection-based linear regression algorithms with a focus on modern machine-learning models and their algorithmic performance. We study the role of the relaxation parameter in generalized Kaczmarz algorithms and establish a priori regret bounds with explicit λ-dependence to quantify how much an algorithm’s performance deviates from its optimal performance. A detailed analysis of relaxation parameter is also provided. Applications include: explicit regret bounds for the framework of Kaczmarz algorithm models, non-orthogonal Fourier expansions, and the use of regret estimates in modern machine learning models, including for noisy data, i.e., regret bounds for the noisy Kaczmarz algorithms. Motivated by machine-learning practice, our wider framework treats bounded operators (on infinite-dimensional Hilbert spaces), with updates realized as (block) Kaczmarz algorithms, leading to new and versatile results.

The paper deals with partial and weak preference relations defined on infinite-dimensional vector spaces and compatible with algebraic operations. By a partial preference we mean an asymmetric and transitive binary relation, while a weak preference is such a partial preference for which the indifference relation corresponding it is transitive (an indifference relation is the complement to the union of a partial preference and the reverse to it). Our first aim is to study the internal structure of compatible partial and weak preferences. We suppose that the internal structure of a compatible partial preference is elementary if its positive cone is relatively algebraic open, and we refer to such compatible partial preferences as relatively open ones. It is proved that an arbitrary compatible partial preference is the disjoint union of the partially ordered family of relatively open compatible partial preferences and, moreover, as an ordered set this family is an upper semilattice. We identify the structure of this upper semilattice with the internal structure of a compatible partial preference. Further we prove that each compatible weak preference is analytically represented by a step-linear function while for the analytical representation of a compatible partial preference one needs the family of step-linear functions.

A bstract We formulate interacting antisymmetric tensor gauge theory in a configuration space consisting of a pair of dual field strengths which has a natural symplectic structure. The field equations are formulated as the intersection of a pair of submanifolds of this infinite-dimensional symplectic configuration space, one of which is a Lagrangian submanifold while the other is either a coisotropic or Lagrangian submanifold, depending on the topology. Chern-Simons interactions give the configuration space an interesting global structure. We consider in detail the example of a six-dimensional theory of a 3-form field strength coupled to Yang-Mills theory via a Chern-Simons interaction. Our approach applies to a broad class of gauge systems.

Abstract The geometric description of incompressible hydrodynamics, as geodesic motion on the infinite-dimensional group of volume-preserving diffeomorphisms, enables notions of curvature in the study of fluids in order to study stability. Formulas for Ricci curvature are often simpler than those for sectional curvature, which typically takes both signs, but the drawback is that Ricci curvature is rarely well-defined in infinite-dimensional spaces. Here we suggest a definition of Ricci curvature in the case of two-dimensional hydrodynamics, based on the finite-dimensional Zeitlin models arising in quantization theory, which gives a natural tool for renormalization. We provide formulae for the finite-dimensional approximations and give strong numerical evidence that these converge in the infinite-dimensional limit, based in part on four new conjectured identities for Wigner 6 j symbols. The suggested limiting expression for (average) Ricci curvature is surprisingly simple and demonstrates an average instability for high-frequency modes which helps explain long-term numerical observations of spherical hydrodynamics due to mixing.

Let [Formula: see text] be a fixed positive integer, and let [Formula: see text] be a fixed odd integer coprime with [Formula: see text] such that [Formula: see text]. We investigate the stable recovery of the source term of the discrete dynamical system indexing over the non-uniform discrete set [Formula: see text] considered by Gabardo and Nashed in the context of one dimensional spectral pairs for infinite-dimensional separable Hilbert spaces. This is inspired by recent work due to Aldroubi et al. on stable recovery of source terms in dynamical systems. Extending results due to Aldroubi et al., firstly, we give a necessary and sufficient condition for the stable recovery of the source term in finitely many iterations. Afterwards, we derive a necessary condition for the stable recovery of the source term in finitely many iterations when it belongs to the closed subspace of an infinite-dimensional separable Hilbert space. Finally, we give a necessary and sufficient condition for the stable recovery of the source term in infinitely many iterations.

