arXiv research

Study of LQ MFGs in infinite-dimensional Hilbert spaces.

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…

Adjoint sampler targets infinite-dimensional function spaces for efficient sampling.

We describe the exponential map from an infinite-dimensional Lie algebra to an infinite-dimensional group of operators on a Hilbert space. Notions of differential geometry are introduced for these groups. In particular, the Ricci curvature, which is understood as the limit of the Ricci curvature of finite-dimensional g…

The paper shows how to use fine shape to understand infinite-dimensional spaces.

Gradient-free method solves infinite-dimensional optimization problems.

Book on infinite-dimensional Lie groups, covering basics and various classes.

Simplified Ricci curvature for spherical fluid dynamics models.

Online learning of linear operators between infinite-dimensional spaces is possible but with limitations.

Differential equations are derived for a continuous limit of iterated Schwarzian reflection of analytic curves, and solutions are interpreted as geodesics in an infinite-dimensional symmetric space geometry.

One of the goals of this article is to define a an unified setting adapted to the description of means (normalized integrals or invariant means) on an infinite product of measured spaces with infinite measure. We first remark that some known examples coming from the theory of metric measured spaces and also from oscill…

New mechanism for pure differential privacy on functional summaries using Laplace-like process.

We study the limits of sequences of spheres and complex projective spaces with unbounded dimensions. A sequence of spheres (resp. complex projective spaces) either is a Levy family, infinitely dissipates, or converges to (resp. the Hopf quotient of) a virtual infinite-dimensional Gaussian space, depending on the size o…

Optimizes learning Hilbert-Schmidt operators between Sobolev spaces.

Study classifies super vector bundles and proves universality.

The paper explores infinite-dimensional nonholonomic and vakonomic systems.

This work bridges stochastic interpolants to infinite-dimensional Hilbert spaces.

This paper extends Lie algebra contractions to infinite-dimensional spaces for better understanding of group limits.

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…

Study on infinite-dimensional Heisenberg groups using hypoelliptic heat kernels.

Develops a new approach to establish universality for any-dimensional machine learning models.

In this paper we derive a scaling limit for an infinite dimensional limit order book model driven by Hawkes random measures. The dynamics of the incoming order flow is allowed to depend on the current market price as well as on a volume indicator. With our choice of scaling the dynamics converges to a coupled SDE-ODE s…

We extend rectified flow to infinite-dimensional Hilbert space.

Proves uniqueness of embedding complex manifold into infinite-dimensional space.

New non-trivial Kaehler-Ricci solitons found in infinite dimensional complex space forms.

Paper investigates optimal transport map estimation in infinite-dimensional spaces.

Derives PF-ODE for infinite-dimensional functions, improving function generation tasks.

The second order tangent bundle $T^{2}M$ of a smooth manifold $M$ consists of the equivalent classes of curves on $M$ that agree up to their acceleration. It is known that in the case of a finite $n$-dimensional manifold $M$, $T^{2}M$ becomes a vector bundle over $M$ if and only if $M$ is endowed with a linear connecti…

Develops robust methods for infinite-dimensional stochastic processes.

We develop a second-order model for limit order books in a single scaling regime.

We solve three open problems concerning infinite-dimensional Lie groups posed in a recent survey article by K.-H. Neeb: (1) There exists a subgroup of some infinite-dimensional Lie group G which does not admit an initial Lie subgroup structure; (2) The pathology cannot occur if G is a direct limit of an ascending seque…

The report analyzes infinite-dimensional output space regression.

Smooth structures on infinite dimensional Grassmannians and non-commutative cross-ratios.

Diffeologies unify infinite-dimensional geometry and PDEs, enhancing classical function spaces.

We study direct limits $(G,K) = \varinjlim (G_n,K_n)$ of compact Gelfand pairs. First, we develop a criterion for a direct limit representation to be a multiplicity--free discrete direct sum of irreducible representations. Then we look at direct limits $G/K = \varinjlim G_n/K_n$ of compact riemannian symmetric spaces, …

This research develops approximation theory for OOMs of infinite-dimensional processes.

Symplectic method solves infinite-dimensional Schrödinger equations.

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…

In this paper we study the homogeneous Kaehler manifolds (h.K.m.) which can be Kaehler immersed into finite or infinite dimensional complex space forms. On one hand we completely classify the h.K.m. which can be Kaehler immersed into a finite or infinite dimensional complex Euclidean or hyperbolic space. Moreover, we e…

Existence of balanced embedding proved for complex manifold into infinite-dimensional space.

This paper consists of two main results. In the first one we describe all Kaehler immersions of a bounded symmetric domain into the infinite dimensional complex projective space in terms of the Wallach set of the domain. In the second one we exhibit an example of complete and non-homogeneous Kaehler-Einstein metric wit…

We consider the Yang-Mills equations for a matrix gauge group $G$ inside the future light cone of 4-dimensional Minkowski space, which can be viewed as a Lorentzian cone $C(H^3)$ over the 3-dimensional hyperbolic space $H^3$. Using the conformal equivalence of $C(H^3)$ and the cylinder $R\times H^3$, we show that, in t…

New algorithms improve GP inference without approximations, achieving better results.

This study describes the Fisher-Rao metric on Gaussian measures in infinite-dimensional spaces.

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…

Spray-invariant sets maintain geodesics on infinite-dimensional manifolds.

The paper proves infinite-dimensional rational homotopy groups for a specific embedding space.

We endow the group of automorphisms of an exact Courant algebroid over a compact manifold with an infinite dimensional Lie group structure modelled on the inverse limit of Hilbert spaces (ILH). We prove a slice theorem for the action of this Lie group on the space of generalized metrics. As an application, we show that…