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.
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…
This research develops approximation theory for OOMs of infinite-dimensional processes.
problem Developing an approximation theory for OOMs of infinite-dimensional processes.
method Establishing an inner product structure and proving continuity of observable operators.
result A fundamental obstacle in making an infinite-dimensional space of future distributions into a Hilbert space is described.
Infinite-dimensional diffusion models tackle generative tasks for complex data.
problem Challenges in applying diffusion models to infinite-dimensional data.
method Directly formulate diffusion-based generative models in infinite dimensions.
result Developed guidelines for designing infinite-dimensional diffusion models.
Infinite-dimensional contact geometry explored.
problem Generalizing contact geometry to infinite dimensions.
method Generalization of cosymplectic, contact, and cocontact manifolds to infinite dimensions.
result Model examples of time-dependent and dissipative Hamiltonian systems calculated.
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.
Infinite-dimensional SBDMs improve image generation across multiple resolutions.
problem Efficient image generation at high resolutions and across different levels.
method Developed SBDMs in infinite-dimensional setting, using trace class operators and operator networks.
result Improved efficiency and generalization across resolution levels.
We define submersions f between manifolds M and N modelled on locally convex spaces. If the range N is finite-dimensional or a Banach manifold, then these coincide with the naive notion of a submersion. We study pre-images of submanifolds under submersions and pre-images under mappings whose differentials have dense im…
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.
Study shows how Poisson brackets factor on infinite dimensional manifolds.
problem Understanding Poisson brackets on infinite dimensional manifolds.
method Analyzes Poisson brackets on smoothly paracompact manifolds with specific properties.
result Dual map of a Poisson bracket factors as a smooth section of a vector bundle.
Existence of balanced embedding proved for complex manifold into infinite-dimensional space.
problem Balanced embedding of non-compact complex manifolds into infinite-dimensional projective space.
method Gradient flow in a Hilbert space, long-time existence established by perturbation, convergence depends on a priori bounds.
result Existence of balanced embedding proved in a model case.
Paper introduces infinite-dimensional generative models using Doob's h-transform.
problem Defining generative models in infinite dimensions.
method Using Doob's h-transform to force a reference diffusion towards a target distribution.
result The forced process can be approximated by minimising a score-matching objective.
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.
Symplectic method solves infinite-dimensional Schrödinger equations.
problem Solving Schrödinger equations on infinite-dimensional Hilbert spaces with unbounded Hamiltonians.
method Analytic vectors, manifolds modelled on normed spaces, symplectic differential geometry, Marsden--Weinstein reduction.
result Mapped t-dependent Schrödinger equations onto projective spaces. 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.
In this paper we propose a new treatment about infinite dimensional manifolds, using the language of category and functor. Our definition of infinite dimensional manifolds is a natural generalization of finite dimensional manifolds in the sense that de Rham cohomology and singular cohomology can be naturally defined an…
We characterize conjugate nonparametric Bayesian models as projective limits of conjugate, finite-dimensional Bayesian models. In particular, we identify a large class of nonparametric models representable as infinite-dimensional analogues of exponential family distributions and their canonical conjugate priors. This c…
The infinite Viterbi alignment is the limiting maximum a-posteriori estimate of the unobserved path in a hidden Markov model as the length of the time horizon grows. For models on state-space Rd satisfying a new ``decay-convexity'' condition, we develop an approach to existence of the infinite Viterbi ali…
The paper proves infinite-dimensional rational homotopy groups for a specific embedding space.
problem The study of rational homotopy groups of the space of long embeddings of codimension 2.
method Utilizing hairy graphs, the authors construct elements in the homotopy groups and prove their nontriviality.
result The rational homotopy groups of the space of long embeddings are infinite-dimensional in infinitely many degrees.
Simplified Ricci curvature for spherical fluid dynamics models.
problem Studying stability in incompressible fluid dynamics on a sphere.
method Definition and calculation of Ricci curvature for two-dimensional hydrodynamics using finite-dimensional Zeitlin models.
result Strong numerical evidence suggests convergence of finite-dimensional approximations to infinite-dimensional limit, indicating average instability for high-frequency modes.
We prove a Frobenius theorem for Banach distributions on manifolds that are modelled over locally convex spaces. Moreover, we recall how Frobenius theorems can be applied to infinite-dimensional Lie groups and obtain, that given a Lie subalgebra of the Lie algebra of a Lie group that is modelled over a locally convex s…
The study finds infinitely many different geometries for odd-dimensional manifolds with positive Ricci curvature.
problem Finding different geometries for odd-dimensional manifolds with positive Ricci curvature.
method Examining closed manifolds in odd dimensions n≥5. result Large classes of manifolds admit infinitely many different geometries of positive Ricci curvature.
