arXiv research

Let S be a non-exceptional oriented surface of finite type. We classify all Radon measures on the space of measured geodesic laminations for S which are invariant under the mapping class group.

Paper proves uniqueness of Ricci flows from nonatomic measures on surfaces.

We prove that a transversely equicontinuous minimal lamination on a locally compact metric space $Z$ has a transversely invariant Radon measure. Moreover if the space $Z$ is compact, then the tranversely invariant Radon measure is shown to be unique up to a scaling.

Technical proofs for Radon-Nikodym derivative identities.

We investigate existence and uniqueness of weak solutions of the Cauchy problem for the porous medium equation on negatively curved Riemannian manifolds. We show existence of solutions taking as initial condition a finite Radon measure, not necessarily positive. We then establish uniqueness in the class of nonnegative …

Proves bounded subsolution theorem for complex Monge-Ampère equation on compact Hermitian manifolds.

Sharp estimates for Bergman metrics derived from Kähler quantization.

The study defines divergence for multivector fields on infinite-dimensional manifolds.

We prove the differentiability of Lipschitz maps X-->V, where X is a complete metric measure space satisfying a doubling condition and a Poincaré inequality, and V is a Banach space with the Radon Nikodym Property (RNP). The proof depends on a new characterization of the differentiable structure on such metric measure …

New model estimates indoor radon distribution with higher spatial resolution.

A new metric HSW derived from hierarchical Radon Transform addresses computational bottlenecks in sliced Wasserstein.

The paper studies invariant measures for specific actions in algebraic groups.

Study measures invariant under horospherical subgroups for finitely generated Kleinian groups.

The paper extends a measure preserving property to bi-Lipschitz maps between Moran sets.

The paper explores properties of the Radon transform in relation to neural networks and ridges.

Solves Ricci flow on Riemann surfaces with measure initial data.

Extends Gaussian process theory to Banach spaces.

There has been growing recent interest in probabilistic interpretations of kernel-based methods as well as learning in Banach spaces. The absence of a useful Lebesgue measure on an infinite-dimensional reproducing kernel Hilbert space is a serious obstacle for such stochastic models. We propose an estimation model for …

New gradient flows improve high-dimensional sampling.

Neural networks with ReLU^k approximate Sobolev functions efficiently via Radon transform.

Problems of interpolation, classification, and clustering are considered. In the tenets of Radon--Nikodym approach $\langle f(\mathbf{x})ψ^2 \rangle / \langleψ^2\rangle$, where the $ψ(\mathbf{x})$ is a linear function on input attributes, all the answers are obtained from a generalized eigenproblem $|f|ψ^{[i]}\rangle =…

This research explains why SGD generalizes better than ADAM in deep learning.

Study invariant measures on measured laminations for subgroups of mapping class group.

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

The article provides representations of exchange option prices under SVJD dynamics.

We interpret the setting for a Radon transform as a submanifold of the space of generalized functions, and compute its extrinsic curvature: it is the Hessian composed with the Radon transform.

This paper studies Brownian motion and heat kernel measure on a class of infinite dimensional Lie groups. We prove a Cameron-Martin type quasi-invariance theorem for the heat kernel measure and give estimates on the $L^p$ norms of the Radon-Nikodym derivatives. We also prove that a logarithmic Sobolev inequality holds …

Uniform convergence of metrics on surfaces with bounded curvature measures proved.

Paper revisits Black-Scholes model, proving solution existence and measuring market uncertainty.

The paper uses Banach spaces to analyze neural networks.

In this paper, we extend the DC Calculus introduced by Perelman on finite dimensional Alexandrov spaces with curvature bounded below. Among other things, our results allow us to define the Hessian and the Laplacian of DC functions (including distance functions as a particular instance) as a measure-valued tensor and a …

Study utility maximization with delayed information in continuous time Gaussian markets.

This revisit gives a survey on the analytical methods for the inverse exponential Radon transform which has been investigated in the past three decades from both mathematical interests and medical applications such as nuclear medicine emission imaging. The derivation of the classical inversion formula is through the re…

Kernel Density Machines learn probability densities without structural assumptions.

Let (M,g) be an analytic, compact, Riemannian manifold with boundary, of dimension n >= 2. We study a class of generalized Radon transforms, integrating over a family of hypersurfaces embedded in M, satisfying the Bolker condition [23]. Using analytic microlocal analysis, we prove a microlocal regularity theorem for ge…

Paper introduces S3W distance for spherical probability distributions.

Developed new Crofton formulas for pseudo-Riemannian spaces.

We define Radon transform and its inverse on the two-dimensional anti-de Sitter space over local fields using a novel construction through a quadratic equation over the local field. We show that the holographic bulk reconstruction of quantum fields in this space can be formulated as the inverse Radon transform, general…

We estimate Radon-Nikodym derivatives using regularization in reproducing kernel Hilbert spaces.

Kernel estimator optimally recovers function from noisy exponential Radon transform.

The paper derives Pizzetti formulae and inverts the Radon transform on spheres.

Extends SW and GSW to compare heterogeneous joint distributions.

We show that the Radon transform related to closed geodesics is injective on a Lie group if and only if the connected components are not homeomorphic to $S^1$ nor to $S^3$. This is true for both smooth functions and distributions. The key ingredients of the proof are finding totally geodesic tori and realizing the Rado…

We study heat kernel measures on sub-Riemannian infinite-dimensional Heisenberg-like Lie groups. In particular, we show that Cameron-Martin type quasi-invariance results hold in this subelliptic setting and give $L^p$-estimates for the Radon-Nikodym derivatives. The main ingredient in our proof is a generalized curvatu…

If $G$ is a finite group, is a function $f:G\to\mathbb C$ determined by its sums over all cosets of cyclic subgroups of $G$? In other words, is the Radon transform on $G$ injective? This inverse problem is a discrete analogue of asking whether a function on a compact Lie group is determined by its integrals over all ge…

Let $M$ be a Riemannian globally symmetric space of compact type, $M'$ its set of maximal flat totally geodesic tori, and $\mathrm{ad}(M)$ its adjoint space. We show that the kernel of the maximal flat Radon transform $τ:L^2(M) \rightarrow L^2(M')$ is precisely the orthogonal complement of the image of the pullback map…

The Wasserstein distance and its variations, e.g., the sliced-Wasserstein (SW) distance, have recently drawn attention from the machine learning community. The SW distance, specifically, was shown to have similar properties to the Wasserstein distance, while being much simpler to compute, and is therefore used in vario…

The main result of this paper is the following: any `weighted' Riemannian manifold $(M,g,μ)$ - i.e. endowed with a generic non-negative Radon measure $μ$ - is `infinitesimally Hilbertian', which means that its associated Sobolev space $W^{1,2}(M,g,μ)$ is a Hilbert space. We actually prove a stronger result: the abstrac…