The Bergman measure converges to the Zhang measure on a hybrid space.
problem Proving convergence of Bergman measures to Zhang measure.
method Analyzing convergence on a hybrid space and metrized curve complex.
result Bergman measure converges to Zhang measure on a hybrid space.
The paper proves a convergence theorem for Wiener measures on holonomy groups.
problem Understanding convergence of Wiener measures on holonomy groups.
method Using stochastic parallel transports along convergent metric connections.
result Proves a convergence theorem for push-forward Wiener measures on holonomy groups.
This paper studies convergence behavior of latent mixing measures that arise in finite and infinite mixture models, using transportation distances (i.e., Wasserstein metrics). The relationship between Wasserstein distances on the space of mixing measures and f-divergence functionals such as Hellinger and Kullback-Leibl…
Study convergence and approximations of entropic regularized Wasserstein distances for Gaussian and RKHS measures.
problem Convergence and approximations of entropic regularized Wasserstein distances in Gaussian and RKHS settings.
method Analysis of convergence and finite sample approximations of entropic regularized Wasserstein distances in Gaussian and RKHS settings.
result Strictly weaker convergence in 2-Sinkhorn divergence for Gaussian measures compared to exact 2-Wasserstein distance.
Paper proves convergence of Kalman filter on Stiefel manifolds with measurement errors.
problem Filtering constant particle with measurement errors on Stiefel manifolds.
method Extended Kalman filter applied to Stiefel manifold-valued observations.
result Convergence of the extended Kalman filter proved for constant system process.
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.
Study provides convergence rates for risk measure estimation.
problem Estimating risk measures from limited data.
method Plug-in estimation using empirical measures.
result Non-asymptotic convergence rates for risk measure estimation.
We introduce a natural definition of Lp-convergence of maps, p≥1, in the case where the domain is a convergent sequence of measured metric space with respect to the measured Gromov-Hausdorff topology and the target is a Gromov-Hausdorff convergent sequence. With the Lp-convergence, we establish a theory of …
The paper shows how MMD metrizes weak convergence for certain kernels.
problem Characterizing MMD metrizing weak convergence for a wide class of kernels.
method Proving MMD metrizes weak convergence for specific kernels on a locally compact space.
result Corrected prior results and identified new kernels metrizing weak convergence.
The paper studies PCA of probability measures with varying sample sizes and finds optimal convergence rates.
problem PCA of multiple probability measures with varying sample sizes.
method Double asymptotic regime analysis with convergence rates n−1/2+m−α for empirical covariance and PCA risk. result Optimal convergence rates for empirical covariance and PCA risk in the dense regime are proven.
Study group actions in metric spaces, proving convergence of lens spaces.
problem Understanding convergence in metric measure spaces with group actions.
method Generalized box and observable distances, applied mass-transport theory.
result Sequence of lens spaces converging to infinite-dimensional complex projective space.
CAVI converges for log-concave measures via optimal transport.
problem Finding the closest product measure to a log-concave measure via CAVI.
method Adapting coordinate descent techniques from Euclidean space to optimal transport for log-concave densities.
result Proves convergence of CAVI for log-concave densities and provides rates of convergence under additional conditions.
New conditions ensure MMDs separate and converge to target distributions.
problem Ensuring MMDs separate and converge to target distributions.
method Deriving new sufficient and necessary conditions for MMDs on separable metric spaces.
result First KSDs that exactly metrize weak convergence to P.
Study on convergence rate of weighted Yamabe flow.
problem Weighted Yamabe problem on smooth metric measure spaces.
method Weighted Yamabe flow and its convergence rate analysis.
result Study and analysis of convergence rate of the weighted Yamabe flow.
Study stability of curvature-dimension condition for negative dimensions.
problem Stability of curvature-dimension condition with negative dimension parameters.
method Introduced CD(K, N)-condition for N < 0, defined distance d_{\mathsf{iKRW}}, proved convergence stability.
result Limit structure of converging metric measure spaces remains CD(K, N) for N < 0.
The paper reconstructs Lorentzian spacetimes from causal sets.
problem Reconstructing Lorentzian spacetimes from causal sets.
method Introduced a concept of isomorphy and three types of convergence.
result Established Gromov's reconstruction theorem in Lorentzian geometry.
Study improves convergence rates for GVI under prior misspecification.
problem Improving convergence rates for GVI under prior misspecification.
method Proves rates of convergence and robustness to prior misspecification in GVI framework.
result Establishes sufficient conditions for existence and uniqueness of GVI posteriors.