Let B ( X ) be the algebra of all bounded linear operators on an infinite-dimensional complex Banach space X. For an operator T ∈ B ( X ) , by K ( T ) we shall denote as usual the analytic core of T. We determine the form of surjective maps ϕ on B ( X ) satisfying K ( ϕ ( T ) ϕ ( S ) + ϕ ( S ) ϕ ( T ) ) = K ( TS + ST ) for all T , S ∈ B ( X ) .

We survey some classical results on the polynomial identities satisfied by Grassmann algebras, with a list of recent new results on their derivations. Other new results concern the differential polynomial identities of the Grassmann algebra \(E\) of an infinite-dimensional vector space over a field of characteristic zero, arising under the derivation action of some Lie subalgebras of \(Der(E)\).

A bstract This paper develops a framework for the Hamiltonian quantization of complex Chern-Simons theory with gauge group $$\text{SL}(2,{\mathbb{C}})$$ at an even level $$k\in {\mathbb{Z}}_{+}$$ . Our approach follows the procedure of combinatorial quantization to construct the operator algebras of quantum holonomies on 2-surfaces and develop the representation theory. The *-representation of the operator algebra is carried by the infinite dimensional Hilbert space $${\mathcal{H}}_{\overrightarrow{\lambda }}$$ and closely connects to the infinite-dimensional *-representation of the quantum deformed Lorentz group $${\mathcal{U}}_{\text{q}}\left(s{l}_{2}\right)\otimes {\mathcal{U}}_{\widetilde{\text{q}}}\left(s{l}_{2}\right)$$ . The quantum group $${\mathcal{U}}_{\text{q}}\left(s{l}_{2}\right)\otimes {\mathcal{U}}_{\widetilde{\text{q}}}\left(s{l}_{2}\right)$$ also emerges from the quantum gauge transformations of the complex Chern-Simons theory. Focusing on a m -holed sphere Σ 0, m , the physical Hilbert space $${\mathcal{H}}_{\text{phys}}$$ is identified by imposing the gauge invariance and the flatness constraint. The states in $${\mathcal{H}}_{\text{phys}}$$ are the $${\mathcal{U}}_{\text{q}}\left(s{l}_{2}\right)\otimes {\mathcal{U}}_{\widetilde{\text{q}}}\left(s{l}_{2}\right)$$ -invariant linear functionals on a dense domain in $${\mathcal{H}}_{\overrightarrow{\lambda }}$$ . Finally, we demonstrate that the physical Hilbert space carries a Fenchel-Nielsen representation, where a set of Wilson loop operators associated with a pants decomposition of Σ 0, m are diagonalized.

Light Transport Operators (LTOs) represent a fundamental concept in computer graphics, modeling single bounces of light within a virtual environment as linears operators on infinite dimensional spaces. While the LTOs play a crucial role in rendering, prior studies have primarily focused on spectral analyses of the light field rather than the operators themselves. This article presents a rigorous investigation into the spectral properties of the LTOs. Due to their non-compact nature, traditional spectral analysis techniques face challenges in this setting. However, many practical rendering methods effectively employ compact approximations, suggesting that non-compactness is not an absolute barrier. We show the relevance of such approximations and establish various path integral formulations of their spectrum. These findings enhance the theoretical understanding of light transport and offer new perspectives for improving rendering efficiency and accuracy.

In this paper, we introduce a generalized framework for conditional probability in stochastic processes taking values in infinite-dimensional normed spaces. Classical definitions, based on measurability with respect to a conditioning σ-algebra, become inadequate when the available information is restricted to a σ-algebra generated by a finite or countable family of random variables. In such settings, many events of interest are not measurable with respect to the conditioning σ-field, preventing the standard definition of conditional probability. To overcome this limitation, we propose an extension of the coherent conditioning model through the use of Hausdorff measures. The key idea is to exploit the non-equivalence of norms in infinite-dimensional spaces, which gives rise to distinct metric structures and corresponding Hausdorff dimensions for the same events. Conditional probabilities are then defined relative to families of Hausdorff outer measures parameterized by their dimensional exponents. This geometric reformulation allows the notion of conditionality to depend explicitly on the underlying metric and topological properties of the space. The resulting model provides a flexible and coherent framework for analyzing conditioning in infinite-dimensional stochastic systems, with potential implications for Bayesian inference in functional spaces.

