We show that for a strongly convergent sequence of geometrically finite Kleinian groups with geometrically finite limit, the Cannon-Thurston maps of limit sets converge uniformly. If however the algebraic and geometric limits differ, as in the well known examples due to Kerckhoff and Thurston, then provided the geometr…
Study geometric quantization on K3 surfaces, showing spectral convergence.
problem Quantization of K3 surfaces from spectral perspective.
method Special Lagrangian fibrations and hyper-Kähler structures.
result Spectral convergence of ∂ˉ-Laplacians on prequantum line bundles. Establishes geometric convergence of iterative optimization algorithms.
problem Analyzes convergence of iterative optimization algorithms under general assumptions.
method General framework for iterative optimization algorithms, proving asymptotic geometric convergence and providing convergence rates.
result Asymptotic geometric convergence of iterative optimization algorithms with exact rate.
Geometric tempering fails for Langevin dynamics, proving convergence limits.
problem Proving convergence and limitations of geometric tempering for Langevin dynamics.
method Theoretical investigation of geometric tempering using Langevin dynamics.
result Geometric tempering can lead to exponential time convergence and poor functional inequalities.
Geometric analysis improves convergence of variational inference.
problem Challenges in analyzing convergence of variational inference due to non-convexity and non-smoothness.
method Exploits exponential family structure and Bregman divergences to geometrically analyze the optimization landscape.
result Establishes non-asymptotic convergence rates for gradient descent algorithms.
The paper proves spectral convergence for a specific type of geometric quantization.
problem Spectral convergence of ∂-Laplacians on toric symplectic manifolds. method Study of a family of compatible complex structures converging to the large complex structure limit.
result Spectral convergence of ∂-Laplacians acting on Lk. This work proves a strong convergence result for a geometric EM scheme on Riemannian manifolds.
problem Convergence of numerical schemes for manifold-valued SDEs.
method Geometric Euler-Maruyama scheme for Riemannian manifolds.
result Strong convergence of order 1/2 for the geometric EM scheme on Riemannian manifolds.
Study equidistribution for flows on geometrically finite convergence group actions.
problem Counting, mixing and equidistribution for flows on geometrically finite convergence group actions.
method Establishing results for finite BMS measures on flow spaces associated to geometrically finite convergence group actions.
result Results apply to flow spaces associated to relatively Anosov groups.
Study on slow convergence in geometric variational problems.
problem Slow convergence of solutions in geometric variational problems.
method Identifying necessary conditions for slowly converging solutions and characterizing their convergence rate and direction.
result Characterization of the rate and direction of convergence for slowly converging solutions.
A recent algorithmic family for distributed optimization, DIGing's, have been shown to have geometric convergence over time-varying undirected/directed graphs. Nevertheless, an identical step-size for all agents is needed. In this paper, we study the convergence rates of the Adapt-Then-Combine (ATC) variation of the DI…
We develop a new approach to geometric quantization using the theory of convergence of metric measure spaces. Given a family of Kähler polarizations converging to a non-singular real polarization on a prequantized symplectic manifold, we show the spectral convergence result of ∂ˉ-Laplacians, as well as th…
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.
Two algorithms converge to dictionary learning with geometric rate for non-uniform data.
problem Dictionary learning for non-uniform data models.
method Derivation of convergence conditions for MOD and ODL.
result Both algorithms converge to the generating dictionary with geometric rate under certain conditions.
Geometric tempering improves sampling from distributions, with exponential convergence rates.
problem Sampling from probability distributions using gradient flow dynamics.
method Geometric tempering of the target distribution in Wasserstein and Fisher-Rao gradient flows.
result Exponential convergence in continuous and discrete time for geometric tempering.
Paper studies convergence of nonnegative scalar curvature metrics to a specific limit space.
problem Understanding convergence of nonnegative scalar curvature metrics.
method Analyzes a sequence of warped product metrics on $\Sph^2 imes \Sph^1$.
result Sequence converges to an extreme limit space in specific senses.