New metrics avoid high-dimensional analysis challenges, proving convergence without 'curse of dimensionality'.
problem High-dimensional analysis challenges in empirical measure convergence.
method Proposed a new class of probability metrics free of the curse of dimensionality.
result Convergence of empirical measures is free of the curse of dimensionality.
Study local minimizers of Ginzburg-Landau functionals in high dimensions, showing energy measures converge to rectifiable measures.
problem Investigating minimizers of Ginzburg-Landau functionals in high dimensions with energy bounds.
method Analyzing minimizers with logarithmic energy bounds and considering the vacuum manifold's homotopy classes.
result Normalized energy measures converge to an (n−2)-rectifiable measure associated with a stationary varifold. Study on convergence of Narasimhan-Simha measures on degenerating families of Riemann surfaces.
problem Analyzing the convergence of measures on degenerating families of Riemann surfaces.
method Hybrid space approach, using metrized curve complex and Hermitian pairing.
result Convergence of measures on hybrid space, extending to singular curves.
The paper analyzes Laplace learning for Gaussian measure data in infinite dimensions, proving convergence.
problem Analyzing Laplace learning for infinite-dimensional Gaussian measure data.
method Minimizes Dirichlet energy on a graph constructed from the full dataset.
result Proves pointwise convergence of the graph Dirichlet energy for Gaussian measure data.
Binary Iterative Hard Thresholding converges with optimal number of 1-bit measurements.
problem Recovering sparse signals from 1-bit compressed measurements.
method Binary Iterative Hard Thresholding (BIHT) algorithm.
result BIHT converges with only O(k/ε) measurements, optimal for recovery.
Paper addresses group synchronization with incomplete measurements and proves linear convergence of GPM.
problem Orthogonal group synchronization with incomplete measurements and additive noise.
method Generalized power method (GPM) with local error bound analysis.
result Linear convergence of GPM to a global maximizer under general additive noise model.
We associate certain probability measures on R to geodesics in the space $\H_L$ of positively curved metrics on a line bundle L, and to geodesics in the finite dimensional symmetric space of hermitian norms on H0(X,kL). We prove that the measures associated to the finite dimensional spaces converge weakly to t…
We study the convergence of volume forms on a degenerating holomorphic family of log-Calabi-Yau varieties to a non-Archimedean measure, extending a result of Boucksom and Jonsson. More precisely, let (X,B) be a holomorphic family of sub log canonical, log-Calabi-Yau complex varieties parameterized by the punctured un…
The weighted Yamabe flow converges on smooth metric measure spaces.
problem Analyzing convergence of the weighted Yamabe flow on metric measure spaces.
method Introduced the weighted Yamabe flow and proved its long-time existence and convergence under certain conditions.
result Long-time existence and convergence of the weighted Yamabe flow on smooth metric measure spaces.
Paper proves convergence of measure transfer schemes using slicing and matching.
problem Iterative schemes for measure transfer and approximation problems.
method Slicing-and-matching procedure, stochastic gradient descent on Wasserstein space.
result Almost sure convergence proof for stochastic slicing-and-matching schemes.
The paper examines convergence of currents and forms under smooth diffeomorphisms.
problem Analyzing convergence of currents and forms under C0-limits of diffeomorphisms. method Geometric analysis, measure theory, homotopy theory.
result Pushforwards of rectifiable currents converge in the flat norm.
Aim of this paper is to discuss convergence of pointed metric measure spaces in absence of any compactness condition. We propose various definitions, show that all of them are equivalent and that for doubling spaces these are also equivalent to the well known measured-Gromov-Hausdorff convergence. Then we show that the…
Study birth-death dynamics for sampling Gibbs measures with nonconvex potentials.
problem Sampling Gibbs measures with nonconvex potentials.
method Birth-death dynamics, Kullback-Leibler divergence, χ2 divergence, kernel-based approximations, Γ-convergence of gradient flows. result Probability density converges exponentially fast to Gibbs equilibrium measure with a universal rate.
Non-ergodic measures found in horocycle flow on Abelian differentials.
problem Finding non-ergodic measures in the horocycle flow on Abelian differentials.
method Analyzing weak convergence of ergodic measures to non-ergodic invariant measures.
result Existence of points with non-equidistributing horocycle flow orbits.
We show that for any weakly convergent sequence of ergodic SL2(R)-invariant probability measures on a stratum of unit-area translation surfaces, the corresponding Siegel-Veech constants converge to the Siegel-Veech constant of the limit measure. Together with a measure equidistribution result due to Eskin-M…
Enhances SGLD for log-concave posteriors with asynchronous computation.
problem Sampling log-concave posterior distributions efficiently.
method Integrates asynchronous computation into SGLD with delayed gradients.
result Convergence in measure is not significantly affected by delayed gradient information.
