Research covers geometry, analysis, and integration on infinite-dimensional spaces.
problem Exploring geometric and analytical structures in infinite-dimensional settings.
method Analyzes numerical schemes, Lie groups, connections, and integration theory.
result Developed new methods for integration and analysis on infinite-dimensional manifolds.
Paper investigates optimal transport map estimation in infinite-dimensional spaces.
problem Estimating optimal transport maps in infinite-dimensional spaces is challenging.
method Characterizes γ-smoothness for optimal transport maps and develops a polynomial-rate estimator. result Shows polynomial-order minimax risk for optimal transport map estimation.
Introduces infinite-dimensional differential geometry using Bastiani calculus.
problem Calculus breakdown in infinite-dimensional settings.
method Uses Bastiani calculus for directional derivatives.
result Develops and connects infinite-dimensional Lie groups and weak Riemannian geometry.
Diffeologies unify infinite-dimensional geometry and PDEs, enhancing classical function spaces.
problem Combining infinite-dimensional geometry and PDEs for optimization problems.
method Review and extension of classical function spaces and mapping spaces.
result Diffeologies provide a unified framework for evolution equations and optimization problems.
Using symplectic techniques and spectral analysis of smooth paths of self-adjoint operators, we characterize the set of conjugate instants along a geodesic in an infinite dimensional Riemannian Hilbert manifold.
Shape analysis and compuational anatomy both make use of sophisticated tools from infinite-dimensional differential manifolds and Riemannian geometry on spaces of functions. While comprehensive references for the mathematical foundations exist, it is sometimes difficult to gain an overview how differential geometry and…
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.
The paper analyzes the stability of an observer error in a vibrating string system.
problem Stability analysis of observer error in a vibrating string system.
method Abstract Cauchy problem reformulation and application of LaSalle's invariance principle for infinite-dimensional systems.
result The observer error is asymptotically stable.
Following the unified approach of A. Kriegl and P.W. Michor (1997) for a treatment of global analysis on a class of locally convex spaces known as convenient, we give a generalization of Rashevsky-Chow's theorem for control systems in regular connected manifolds modelled on convenient (infinite-dimensional) locally con…
Optimizes shapes on non-standard manifolds.
problem Optimization on non-standard infinite-dimensional manifolds.
method Develops gradient descent on weak Riemannian manifolds.
result Establishes foundational properties for optimization on various weak Riemannian manifolds.
Extended elliptical slice sampling for infinite-dimensional spaces, proving reversibility.
problem Proving reversibility of elliptical slice sampling in infinite-dimensional spaces.
method Extended elliptical slice sampling to infinite-dimensional separable Hilbert spaces, providing an alternative proof of reversibility.
result The approach yields a positive semi-definite Markov operator, proving reversibility.
Geometric analysis on diffeomorphism groups for fluid dynamics and information geometry.
problem Geometric analysis of fluid flows and optimal mass transport.
method Review of metrics and topology on diffeomorphism groups.
result Introduction of new metrics and topology for diffeomorphism groups.
This paper conditions non-linear infinite-dimensional diffusion processes.
problem Conditioning non-linear and infinite-dimensional diffusion processes.
method Infinite-dimensional Girsanov's theorem to condition function-valued stochastic processes.
result Conditioning of non-linear infinite-dimensional diffusion processes is achieved.
Study stabilizes adversarial training in neural networks over infinite-dimensional spaces.
problem Stability issues in adversarial training of neural networks.
method Functional analysis of minimax optimization over infinite-dimensional spaces of continuous functions and probability measures.
result Convergence property of minimax problems under certain conditions, interpreted as stabilization techniques.
In the elastic shape analysis approach to shape matching and object classification, plane curves are represented as points in an infinite-dimensional Riemannian manifold, wherein shape dissimilarity is measured by geodesic distance. A remarkable result of Younes, Michor, Shah and Mumford says that the space of closed p…
Develops robust methods for infinite-dimensional stochastic processes.
problem Measuring covariations in stochastic evolution equations in infinite dimensions.
method Asymptotic theory for jump robust measurement of covariations.
result Identifies scaling limits for realized covariations.