ABSTRACT We study the optimal linear prediction of a random function that takes values in an infinite dimensional Hilbert space. We begin by characterizing the mean square prediction error (MSPE) associated with a linear predictor and discussing the minimal achievable MSPE. This analysis reveals that, in general, there are multiple non‐unique linear predictors that minimize the MSPE, and even if a unique solution exists, consistently estimating it from finite samples is generally impossible. Nevertheless, we can define asymptotically optimal linear operators whose empirical MSPEs approach the minimal achievable level as the sample size increases. We show that, interestingly, standard post‐dimension reduction estimators, which have been widely used in the literature, attain such asymptotic optimality under minimal conditions.

In this paper, we propose three variants of Mann-type inertial forward–backward iterative methods for approximating the minimum-norm solution of the monotone inclusion problem and the fixed points of nonexpansive mappings. In the first two methods, we compute the Mann-type iteration together with the inertial extrapolation and fixed-point iteration in the initiation of the process, while the last method computes only the Mann-type iteration with inertial extrapolation at the start of the process. We establish the strong convergence results for each method with appropriate assumptions and discuss some applications of the presented methods. Finally, we present numerical examples in both finite- and infinite-dimensional Hilbert spaces to demonstrate their efficiency. A comparative analysis with existing methods is also provided.

Operator learning is a recent development in the simulation of partial differential equationsby means of neural networks. The idea behind this approach is to learn the behavior of an operator, such that the resulting neural network is an approximate mapping in infinite-dimensional spaces that is capable of (approximately) simulating the solution operator governed by the partial differential equation. In our work, we study some general approximation capabilities for linear differential operators by approximating the corresponding symbol in the Fourier domain. Analogous to the structure of the class of Hörmander symbols, we consider the approximation with respect to a topology that is induced by a sequence of semi-norms. In that sense, we measure the approximation error in terms of a Fréchet metric, and our main result identifies sufficient conditions for achieving a predefined approximation error. We then focus on a natural extension of our main theorem, in which we reduce the assumptions on the sequence of seminorms. Based on existing approximation results for the exponential spectral Barron space, we then present a concrete example of symbols that can be approximated well.

Vortex beams, whose topological charges form an infinite-dimensional Hilbert space, theoretically offer unlimited communication capacity and hold great potential for future 6G applications. Addressing the challenges of time-consuming and inefficient traditional terahertz metasurface design methods, we propose a bidirectional deep neural network approach for designing all-dielectric transmissive vortex beam metasurfaces. The proposed metasurface was fabricated using 3D printing technology, with experimental results confirming its effective generation of terahertz vortex beams carrying the intended topological charge number l = -2. The study demonstrates that the trained bidirectional deep neural network can predict the phase and transmission coefficient for a single metasurface unit within 4 μs, significantly reducing design complexity. This approach saves substantial time and computational resources for designing terahertz vortex beam metasurfaces, providing an efficient pathway for rapid terahertz metasurface development.

In this paper, we design some generalized split problems which can be seen as an extended form of the split variational inequality problems. We present several iterative algorithms for solving generalized split problems and demonstrate the weak convergence results under some appropriate assumptions within the context of real Hilbert spaces. Finally, we support these results with the help of numerical examples in both the finite and infinite dimensional spaces. As a result of this work, a new direction will be opened in studying split problems.

