Researchers develop neural networks for approximating functions in Banach spaces.
problem Approximating Banach space valued continuous functions.
method Quasi-interpolation Banach space valued neural network operators using algebraic sigmoid functions.
result Jackson type inequalities for function approximation.
Novel neural operator predicts complex spatiotemporal dynamics from partial observations.
problem Capturing complex operator dynamics in infinite-dimensional function spaces.
method Integrates Koopman operator theory with deep neural networks to approximate nonlinear operators between Banach spaces.
result BNO achieves robust zero-shot super-resolution in unsteady flow prediction and outperforms conventional methods.
The paper introduces Causal Neural Operators to approximate operators in stochastic analysis.
problem Leveraging temporal structure in non-linear operators for deep learning models.
method Designing a deep learning model framework for infinite-dimensional linear metric spaces.
result Causal Neural Operators can uniformly approximate Hölder or smooth trace class operators.
Develops vector-valued RKBS for neural networks and operators.
problem Understanding function spaces of Rd-valued neural networks and neural operators. method Defines and constructs vector-valued RKBS (vv-RKBS) without restrictive assumptions.
result Establishes Representer Theorem for neural architectures.
New neural architectures with multivariate nonlinearities are optimal in function space.
problem Optimality of neural architectures with multivariate nonlinearities.
method Construction of Banach spaces via k-plane transform and sparsity-promoting norm, proving representer theorem. result Neural architectures with multivariate nonlinearities are optimal in function space.
Banach fibrations and Nijenhuis operators studied for vanishing torsion.
problem Understanding Nijenhuis operators on Banach fibrations and their properties.
method Analyzing Nijenhuis operators on Banach fibrations and their projectability.
result Vanishing of Nijenhuis torsion on N0 implies vertical values for N. Extends Gaussian process theory to Banach spaces.
problem Extending Gaussian process theory to Banach spaces.
method Investigates the connection between Gaussian processes and Gaussian random elements in reproducing kernel Banach spaces.
result Characterizes positive definite functions that arise from covariance operators in Banach space setting.
Random feature method approximates operators with theoretical guarantees and reduced computation.
problem Approximating operators between infinite dimensional Banach spaces using machine learning.
method Random feature operator learning method with theoretical guarantees and error bounds.
result The random feature method can achieve similar or better test errors than kernel-based methods and neural networks with significantly reduced training times.
The paper explores properties of the Radon transform in relation to neural networks and ridges.
problem Understanding the Radon transform and its application to neural networks and ridges.
method Investigates properties of the Radon transform, introduces new subspaces, and characterizes ridges for any distributional profile.
result Clarifies and simplifies results on the optimality of ReLU networks using the Radon transform.
Study spectral and index properties of Hodge-Dirac operator on compact manifolds.
problem Investigate spectral and index-theoretic properties of Hodge-Dirac operator on compact Riemannian manifolds.
method Establish bisectoriality and H∞ functional calculus without curvature assumptions. result Prove compact Banach spectral triple and recover classical topological invariants as Lp-indices. Develops an L^p theory for Dolbeault-Dirac operators on compact Kähler manifolds.
problem Analyzing Dolbeault-Dirac operators on compact Kähler manifolds with Banach space coefficients.
method Establishes an L^p theory for Dolbeault-Dirac operators, proving bisectoriality, H^\infty functional calculus, and Gaffney-type estimates.
result Identifies the index of the associated Fredholm operator with the holomorphic Euler characteristic, independent of p.
Study Nijenhuis operators on Banach homogeneous spaces, extending previous work.
problem Characterize Nijenhuis torsion and integrability of almost complex structures on homogeneous spaces.
method Analyze bounded operators on Lie(G) to define homogeneous vector bundles and their Nijenhuis torsion.
result Equivalence of Nijenhuis torsion vanishing and Nijenhuis torsion values in Lie(K).
Study Cowen-Douglas operators from analytic function spaces.
problem Analytic continuation and spectrum of Cowen-Douglas operators.
method Investigate Banach spaces of analytic functions and their operators.
result Analytic continuations of functions relate to the spectrum of Cowen-Douglas operators.
The paper uses Banach spaces to analyze neural networks.
problem Understanding the function spaces of neural networks.
method Theory of reproducing kernel Banach spaces.
result Representer theorem for wide class of Banach spaces.
Study variance-reduced method for estimating fixed points in Banach spaces.
problem Estimating fixed points of contractive operators in Banach spaces with noisy evaluations.
method Variance-reduced stochastic approximation scheme in Banach spaces.
result Establish non-asymptotic bounds for operator defect and estimation error.
