arXiv research

Study small-time CLTs for stochastic Volterra equations with various kernels.

In this paper, we investigate the mean curvature flow having equifocal submanifolds as initial data. The investigation are performed by investigating the mean curvature flow having the lifted submanifolds to a Hilbert space through a Riemannian submersion as initial data.

We lift ambit fields as introduced by Barndorff-Nielsen and Schmiegel to a class of Hilbert space-valued volatility modulated Volterra processes. We name this class Hambit fields, and show that they can be expressed as a countable sum of weighted real-valued volatility modulated Volterra processes. Moreover, Hambit fie…

A commuting $n$-tuple $(T_1, \ldots, T_n)$ of bounded linear operators on a Hilbert space $\clh$ associate a Hilbert module $\mathcal{H}$ over $\mathbb{C}[z_1, \ldots, z_n]$ in the following sense: \[\mathbb{C}[z_1, \ldots, z_n] \times \mathcal{H} \rightarrow \mathcal{H}, \quad \quad (p, h) \mapsto p(T_1, \ldots, T_n)h…

We define and study isoparametric submanifolds of general ambient spaces and of arbitrary codimension. In particular we study their behaviour with respect to Riemannian submersions and their lift into a Hilbert space. These results are used to prove a Chevalley type restriction theorem which relates by restriction eige…

We introduce a novel data-driven order reduction method for nonlinear control systems, drawing on recent progress in machine learning and statistical dimensionality reduction. The method rests on the assumption that the nonlinear system behaves linearly when lifted into a high (or infinite) dimensional feature space wh…

GOPO optimizes large models in Hilbert space, avoiding Kullback-Leibler's curvature.

SOBER framework optimizes Bayesian optimization tasks efficiently.

Parallel transport map over reductive spaces is an affine submersion.

Study uses Bayes Hilbert framework to recover probability measure flows from sensors.

It is known that principal orbits of Hermann actions on a symmetric space of non-compact type are curvature-adapted isoparametric submanifolds having no focal point of non-Euclidean type on the ideal boundary of the ambient symmetric space. In this paper, we investigate the mean curvature flows for such a curvature-ada…

The paper extends Riemann-Hilbert correspondence to foliations.

In this paper, we investigate the mean curvature flows for an equifocal submanifold in a symmetric space of compact type and its focal submanifolds as initial data. It is known that equifocal submanifolds of codimension greater than one in irreducible symmetric spaces of compact type occur as principal orbits of Herman…

Kernel-based Bayesian filter for nonlinear systems using infinite-dimensional operators.

New method predicts state evolution for non-first-order algorithms on nonconvex problems.

We introduce a data-driven order reduction method for nonlinear control systems, drawing on recent progress in machine learning and statistical dimensionality reduction. The method rests on the assumption that the nonlinear system behaves linearly when lifted into a high (or infinite) dimensional feature space where ba…

New methods avoid spectral pollution in transfer operators for accurate analysis.

New algorithm reduces online regression error in RKHS.

Kernel DRO uses RKHS to optimize under distributional uncertainty.

New method shows unitarity in quantization for toric manifolds.

Survey of twistor lifts of surfaces in 4-dimensional spaces.

Develops a new algebraic framework for differential geometry of infinite dimensional spaces.

Lie groupoids and their orbit spaces are linked through equivalence classes.

Maps can be embedded in higher dimensions if they lift to embeddings in product spaces.

NOs can learn any finite collection of classes in functional data.

An inverse limit of a sequence of covering spaces over a given space $X$ is not, in general, a covering space over $X$ but is still a lifting space, i.e. a Hurewicz fibration with unique path lifting property. Of particular interest are inverse limits of finite coverings (resp. finite regular coverings), which yield fi…

Lifts isometries in orbit spaces for compact groups.

Metric spaces uniquely split into Hilbert and non-line-split parts.

New Hilbert bundles with ends defined from indexed bases.

Study geometry of tetrahedra in complex hyperbolic space and Hilbert spaces.

We give first examples of finitely generated groups having an intermediate, with values in (0,1), Hilbert space compression (which is a numerical parameter measuring the distortion required to embed a metric space into Hilbert space). These groups include certain diagram groups. In particular, we show that the Hilbert …

The paper introduces austere and arid submanifolds in Hilbert spaces.

We review and then combine two aspects of the theory of bundle gerbes. The first concerns lifting bundle gerbes and connections on those, developed by Murray and Gomi. Lifting gerbes represent obstructions against extending the structure group of a principal bundle. The second is the transgression of gerbes to loop spa…

An important geometric invariant of links in lens spaces is the lift in the 3-sphere of a link $L$ in $L(p,q)$, that is the counterimage $\widetilde L$ of $L$ under the universal covering of $L(p,q)$. If lens spaces are defined as a lens with suitable boundary identifications, then a link in $L(p,q)$ can be represented…

The study proves unique path lifting properties and their implications on quotient spaces and covering maps.

We deal here with the geometry of the twistor fibration $\mathcal{Z} \to \bb{S}^3_1$ over the De Sitter 3-space. The total space $\mathcal{Z}$ is a five dimensional reductive homogeneous space with two canonical invariant almost CR structures. Fixed the normal metric on $\mathcal{Z}$ we study the harmonic map equation …

Theory developed for Hilbert geometry over valued fields, linking real and non-Archimedean geometries.

The article analyzes LCE in Hilbert space, deriving new formulas and regularisation methods.

We show that one can lift locally real analytic curves from the orbit space of a compact Lie group representation, and that one can lift smooth curves even globally, but under an assumption.

Study Markov chain gradient descent in Hilbert spaces for quadratic loss.

In the present paper, we study complete and vertical lifts of tensor fields from a smooth manifold $M$ to its Weil bundle $T^A M$ defined by a Frobenius Weil algebra $A$. For a Poisson manifold $(M,w)$, we show that the complete lift $w^C$ and the vertical lift $w^V$ of the Poisson tensor $w$ are Poisson tensors on $T^…

No exceptional orbits found in Hilbert spaces actions.

This paper introduces Bayes Hilbert spaces for efficient posterior approximation.

In the paper "Direct Images, Fields of Hilbert Spaces, and Geometric Quantization", Lempert and Szőke proved that any flat analytic Hilbert field will induce a hermitian Hilbert bundle and gave an example of a flat Hilbert field that does not induce any Hilbert bundle. In this paper, we will provide an example of an an…

We introduce a notion of fibred coarse embedding into Hilbert space for metric spaces, which is a generalization of Gromov's notion of coarse embedding into Hilbert space. It turns out that a large class of expander graphs admit such an embedding. We show that the maximal coarse Baum-Connes conjecture holds for metric …

When there is a family of complex structures on the phase space, parametrized by a set $S$, the prequantum Hilbert spaces produced by geometric quantization, using the half-form correction, also depends on these parameters. This way we obtain a field of Hilbert spaces $p:H^{pr Q}\rightarrow S$. We show that this field …

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

In the present work we construct a lift of a metric $g$ on a 2-dimensional oriented Riemannian manifold $M$ to a metric $\hat{g}$ on the total space $P$ of the orthonormal frame bundle of $M$. We call this lift the \textit {Wagner lift}. Viktor Vladimirovich Wagner (1908 -1981) proposed a technique to extend a metric d…