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 variation spaces for neural networks, linking them to approximation theory.
problem Understanding the variation spaces of shallow neural networks.
method Examined variation spaces defined by convex hulls and integral representations for a dictionary of functions.
result Found that Barron space, spectral Barron space, and Radon BV space are variation spaces for certain neural networks.
Optimizes sampling from target distributions with applications to online learning.
problem Optimizing the total variation distance between target and sampled distributions.
method Analyzes the sample complexity of approximate rejection sampling and its applications.
result The optimal total variation distance is given by $ ildeΘ( rac{D}{f'(n)})$.
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. 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.
The paper uses Banach spaces to analyze neural networks.
problem Understanding the function spaces of neural networks.
method Theory of reproducing kernel Banach spaces.
result Representer theorem for wide class of Banach spaces.
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.
New framework explains deep neural networks using variational spline theory.
problem Understanding functions learned by deep neural networks.
method Developed a variational framework and function space.
result Deep ReLU networks are solutions to regularized data fitting problems over the proposed function space.
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.
Decision trees and shallow neural networks have different geometric complexities, impacting their interpretability and accuracy.
problem The geometric simplicity of decision boundaries in decision trees conflicts with the approximation capabilities of shallow neural networks.
method Analysis of the Radon total variation (RTV) seminorm to compare geometric complexity of decision regions and neural network approximations.
result Smooth barrier scores can approximate decision regions with finite RTV, but their performance depends on the tube-mass condition near the decision boundary.
A support theorem for the horocycle Radon transform f \to \hat{f} is a property of the form \hat{f} of compact support \rightarrow f of compact support. Here we prove a variation of this result where support (\hat{f}) is outside a fixed horocycle in hyperbolic space.
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.
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.
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.
Paper introduces S3W distance for spherical probability distributions.
problem Comparing spherical probability distributions efficiently and accurately.
method S3W distance using stereographic projection and generalized Radon transform.
result Extensive theoretical analysis and evaluation of S3W performance.
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.
By employing the differential structure recently developed by N. Gigli, we first give a notion of functions of bounded variation (BV) in terms of suitable vector fields on a complete and separable metric measure space (X,d,μ) equipped with a non-negative Radon measure μ finite on bounded sets. Then, we e…
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…
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.
New algorithms ensure generated objects evolve and fill a distribution, unlike static neural networks.
problem Ensure generated objects evolve and fill a distribution, unlike static neural networks.
method Propose a numerical paradigm based on Radon-Sobolev statistical distances to ensure objects do not repeat and evolve.
result Objects created by VAEs evolve and fill the target probability distribution, unlike static neural networks.
The paper improves sample reweighting methods for adapting to covariate shifts.
problem Improving accuracy in reproducing kernel Hilbert spaces when data distributions differ.
method Combining known error bounds for reweighted kernel regression in RKHS to show reduced sample size needed for accuracy.
result Under weak smoothness conditions, fewer samples are needed for the same accuracy as standard supervised learning.
This paper concerns integral varifolds of arbitrary dimension in an open subset of Euclidean space with its first variation given by either a Radon measure or a function in some Lebesgue space. Pointwise decay results for the quadratic tilt-excess are established for those varifolds. The results are optimal in terms of…
New model classes for function approximation by neural networks defined on domains.
problem Defining novel model classes for function approximation on bounded domains.
method Introducing weighted variation spaces to define new model classes on domains.
result New model classes are strictly larger than classical ones but maintain the same NNA rates.
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…
Uniform convergence of metrics on surfaces with bounded curvature measures proved.
problem Proving uniform convergence of metrics on Alexandrov surfaces with bounded integral curvature.
method Weak convergence of measures and analytic approximation of metrics.
result Uniform convergence of metrics on Alexandrov surfaces proved.
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…
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.
Study eigenvalues and shapes, proving sharp inequalities for Steklov eigenvalues.
problem Eigenvalue continuity and shape optimization for Laplace and Steklov problems.
method Variational eigenvalue analysis, Sobolev space convergence, shape optimization techniques.
result Sharp isoperimetric inequalities for Steklov eigenvalues, upper bound 8πk for k-th perimeter-normalized eigenvalue. 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 $\…
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 …
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.