FDPs generalize diffusion models to function spaces, enabling efficient image generation.
problem Efficiently generating images from continuous data.
method Introducing FDPs with new mathematical frameworks and training objectives.
result FDPs achieve high-quality image generation with fewer parameters.
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.
This paper tackles infinite-dimensional diffusion bridge simulation using operator learning.
problem Challenges in simulating diffusion bridges for modeling natural data due to intractable drift terms and continuous data representations.
method Merges score matching techniques with operator learning to directly learn infinite-dimensional bridges.
result Demonstrates high efficacy in simulating diffusion bridges for various applications, including real-world biological data.
The study defines divergence for multivector fields on infinite-dimensional manifolds.
problem Defining divergence for multivector fields on infinite-dimensional manifolds.
method Definition of divergence consistent with finite-dimensional geometry, properties transferred from finite to infinite dimensions.
result Natural properties of divergence are preserved in infinite dimensions.
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 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.
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.
We introduce an asymptotic small noise expansion, a so called vol-of-vol expansion, for potentially infinite dimensional and rough stochastic volatility models. Thereby we extend the scope of existing results for finite dimensional models and validate claims for infinite dimensional models. Furthermore we provide new, …
Solves infinite horizon portfolio problem with path-dependent labor income.
problem Infinite horizon portfolio choice with path-dependent labor income.
method Solves an infinite dimensional stochastic optimal control problem using explicit solutions to the HJB equation.
result Explicit solutions to the optimal controls in feedback form are found.
New method for Bayesian inference in infinite dimensions using SDMs.
problem Bayesian inference in infinite-dimensional function spaces.
method Amortized conditional score-based diffusion models (SDMs).
result Validated approach for large-scale, discretization-invariant Bayesian inference.
Proposes an algorithm for infinite-dimensional sparse learning in system identification.
problem System identification without known model structures.
method Atomic norm regularization and greedy algorithm for solving an infinite-dimensional group lasso problem.
result The proposed algorithm outperforms benchmark methods in impulse response fitting and pole location estimation.
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.
Develops infinite-dimensional Kempf-Ness theory for complexification-free groups.
problem Infinite-dimensional Kempf-Ness theorem challenges due to lack of complexification.
method Uses Cartan bundles to generalize theory, defining essential objects.
result Establishes convexity properties and generalized Futaki character.
New non-trivial Kaehler-Ricci solitons found in infinite dimensional complex space forms.
problem Non-trivial Kaehler-Ricci solitons in infinite dimensional complex space forms.
method Exhibited families of non trivial radial Kaehler-Ricci solitons in infinite dimensional complex space forms.
result Non-trivial Kaehler-Ricci solitons exist in infinite dimensional complex space forms, contradicting previous finite dimensional results.
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.
This paper is one step toward infinite energy gauge theory and the geometry of infinite dimensional moduli spaces. We generalize a gluing construction in the usual Yang-Mills gauge theory to an ``infinite energy'' situation. We show that we can glue an infinite number of instantons, and that the resulting instantons ha…
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.
Constructs infinite-dimensional Siegel disc as symplectic and Kaehler quotient.
problem No specific problem stated; focuses on mathematical construction.
method Symplectic and Kaehler quotient construction.
result Infinite-dimensional Siegel disc constructed as symplectic and Kaehler quotient.
New approach to quantify posterior concentration rates using Wasserstein dynamics.
problem Quantifying the speed of posterior distribution concentration in Bayesian statistics.
method Combining local Lipschitz-continuity with dynamic formulation of Wasserstein distance.
result Optimal posterior contraction rates in finite and infinite-dimensional models.
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.
Study pricing options on forward contracts using infinite-dimensional affine models.
problem Pricing European-style options on forward contracts in complex stochastic volatility models.
method Model forward price curves using stochastic partial differential equations modulated by stochastic volatility processes. Analyze two classes of affine stochastic volatility models: Gaussian and pure-jump. Derive conditions for existence of exponential moments and develop semi-closed pricing formulas.
result Developed semi-closed Fourier-based pricing formulas for vanilla call and put options in infinite-dimensional affine models.
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…
Adjoint sampler targets infinite-dimensional function spaces for efficient sampling.
problem Limited theory and algorithms for sampling infinite-dimensional function spaces.
method Adjoint Sampler for infinite-dimensional function spaces based on stochastic maximum principle.
result FAS achieves superior performance in synthetic and real systems.
Infinite dimensional measure-valued processes modeled as polynomial diffusions.
problem Modeling term structure in energy markets using measure-valued polynomial diffusions.
method Introduced measure-valued polynomial diffusions, derived moment formulas, and characterized infinitesimal generators.
result Recovery of measure-valued affine diffusions as a special case.
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…