Kernel methods outperform neural nets in operator learning tasks.
problem Learning operators between Banach spaces from partial observations.
method Kernel-based framework with a priori error analysis and numerical comparisons.
result Kernel methods are competitive with neural nets in cost-accuracy trade-off.
Deep neural networks define suitable reproducing kernel Banach spaces.
problem Characterizing the function spaces of deep neural networks.
method Reproducing kernel Banach spaces and variational results.
result Deep neural networks define suitable reproducing kernel Banach spaces.
New analysis shows a gap between Gaussian RKHS and neural networks on unbounded domains.
problem Understanding the function space bias of neural networks compared to Gaussian RKHS.
method Infinite-center asymptotic analysis of neural network Banach space and Gaussian RKHS on unbounded domains.
result Certain functions in Gaussian RKHS have infinite norm in neural network Banach space on unbounded domains.
Study Nijenhuis operators on homogeneous spaces related to C*-algebras.
problem Characterize Nijenhuis operators on homogeneous spaces of C*-algebras.
method Analyze vector bundle maps induced by admissible operators on C*-algebras.
result Identify conditions for vector bundle maps to be Nijenhuis operators.
The paper defines projective structures for Lie bialgebras and Poisson-Lie groups.
problem Defining projective analogues of Lie bialgebras and Poisson-Lie groups.
method Introducing projective tensor products and adapting classical notions to these structures.
result Every quasi-triangular projective r-matrix gives rise to a projective Banach Lie bialgebra.
Improved DeepONets for PDE solution operators with adaptive re-weighting and new architecture.
problem Training DeepONets for PDE solution operators without paired data.
method Adaptive re-weighting of training examples and novel network architecture.
result Consistently improved predictive accuracy by a factor of 10-50x.
Study solves optimal portfolio selection using HJB equation.
problem Optimal portfolio selection problem.
method Maximal monotone operator method, Banach fixed-point theorem, Fourier transform, monotone operators technique.
result Existence and uniqueness of solution to HJB equation.
Geometrically studies Moore-Penrose inverse and polar decomposition continuity.
problem Continuity of Moore-Penrose inverse for perturbations by operator ideals.
method Geometric construction using essential codimension and Banach-Lie group action.
result Moore-Penrose inverse is a real analytic map between manifolds.
Random feature models approximate functions in Banach spaces efficiently.
problem Approximating functions in Banach spaces efficiently.
method Randomly initialized feature maps and linear readout training.
result Universal approximation in Bochner spaces for Banach space-valued models.
FunDPS improves PDE solution recovery from sparse data.
problem Recovering whole solutions from sparse or noisy measurements in PDEs.
method Function-space diffusion model with gradient-based guidance.
result FunDPS achieves 32% accuracy improvement over state-of-the-art methods.
Let A be a positive injective operator in a Hilbert space (\h, <,>), and denote by [,] the inner product defined by A: [f,g]=<Af,g>. A closed subspace $\s \subset \h$ is called A-compatible if there exists a closed complement for $\s$, which is orthogonal to $\s$ with respect to the inner product [,]. Equivalently, i…
Represents neural networks as solutions to inverse problems in Banach spaces.
problem Understanding the function learned by neural networks.
method Variational framework, representer theorem, polynomial ridge splines.
result Neural networks are solutions to inverse problems in Banach spaces.
Let E, F be separable Hilbert spaces, and assume that E is infinite-dimensional. We show that for every continuous mapping f:E→F and every continuous function ε:E→(0,∞) there exists a C∞ mapping g:E→F such that ∥f(x)−g(x)∥≤ε(x) and Dg(x):E→F is a sur…
We consider a curve of Fredholm pairs of Lagrangian subspaces in a fixed Banach space with continuously varying (weak) symplectic structures. Assuming vanishing index, we obtain intrinsically a continuously varying splitting of the total Banach space into pairs of symplectic subspaces. Using such decompositions we defi…
Develops a mathematical framework for causal fermion systems in infinite dimensions.
problem Analysis of causal fermion systems in infinite-dimensional settings.
method Introduces Banach manifold structure and expedient differential calculus.
result Establishes Hölder continuity of causal Lagrangian and integrated causal Lagrangian.
Explicit BCH series radii found for special Banach-Malcev shift algebras.
problem Finding convergence radii for BCH series in specific algebraic structures.
method Established explicit convergence radii using continuity estimates and algebraic properties.
result Explicit formula for convergence radii derived and validated for various shift algebras.
