New model estimates indoor radon distribution with higher spatial resolution.
problem Accurate estimation of indoor radon concentration for health assessment.
method Quantile regression forest and probabilistic Monte Carlo sampling.
result Approximate lognormal distribution of indoor radon in Germany with specific exceedance probabilities.
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.
Technical proofs for Radon-Nikodym derivative identities.
problem Formalizing and proving theorems on Radon-Nikodym derivatives.
method Careful consideration of conditional and marginal probability measures.
result New interpretation of mutual and lattum information sums.
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…
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.
problem Estimating Radon-Nikodym derivatives in various applications.
method General regularization scheme in reproducing kernel Hilbert spaces.
result High order accuracy in reconstructing Radon-Nikodym derivatives at any point.
Kernel estimator optimally recovers function from noisy exponential Radon transform.
problem Inverting noisy exponential Radon transform of a function.
method Proposed a kernel estimator to estimate the true function.
result The estimator converges to the true function at minimax optimal rate.
The paper derives Pizzetti formulae and inverts the Radon transform on spheres.
problem Inverting the Radon transform on spheres.
method Obtained Pizzetti-type formulae on sphere regions, used delta distributions, and derived inversion formulae.
result Derived Pizzetti formulae and inversion formulae for the Radon transform on spheres.
Extends SW and GSW to compare heterogeneous joint distributions.
problem Limited applicability of SW and GSW to heterogeneous joint distributions.
method Introduces HHRT and PGRT to extend SW and GSW.
result H2SW distance for 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 S1 nor to S3. This is true for both smooth functions and distributions. The key ingredients of the proof are finding totally geodesic tori and realizing the Rado…
Paper proves uniqueness of Ricci flows from nonatomic measures on surfaces.
problem Existence and uniqueness of Ricci flows from nonatomic Radon measures.
method Combining previous work, established existence and proved uniqueness.
result Uniqueness of Ricci flows from nonatomic Radon measures on Riemann surfaces.
The paper explores properties of the Radon transform in relation to neural networks and ridges.
problem Understanding the Radon transform and its application to neural networks and ridges.
method Investigates properties of the Radon transform, introduces new subspaces, and characterizes ridges for any distributional profile.
result Clarifies and simplifies results on the optimality of ReLU networks using the Radon transform.
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.
A new metric HSW derived from hierarchical Radon Transform addresses computational bottlenecks in sliced Wasserstein.
problem Computational inefficiency of sliced Wasserstein in high-dimensional settings with few supports.
method Hierarchical Radon Transform (HRT) and bottleneck projections to reduce projection number.
result HSW metric derived from HRT is computationally efficient and maintains metric properties.
If G is a finite group, is a function f:G→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…
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.
Let M be a Riemannian globally symmetric space of compact type, M′ its set of maximal flat totally geodesic tori, and ad(M) its adjoint space. We show that the kernel of the maximal flat Radon transform τ:L2(M)→L2(M′) is precisely the orthogonal complement of the image of the pullback map…
Survey on inverse exponential Radon transform methods.
problem Analytical methods for inverse exponential Radon transform.
method Derivation of classical inversion formula, finite Hilbert transform, exact reconstruction from partial measurements, diverging-beam data.
result Exact reconstruction from 180 degree data using finite Hilbert transform.
New gradient flows improve high-dimensional sampling.
problem Sampling from high-dimensional target densities.
method Introducing Radon--Wasserstein gradient flows.
result Linear scaling in particles and dimensions.
The article studies mapping properties of Radon transform and backprojection on a unit ball.
problem Polyhomogeneous mapping properties of Radon transform and backprojection operator on the unit ball.
method Constructs a double b-fibration to desingularize the point-hyperplane relation, provides formulas and sharper estimates.
result Sharper estimates on polyhomogeneous mapping properties of Radon transform and backprojection compared to classic estimates.
Study proper actions of Lie groups on symmetric spaces, finding rigidity results and Hurwitz-Radon numbers.
problem Proper actions of non-compact semisimple Lie groups on pseudo-Riemannian symmetric spaces.
method Analysis of symmetric spaces and rigidity results.
result Any connected non-compact semisimple Lie group acting properly on these spaces must be globally isomorphic to Spin(n,1) up to compact factors. Problems of interpolation, classification, and clustering are considered. In the tenets of Radon--Nikodym approach ⟨f(x)ψ2⟩/⟨ψ2⟩, where the ψ(x) is a linear function on input attributes, all the answers are obtained from a generalized eigenproblem $|f|ψ^{[i]}\rangle =…
In this paper we prove a new inversion theorem and a refinement of an old support theorem for two Radon transforms on a symmetric space. Included are some new identities for the Abel transform and some results about the Fourier transform from a joint work with Rawat, Sengupta and Sitaram.
We establish a necessary and sufficient condition for a heptagonal knot to be figure-8 knot. The condition is described by a set of Radon partitions formed by vertices of the heptagon. In addition we relate this result to the number of nontrivial heptagonal knots in linear embeddings of the complete graph K7 into $\…
Neural networks with ReLU^k approximate Sobolev functions efficiently via Radon transform.
problem Approximating functions from Sobolev spaces using shallow ReLU^k neural networks.
method Utilizing the Radon transform and discrepancy theory, we provide nearly optimal approximation rates.
result Optimal approximation rates for smoothness up to order s = k + (d+1)/2.