We consider stochastic versions of Euler--Arnold equations using the infinite-dimensional geometric approach as pioneered by Ebin and Marsden. For the Euler equation on a compact manifold (possibly with smooth boundary) we establish local existence and uniqueness of a strong solution (in the stochastic sense) in spaces…
Method solves Bayesian inverse problems in function space without assuming log-concavity.
problem Bayesian inverse problems in infinite-dimensional nonlinear settings.
method Score-based diffusion models as a prior, Langevin-type MCMC on function spaces.
result Provable convergence bound for posterior sampling, dependent on score approximation.
Study of LQ MFGs in infinite-dimensional Hilbert spaces.
problem Mean field games in infinite-dimensional settings with stochastic dynamics.
method Analysis of coupled semilinear infinite-dimensional stochastic evolution equations, development of Nash equilibrium.
result Characterization of unique Nash equilibrium in the limit of many agents.
Cyclohedra are a well-known infinite familiy of finite-dimensional polytopes that can be constructed from centrally symmetric triangulations of even-sided polygons. In this article we introduce an infinite-dimensional analogue and prove that the group of symmetries of our construction is a semidirect product of a degre…
Paper analyzes Langevin dynamics for solving infinite-dimensional Bayesian inverse problems.
problem Solving high-dimensional Bayesian inverse problems in infinite-dimensional function spaces.
method Preconditioned Langevin dynamics with score-based generative models (SGMs).
result Derives error estimates and sufficient conditions for global convergence in Kullback-Leibler divergence.
We introduce cylindrical projections to simulate infinite-dimensional occupation flows of diffusions.
problem Computational intractability of infinite-dimensional occupation flows of diffusions.
method Introduce cylindrical projections to approximate the occupation flow via a finite-dimensional system.
result Strong convergence of cylindrical projections to the initial process with derived rates.
This paper tackles the topology of infinite-dimensional fiber spaces.
problem Classifying and understanding the topological structure of infinite-dimensional fiber spaces.
method Algebraic and differential topology, focusing on the homotopy type of the moduli space of fibrations.
result Explicit homotopy calculations for low-dimensional cases, completing the solution for dimensions up to three.
We introduce a class of non-commutative Heisenberg like infinite dimensional Lie groups based on an abstract Wiener space. The Ricci curvature tensor for these groups is computed and shown to be bounded. Brownian motion and the corresponding heat kernel measures, {νt}t>0, are also studied. We show that these he…
GP-PCA reduces infinite-dimensional GP posteriors to a finite space for meta-learning.
problem How to define a structure for a set of Gaussian process posteriors.
method Information geometric framework and variational inference.
result GP-PCA improves meta-learning performance through reduced GP posteriors.
The study of universal approximation of arbitrary functions f:X→Y by neural networks has a rich and thorough history dating back to Kolmogorov (1957). In the case of learning finite dimensional maps, many authors have shown various forms of the universality of both fixed depth and fixed width…
New rates for GLD and SGLD in infinite-dimensional spaces without dimensionality issues.
problem Gradient Langevin dynamics and SGLD convergence rates in high-dimensional spaces.
method Analysis of GLD and SGLD in infinite-dimensional Hilbert spaces, using stochastic differential equations and Markov chains.
result Derivation of dimension-free convergence rates for GLD and SGLD.
Most Finsler metrics have infinite-dimensional holonomy groups.
problem Understanding the holonomy groups of Finsler metrics.
method Analyzing the set of Finsler metrics on a manifold.
result An open dense subset of Finsler metrics have infinite-dimensional holonomy groups.
New algorithms improve machine learning performance with explicit regret bounds.
problem Improving machine learning performance with explicit regret bounds.
method Projection-based linear regression algorithms with a focus on modern machine-learning models and their algorithmic performance.
result Established a priori regret bounds with explicit λ-dependence.
CGNNs use wavelets for continuous function generation in infinite-dimensional spaces.
problem Generating continuous functions in infinite-dimensional spaces for applications like inverse problems.
method Inspired by DCGAN, CGNNs use wavelet multiresolution analysis with convolutional and nonlinear layers.
result CGNNs can be injective under certain conditions on filters and nonlinearity, leading to Lipschitz stability estimates.
These are the proceedings of the workshop "Math in the Black Forest", which brought together researchers in shape analysis to discuss promising new directions. Shape analysis is an inter-disciplinary area of research with theoretical foundations in infinite-dimensional Riemannian geometry, geometric statistics, and geo…
Derives PF-ODE for infinite-dimensional functions, improving function generation tasks.
problem Efficient inference in infinite-dimensional diffusion models.
method Derives PF-ODE in infinite-dimensional function spaces.
result Reduces function evaluations while maintaining sample quality.