LSCI provides locally adaptive prediction sets for operator models with tighter coverage.
problem Generating robust, calibrated uncertainty quantification for operator models.
method Local Sliced Conformal Inference (LSCI) for operator models.
result LSCI yields tighter prediction sets with stronger adaptivity compared to conformal baselines.
The notion of nonpositive curvature in Alexandrov's sense is extended to include p-uniformly convex Banach spaces. Infinite dimensional manifolds of semi-negative curvature with a p-uniformly convex tangent norm fall in this class on nonpositively curved spaces, and several well-known results, such as existence and uni…
New spectral theory for non-associative algebras with applications to Moufang dynamics.
problem Spectral theory of non-associative algebras and their applications.
method Introducing almost periodic Banach--Malcev algebras and analyzing their spectral properties.
result Spectral characterization and continuous functional calculus for almost periodic derivations.
New Banach spaces for ReLU networks enable better function approximation and gradient dynamics analysis.
problem Function approximation and gradient dynamics in multi-layer ReLU networks.
method Developed Banach spaces for ReLU networks, defined new function representations, and analyzed gradient flow dynamics.
result Gradient flow dynamics of the new representation is the continuous analog of gradient descent for ReLU networks.
Anosov maps study with new Banach space and foliation method.
problem Understanding statistical properties of Anosov maps.
method Constructing a new Banach space and using a new foliation method.
result New Banach space provides insights into foliation absolute continuity.
Transformers are explained as infinite-dimensional kernel machines.
problem Understanding the mechanics of Transformers in AI.
method Characterized Transformers' attention mechanism as a kernel learning method on Banach spaces.
result Transformer's kernel has infinite feature dimension and can learn any binary non-Mercer reproducing kernel Banach space pair.
Estimates neural network error approximating compact sets.
problem Approximating compact subsets from Banach spaces with neural networks.
method Estimates error rates for neural networks of varying width and depth.
result Depth is crucial for better approximation rates, width alone does not improve.
Study on measure-valued CARMA processes in Banach spaces.
problem Modeling dynamics of functionals of spatio-temporal random fields.
method Defined measure-valued CARMA processes and derived conditions for stationarity.
result Positive measure-valued CARMA processes can model spatio-temporal random fields.
Some recent work in Frechet geometry is briefly reviewed. In particular an earlier result on the structure of second tangent bundles in the finite dimensional case was extended to infinite dimensional Banach manifolds and Frechet manifolds that could be represented as projective limits of Banach manifolds. This led to …
Sharp bounds for max-sliced Wasserstein distances derived for empirical distributions.
problem Estimating the expected max-sliced Wasserstein distance between a probability measure and its empirical distribution.
method Banach space version and operator norm approach for upper bounds.
result Upper bounds for max-sliced Wasserstein distances are essentially matching and sharp up to a log factor.
Neural operators improve solving Helmholtz equation for various wave speeds.
problem Neural operators struggle with out-of-distribution scenarios for high-frequency waves.
method Proposed a subfamily of neural operators with stochastic depth for enhanced approximation of the Helmholtz equation.
result Neural operators with stochastic depth outperform standard models in out-of-distribution scenarios.
The aim of this paper is the study of the geodesic distance in operator groups with several Riemannian metrics. More precisely we study the geodesic distance in self-adjoint operator groups with the left invariant Riemannian metric induced by the infinite trace and extend known results about the completeness of some cl…
New framework explains deep neural networks using variational spline theory.
problem Understanding functions learned by deep neural networks.
method Developed a variational framework and function space.
result Deep ReLU networks are solutions to regularized data fitting problems over the proposed function space.
Random feature model approximates PDE solutions efficiently.
problem Approximating solutions to PDEs with high-dimensional inputs and outputs.
method Random feature model applied to infinite-dimensional operators.
result Efficient and accurate approximation of PDE solutions.
We investigate infinitesimal properties of sets of ordered n-uples of idempotents in a symmetric Banach ∗-algebra. These sets are called flag manifolds and carry several interesting bundles that hold an important role in some areas of operator theory. In this direction, we introduce and study Stiefel bundles on fla…
Let X be a (real or complex) Banach space, and I(X) be the set of all (non-zero and non-identity) idempotents; i.e., bounded linear operators on X whose squares equal themselves. We show that the Banach submanifold I(X) of L(X) is a locally trivial analytic affine-Banach bundle o…
Paper establishes DRL for high-dimensional rewards.
problem Intractable reinforcement learning with high-dimensional rewards.
method Theoretical foundations and a novel DRL algorithm.
result Bellman operator contraction in high-dimensional spaces.