Proves bounded subsolution theorem for complex Monge-Ampère equation on compact Hermitian manifolds.
problem Complex Monge-Ampère equation with positive Radon measure on compact Hermitian manifolds.
method Proves bounded subsolution theorem.
result Establishes bounded subsolution theorem for complex Monge-Ampère equation.
Sharp estimates for Bergman metrics derived from Kähler quantization.
problem Estimating Bergman metrics in Kähler quantization.
method Upper and lower bounds on the Bergman metric expressed in terms of φ. result Optimal C1,1ˉ-convergence for quantization of Kähler currents. 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.
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 …
We exhibit relations between van Kampen-Flores, Conway-Gordon-Sachs and Radon theorems, by presenting direct proofs of some implications between them. The key idea is an interesting relation between the van Kampen and the Conway-Gordon-Sachs numbers for restrictions of a map of (d+2)-simplex to Rd to the $…
Adapts to estimate functions from noisy ERT data.
problem Estimating functions from noisy Exponential Radon Transform data.
method Locally adaptive kernel type estimator for functions of varying smoothness.
result Achieves minimax optimal rate up to a log(n) factor for Sobolev functions.
We show injectivity of the X-ray transform and the d-plane Radon transform for distributions on the n-torus, lowering the regularity assumption in the recent work by Abouelaz and Rouvière. We also show solenoidal injectivity of the X-ray transform on the n-torus for tensor fields of any order, allowing the tensor…
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 …
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 …
Unified proof of end-point estimates for Radon transform on curved spaces.
problem Proving end-point estimates for the totally-geodesic Radon transform on spaces of constant curvature.
method Unified geometric approach to prove end-point estimates for Radon transform on spaces of constant curvature.
result Unified formula for the k-plane transform of radial functions on spaces of constant curvature. This paper reinterprets and generalizes Hurwitz--Radon numbers using Lie groups and manifolds.
problem Understanding and generalizing Hurwitz--Radon numbers in Lie algebra settings.
method Defining new numbers ρG,s(M,σ) and ρG,s±(M,σ,abla) for G-manifolds and their affine connections. result New numbers coincide with Kannaka--Tojo's ρ(1)(g,ι) and ρ(2)(g,ι) in certain cases. Connections between integration along hypersufaces, Radon transforms, and neural networks are exploited to highlight an integral geometric mathematical interpretation of neural networks. By analyzing the properties of neural networks as operators on probability distributions for observed data, we show that the distribu…
Improved bounds for neural network approximations of functions.
problem Bounding the width of neural networks for function approximation.
method Extending Radon-based norms to bounded open sets and deriving new approximation bounds.
result Improved sparse approximation bounds for neural networks.
Extends Gaussian process theory to Banach spaces.
problem Extending Gaussian process theory to Banach spaces.
method Investigates the connection between Gaussian processes and Gaussian random elements in reproducing kernel Banach spaces.
result Characterizes positive definite functions that arise from covariance operators in Banach space setting.
New test uses neural networks to compare distributions, outperforming traditional methods.
problem Comparing distributions in high dimensions and higher orders of smoothness.
method Integral probability metrics with Radon bounded variation functions and neural networks.
result The Radon-Kolmogorov-Smirnov (RKS) test outperforms traditional methods in distinguishing distributions.
Study measures invariant under horospherical subgroups for finitely generated Kleinian groups.
problem Investigating measures invariant under horospherical subgroups for finitely generated Kleinian groups.
method Combining results from Landesberg and Lindenstrauss with 3-manifold theory, including the Tameness Theorem.
result Identified all Radon measures on quotient space that are ergodic and invariant under horospherical subgroup.
This research explains why SGD generalizes better than ADAM in deep learning.
problem Understanding the generalization gap between SGD and ADAM in deep learning.
method Analyzing local convergence behaviors through Levy-driven stochastic differential equations (SDEs).
result SGD is more locally unstable and better escapes from sharp minima to flatter ones, leading to better generalization.
The paper studies invariant measures for specific actions in algebraic groups.
problem Investigating invariant measures for horospherical actions and Anosov groups.
method Analyzing the space of invariant measures for NM-actions on Γ\G. result The space of invariant measures is homeomorphic to RextrankG−1. Study utility maximization with delayed information in continuous time Gaussian markets.
problem Maximizing utility with delayed information in continuous time Gaussian markets.
method Purely probabilistic approach based on Radon-Nikodym derivatives of Gaussian measures.
result Solution for optimal control and value in a specific Gaussian framework.
This paper deals with various topics in analysis on hyperbolic spaces. It surveys some recent progress in non-Euclidean Fourier Analysis and proves some new results for the geodesic Radon transform on hyperbolic spaces.
The abstract shows how embeddings inscribe trapezoids or map three points to a line, proving nonexistence of certain maps.
problem Proving the nonexistence of affinely 3-regular maps in infinitely many dimensions.
method Elementary proof using embeddings and nonsingular bilinear maps.
result Recovery of nonexistence results for affinely 3-regular maps without complex algebraic techniques.
A general method for analytic inversion in integral geometry is proposed. All classical and some new reconstruction formulas of Radon-John type are obtained by this method. No harmonic analysis and PDE is used.
Solves Ricci flow on Riemann surfaces with measure initial data.
problem Existence and smoothness of Ricci flow on Riemann surfaces.
method Formulation and solution of existence problem using Ricci flow.
result New examples of nongradient expanding Ricci solitons.