New HMC method uses asymmetrical momentum distributions and improves performance.
problem Rigorous convergence guarantees for HMC with Gaussian momentum distributions.
method New convergence analysis for HMC with general asymmetrical momentum distributions, proposing AD-HMC.
result AD-HMC exhibits geometric convergence in Wasserstein distance under certain conditions.
This paper uses combinatorial Ricci flow to tackle Thurston's triangulation conjecture.
problem Thurston's triangulation conjecture for hyperbolic 3-manifolds.
method Combinatorial Ricci flow approach to prove convergence and geometric decompositions.
result Combinatorial Ricci flow converges if and only if the triangulation is geometric.
The paper studies degenerations of rational maps and their limits as geometrically finite rational maps.
problem Understanding the limits of quasi post-critically finite degenerations of rational maps.
method Constructing limits as geometrically finite rational maps on a tree of Riemann spheres, proving boundedness, and giving convergence criteria.
result Progress towards Thurston's compactness theorem and double limit theorem in complex dynamics.
Study geometric structure of Ricci shrinker ends without global curvature assumptions.
problem Understand the geometric structure of Ricci shrinker ends without global curvature constraints.
method Analyze blow-up sequences of Ricci shrinkers at points with Type I scalar curvature bound, extending F-convergence theory.
result Limits of Ricci shrinkers at points with Type I scalar curvature bound split a line in four dimensions.
Yau's Affine Normal Descent optimizes smooth unconstrained problems with geometrically adapted directions.
problem Optimizing smooth unconstrained problems with geometrically adapted directions.
method Yau's Affine Normal Descent (YAND) uses the equi-affine normal of level-set hypersurfaces as search directions.
result YAND converges globally under standard smoothness assumptions and locally quadratically near nondegenerate minimizers.
Study continuity of limit sets in symmetric spaces.
problem Continuity of limit sets for geometrically finite subgroups in symmetric spaces.
method Extended geometrically finite representations theory.
result Limit sets vary continuously with respect to Hausdorff distance under strong convergence.
This paper proposes a geometric estimator of dependency between a pair of multivariate samples. The proposed estimator of dependency is based on a randomly permuted geometric graph (the minimal spanning tree) over the two multivariate samples. This estimator converges to a quantity that we call the geometric mutual inf…
Unified geometric flows improve deep learning efficiency and simplify neural network topologies.
problem Improving deep learning performance and simplifying neural network structures.
method Proposes a thermodynamically coupled Ricci flow that dynamically adapts parameter space geometry to loss landscape topology, enabling automated singularity resolution and providing entanglement entropy bounds.
result Demonstrates 2.1× convergence acceleration and 63% topological simplification while maintaining O(NlogN) complexity, outperforming Riemannian baselines by 15.2% in few-shot accuracy. BBVI with STL converges geometrically under perfect specification, with quadratic variance bound.
problem Convergence rate of BBVI with STL estimator.
method Proved geometric convergence rate with quadratic variance bound for BBVI with STL estimator.
result BBVI with STL converges geometrically under perfect variational family specification.
This research smooths out fluid equations to avoid sudden shocks.
problem Formation of shock singularities in compressible fluid equations.
method Information geometric regularization of unidimensional pressureless Euler equations.
result Smooth global solutions without artificial viscosity.
The principle of convergence stability for geometric flows is the combination of the continuous dependence of the flow on initial conditions, with the stability of fixed points. It implies that if the flow from an initial state g0 exists for all time and converges to a stable fixed point, then the flows of solutions…
Elliptical slice sampling converges geometrically, providing reliable sampling for Bayesian learning.
problem Sampling from posterior distributions in Bayesian learning.
method Elliptical slice sampling, geometric ergodicity.
result Elliptical slice sampling yields geometric convergence guarantees under weak regularity assumptions.
Generalized Tanaka prolongation ensures convergence of formal embeddings of complex manifolds.
problem Ensuring convergence of formal embeddings of complex manifolds under weaker conditions.
method Formulated and proved generalized Tanaka prolongation for geometric structures.
result Convergence of formal embeddings holds under weaker semi-positive normal bundle conditions.