In biomedical applications, microbubbles are often encapsulated with lipid or protein compounds to increase their longevity within the bloodstream. These encapsulated microbubbles (EMBs) are used in ultrasound imaging and drug delivery. Here, a data-driven method is presented for controlling EMBs with an acoustic field based on Koopman operator theory, which is a mathematical framework for transforming nonlinear dynamical systems into linear systems on an infinite-dimensional function space. This linearization, in turn, allows classical linear control methods to be directly applied to strongly nonlinear systems. In this work, we use a Koopman linear quadratic regulator (KLQR) to design acoustic control signals for a spherical EMB based on the Marmottant model. It is shown that KLQR is able to effectively drive EMBs to specific target behaviors, such as amplifying a subharmonic resonance or exhibiting a quasiperiodic oscillation. These results are compared to previous work by the authors on unencapsulated microbubbles, and it is discovered that EMBs present unique difficulties caused by the presence of a slow manifold in their dynamics. This slow manifold disrupts the controller when a target trajectory nears it; as a result, the controller must be built using Koopman eigenfunctions that are carefully constructed so as to capture the relevant dynamics.

Abstract Reconstructing high-fidelity fluid flow fields from sparse sensor measurements is vital for many science and engineering applications but remains challenging because of the dimensional disparities between state and observational spaces. Due to such dimensional differences, the measurement operator becomes ill conditioned and noninvertible, making the reconstruction of flow fields from sensor measurements extremely difficult. Although sparse optimization and machine learning address the above problems to some extent, questions about their generalization and efficiency remain, particularly regarding the discretization dependence of these models. In this context, deep operator learning offers a better solution as this approach models mappings between infinite-dimensional function spaces, enabling superior generalization and discretization-independent reconstruction. We introduce a deep operator-learning model that is trained to reconstruct fluid flow fields from sparse sensor measurements. Our deep-learning model employs a branch–trunk network architecture to represent the inverse measurement operator that maps sensor observations to the original flow field, a continuous function of both space and time. Our validation has demonstrated that the proposed deep-learning method consistently achieves high levels of reconstruction accuracy and robustness, even in scenarios where sensor measurements are inaccurate or missing. Furthermore, the operator-learning approach enables the capability to perform zero-shot super-resolution in both spatial and temporal domains, offering a solution for rapid reconstruction of high-fidelity flow fields.

We investigate algebraic and topological properties of subsymmetric polynomials on finite- and infinite-dimensional spaces. In particular, we focus on the problem of the existence of an algebraic basis in the algebra of subsymmetric polynomials, as well as possible extensions of subsymmetric polynomials and analytic functions to larger spaces. We consider algebras of subsymmetric analytic functions of bounded type and their spectra, and study linear subspaces in the zero-sets of subsymmetric polynomials, as well as subspaces where a subsymmetric polynomial is symmetric. In addition, we propose some possible applications of subsymmetric polynomials in cryptography and in operator theory.

This article concerns evolution equations involving \(\psi\)-Hilfer fractional derivative in a Banach space. By using the theory of fractional calculus and \(\psi\)-Laplace transform, we firstly derive a definition of mild solutions for these equations. Then we establish theorems for the existence and uniqueness of solutions and the approximate controllability (not the exact controllability) of the \(\psi\)-Hilfer fractional differential system under appropriate conditions. We focus on the approximate controllability rather than the exact controllability bcause the exact controllability cannot be achieved generally for the system in infinite-dimensional spaces. We present a new multidimensional Gronwall-type inequality with multiple singular kernels involving exponential factors, which extends essentially many existing results. We also use the new Gronwall-type inequality to study the dependence of the solution on the order and the initial condition for the fractional integro-differential equations involving \(\psi\)-Hilfer fractional derivative. Finally, an example is given to illustrate our main results. For more information and the latex file, see https://ejde.math.txstate.edu/Volumes/2025/109/abstr.html

The aim of this paper is to generalize the core-nilpotent decomposition of endomorphisms on arbitrary vector spaces. To this end, we introduce the degree of an endomorphism as a natural extension of the classical notion of index. We then provide several characterizations of generalized core-nilpotent endomorphisms from distinct perspectives.