Optimizes learning Hilbert-Schmidt operators between Sobolev spaces.
problem Statistical limits of learning mappings between infinite-dimensional function spaces.
method Minimax optimal regularization and multilevel training.
result Multilevel kernel operator learning achieves optimal learning rate.
Study on infinite-dimensional Heisenberg groups using hypoelliptic heat kernels.
problem Properties of hypoelliptic heat kernels on infinite-dimensional reduced Heisenberg groups.
method Construction and study of hypoelliptic heat kernels on infinite-dimensional reduced Heisenberg groups.
result Hypoelliptic logarithmic Sobolev inequalities on the space.
Generalizes insider trading model to multiple assets.
problem Modeling informed trading in a multi-asset context.
method Formulated an infinite-dimensional Bayesian trading game.
result Obtained a parsimonious equilibrium with closed-form solutions.
Toolbox for stochastic Euler equations using Ebin-Marsden theory.
problem Applying geometric framework to stochastic PDEs.
method Combining infinite-dimensional geometry and stochastic analysis.
result Local well-posedness of maximal solutions for incompressible Euler equation with noise.
Operator-Valued Kernels (OVKs) and associated vector-valued Reproducing Kernel Hilbert Spaces provide an elegant way to extend scalar kernel methods when the output space is a Hilbert space. Although primarily used in finite dimension for problems like multi-task regression, the ability of this framework to deal with i…
Develops derived differential geometry for supermanifolds.
problem Handling non-transverse intersections and singular moduli problems in geometry and physics.
method Extends existing work on derived manifolds to supergeometric and infinite-dimensional contexts.
result Establishes foundational results relating derived differential geometry to differential operators and PDE theory.
Extends metrics for SPD matrices to infinite dimensions.
problem Lack of generalized forms for Riemannian metrics.
method Unitized Hilbert-Schmidt operators and extended Mahalanobis norm.
result Improved performance in high-dimensional comparisons.
Universal algorithm learns unknown distribution for various decision-making problems.
problem Various statistical measures in contextual sequential decision-making.
method Infinite-dimensional functional regression oracle for cumulative distribution functions.
result Utility regret rate bounded by polynomial decay of eigenvalue sequence.
Smooth structures on infinite dimensional Grassmannians and non-commutative cross-ratios.
problem Smooth structures on infinite dimensional Grassmannians and non-commutative cross-ratios.
method Analyzing and expanding the notion of non-commutative cross-ratios, proving their smoothness.
result Smoothness of non-commutative cross-ratios.
Proves uniqueness of embedding complex manifold into infinite-dimensional space.
problem Balanced embedding of non-compact complex manifold into infinite-dimensional projective space.
method Fine estimates of asymptotics of a balanced embedding.
result Uniqueness of embedding proven.
The paper extends Cartan development to infinite dimensional Lie groups.
problem Generalizing Cartan development to infinite dimensional Lie groups.
method Generalization of Cartan development to infinite dimensional manifolds and Lie groups.
result The tangent mapping of a Cartan development is another Cartan development.
The paper explores infinite-dimensional nonholonomic and vakonomic systems.
problem Understanding dynamics of infinite-dimensional systems with constraints.
method Visualizing and revisiting classical and new examples of nonholonomic and vakonomic systems.
result Infinite-dimensional systems exhibit both nonholonomic and vakonomic dynamics.
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.
We construct a infinite-dimensional manifold structure adapted to analytic Lie pseudogroups of infinite type. More precisely, we prove that any isotropy subgroup of an analytic Lie pseudogroup of infinite type is a regular infinite-dimensional Lie group, modelled on a locally convex strict inductive limit of Banach spa…
Reduced-rank method improves least-squares regression under output regularity.
problem Least-squares regression with infinite dimensional outputs.
method Reduced-rank method for solving least-squares problems with output regularity assumptions.
result Learning bounds and improved statistical performance compared to full-rank method.
We extend rectified flow to infinite-dimensional Hilbert space.
problem Extending rectified flow to infinite-dimensional spaces.
method Established a rigorous functional formulation using the superposition principle for continuity equations.
result Demonstrated superior performance compared to existing models.