Study examines risk premium convergence rates in risk sharing contracts.
problem Analyzing risk premium convergence rates in risk sharing contracts.
method Examines the limiting behavior of risk premium associated with Pareto optimal risk sharing contracts under general law-invariant risk measures.
result Risk premium convergence rate is typically n1/2, not n. The paper studies curvature measures and volume-preserving flows on convex bodies.
problem Characterizing and understanding convex bodies through anisotropic curvature measures.
method Developed anisotropic curvature measures, used Minkowski formulas and Heintze-Karcher inequalities, and analyzed volume-preserving flows.
result Characterized Wulff shapes via anisotropic curvature measures and proved convergence of volume-preserving flows.
The paper analyzes mirror descent in measure spaces and its applications.
problem Optimizing convex functionals over measure spaces.
method Derives convergence of mirror descent for relatively smooth and convex pairs of functionals.
result Shows Sinkhorn's primal iterations and EM as mirror descent.
We study the convergence of earthquake paths and horocycle paths in the Gardiner-Masur compactification of Teichmüller space. We show that an earthquake path directed by a uniquely ergodic or simple closed measured geodesic lamination converges to the Gardiner-Masur boundary. Using the embedding of flat metrics into th…
Introduces fractional k-dimensional measure bridging fractional length and area.
problem Defining fractional measures for dimensions between 0 and n-1.
method Introduces a parameterized fractional measure σ that converges to Hausdorff measure. result Fractional measure converges to Hausdorff measure with a known constant factor.
Study of irreversible metric-measure spaces, proving convergence and stability results.
problem Understanding Gromov-Hausdorff convergence and stability in noncompact irreversible metric-measure spaces.
method Introducing a nondecreasing function to bound reversibility of larger balls, proving convergence/stability results in Gromov-Hausdorff topology.
result Satisfactory convergence/stability results in Gromov-Hausdorff topology for various irreversible spaces, including Finsler manifolds.
Preconditioned non-convex gradient descent improves noisy matrix estimation.
problem Estimating low-rank matrices from noisy measurements.
method Preconditioned non-convex gradient descent for noisy measurements.
result Preconditioned method converges to minimax optimal estimate at a linear rate.
The rate of convergence of weighted kernel herding (WKH) and sequential Bayesian quadrature (SBQ), two kernel-based sampling algorithms for estimating integrals with respect to some target probability measure, is investigated. Under verifiable conditions on the chosen kernel and target measure, we establish a near-geom…
We show that the Mahler measures of the Jones polynomial and of the colored Jones polynomials converge under twisting for any link. Moreover, almost all of the roots of these polynomials approach the unit circle under twisting. In terms of Mahler measure convergence, the Jones polynomial behaves like hyperbolic volume …
Study on convergence of exponential probability measures with applications to maximum entropy models and SGLD.
problem Characterizing the limit of probability measures with exponential densities as temperature approaches zero.
method Quantitative bounds on Wasserstein distance using geometric measure theory tools.
result Established quantitative convergence results for norm-like potentials under invertibility conditions.
This paper analyzes OCBA algorithms' convergence rates for DEDS optimization.
problem Optimizing discrete-event dynamic systems with limited computing resources.
method Characterizes convergence rates of two OCBA algorithms under different performance measures.
result OCBA algorithms achieve optimal convergence rates under probability of correct selection and expected opportunity cost measures.
Paper finds sparse representation of functions using inverse scale space flow.
problem Finding sparse representation of L2 functions. method Inverse scale space flow to minimize L2 loss. result Convergence to optimal solution in ideal and noisy cases.
Paper extends MLFD to signed measures via bilevel approach.
problem Risk minimization for infinite width neural networks and sparse deconvolution.
method Bilevel reduction to extend MLFD to signed measures, investigating convergence rates.
result Improved convergence rates for bilevel MFLD in low-noise regime and local exponential convergence for single neuron learning.
The study examines stability of metric measure spaces with integral Ricci curvature bounds.
problem Stability and compactness of metric measure spaces with integral Ricci curvature bounds.
method Proves convergence to metric measure spaces satisfying CD(K,n) condition under certain curvature bounds. result Proves convergence of sequences of Riemannian manifolds to metric measure spaces satisfying CD(K,n) condition. We prove some results concerning the boundary of a convex set in $\H^n$. This includes the convergence of curvature measures under Hausdorff convergence of the sets, the study of normal points, and, for convex surfaces, a generalized Gauss equation and some natural characterizations of the regular part of the Gaussian …