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.
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.
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 =…
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.
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 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.
Kernel Density Machines learn probability densities without structural assumptions.
problem Learning probability densities under minimal assumptions.
method Kernel-based framework, agnostic to structural requirements.
result Consistency and functional central limit theorem for sample estimator.
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.
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 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.
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.
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 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.
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 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 …
The paper reviews historical and modern approaches to asset pricing probability measures.
problem Constructing or selecting probability measures for asset pricing.
method Historical review of various approaches including state price theory, martingale measures, and modern data-driven methods.
result Modern asset pricing involves constructing, transforming, or selecting probability measures to represent market prices.
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.
The results on the mean-variance hedging problem in Gouriéroux, Laurent and Pham (1998), Rheinländer and Schweizer (1997) and Arai (2005) are extended to discontinuous semimartingale models. When the numéraire method is used, we only assume the Radon-Nikodym derivative of the variance-optimal signed martingale measure …
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 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 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.
The paper extends a measure preserving property to bi-Lipschitz maps between Moran sets.
problem Extending the measure preserving property to Moran sets.
method Analyzing bi-Lipschitz maps between Moran sets.
result Bi-Lipschitz maps between Moran sets preserve measure locally.
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.
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.
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.
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…
In this article, we provide representations of European and American exchange option prices under stochastic volatility jump-diffusion (SVJD) dynamics following models by Merton (1976), Heston (1993), and Bates (1996). A Radon-Nikodym derivative process is also introduced to facilitate the shift from the objective mark…
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 …
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.
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.
Study invariant measures on measured laminations for subgroups of mapping class group.
problem Classify invariant Radon measures on space of measured laminations for subgroups of mapping class group.
method Geometric approach, focusing on recurrent measured laminations, explicitly constructing ergodic measures.
result Show uniquely ergodic for divergence-type subgroups, generalize results for full mapping class group.
Develops a new algebraic framework for differential geometry of infinite dimensional spaces.
problem Creating a differential geometry for infinite dimensional spaces without topology or local coordinates.
method Introduces a general algebraic framework for lifted geometry applicable to various infinite dimensional spaces.
result Stokes' Theorem appears as a form of differentiability in the lifted geometry of spaces of submanifolds.
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 Lp norms of the Radon-Nikodym derivatives. We also prove that a logarithmic Sobolev inequality holds …
For Machine Learning (ML) classification problem, where a vector of x--observations (values of attributes) is mapped to a single y value (class label), a generalized Radon--Nikodym type of solution is proposed. Quantum--mechanics --like probability states ψ2(x) are considered and "Cluster Cente…
In a market of deterministic cash flows, given as an additive, symmetric relation of exchangeability on the finite signed Borel measures on the non-negative real time axis, it is shown that the only arbitrage-free price functional that fulfills some additional mild requirements is the integral of the unit zero-coupon b…
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.
Paper revisits Black-Scholes model, proving solution existence and measuring market uncertainty.
problem Proving existence of solution in inverse Black-Scholes model.
method Rigorous proof and empirical study using finite element method.
result New measure of market uncertainty developed.
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.
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.
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 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…
Develops a new framework for estimating joint probability distributions.
problem Estimating joint probability distributions from large sample sizes.
method Tensor product reproducing kernel Hilbert spaces (RKHS) with normalized and positive model.
result Fast computation and applicability to prediction and classification problems.
Sliced Optimal Transport simplifies OT for fast computation.
problem Efficient computation of distances and barycenters for probability measures.
method Combines OT, integral geometry, and statistics for fast computation.
result Retains rich geometric structure while speeding up computations.
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…
Developed new Crofton formulas for pseudo-Riemannian spaces.
problem Computing volumes and curvature integrals in pseudo-Riemannian space forms.
method Introduced Crofton formulas using distributions and Alesker's Radon transform.
result Explicit Crofton formulas for all isometry-invariant valuations on pseudo-Riemannian spaces.