Gradient descent converges to perfect classification in neural nets for non-separable data.
problem Classifying linearly non-separable data using neural networks.
method Analysis of gradient descent dynamics in neural networks with sufficient but not large number of neurons.
result Gradient descent converges to global minima with perfect classification in the landscape of minimization problems.
In this note, we extend our previous work on the inverse σk problem. Inverse σk problem is a fully nonlinear geometric PDE on compact Kähler manifolds. Given a proper geometric condition, we prove that a large family of nonlinear geometric flows converges to the desired solution of the given PDE.
We explore the connection between two problems that have arisen independently in the signal processing and related fields: the estimation of the geometric mean of a set of symmetric positive definite (SPD) matrices and their approximate joint diagonalization (AJD). Today there is a considerable interest in estimating t…
SGD and stochastic gradient descent converge at optimal rates for certain non-convex functions.
problem Optimal convergence rates for non-convex functions under gradient noise.
method Geometric interpretation of the PL-condition to analyze convergence rates.
result Convergence rates of SGD and stochastic gradient descent match those of strongly convex quadratics.
New rates for GLD and SGLD in infinite-dimensional spaces without dimensionality issues.
problem Gradient Langevin dynamics and SGLD convergence rates in high-dimensional spaces.
method Analysis of GLD and SGLD in infinite-dimensional Hilbert spaces, using stochastic differential equations and Markov chains.
result Derivation of dimension-free convergence rates for GLD and SGLD.
In this paper, we will determine the topological types of hyperbolic 3-anifolds H^3/G such that G is a geometric limit of any algebraically convergent sequence of quasi-Fuchsian groups.
Study of geometric actions on CAT(0) spaces and their limits.
problem Understanding limits of geometric actions on CAT(0) spaces.
method Analysis of convergence and splitting/collapsing phenomena in CAT(0) lattices and orbispaces.
result Proof of compactness theorem for CAT(0) homology orbifolds.
Unfolding paths in Outer space accumulate on a simplex, not converge.
problem Understanding accumulation points in Outer space.
method Constructing an unfolding path in Outer space.
result Unfolding paths accumulate on a 1-simplex, not converge.
We show that for a strongly convergent sequence of purely loxodromic finitely generated Kleinian groups with incompressible ends, Cannon-Thurston maps, viewed as maps from a fixed base limit set to the Riemann sphere, converge uniformly. For algebraically convergent sequences we show that there exist examples where eve…
SGD converges with perturbed forward-backward passes, explained by geometric amplification.
problem Analyzing convergence of SGD with perturbed forward-backward passes in composite optimization.
method Characterized propagation and amplification of perturbations, derived convergence guarantees for non-convex and PL objectives.
result Perturbations cascade through the computational graph, affecting convergence order under specific conditions.
A lightweight framework improves convergence and stability of PINNs for complex PDEs.
problem Training instability and reduced accuracy in PINNs for complex PDEs.
method Adaptive curvature correction using secant information to optimize first-order optimizers.
result Consistent improvements in convergence speed, stability, and accuracy over standard optimizers.
Gradient clipping helps private SGD converge despite potential bias.
problem Gradient clipping in private SGD can bias convergence.
method Theoretical analysis and empirical evaluation of gradient clipping effects.
result Gradient clipping can prevent convergence to stationary points and introduces bias.
In this paper, we examine a geometrical projection algorithm for statistical inference. The algorithm is based on Pythagorean relation and it is derivative-free as well as representation-free that is useful in nonparametric cases. We derive a bound of learning rate to guarantee local convergence. In special cases of m-…
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.
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…
In this study, we present and analyze a framework for geometric and topological estimation for mapping of unknown environments. We consider agents mimicking motion behaviors of cyborg insects, known as biobots, and exploit coordinate-free local interactions among them to infer geometric and topological information abou…
The paper explains how ReLU nets converge globally in high dimensions without strict assumptions.
problem Understanding global convergence of ReLU nets in very high dimensions.
method Fine-grained analysis of random activation matrices and detailed gradient norm and curvature analysis.
result Empirical loss function has favorable geometrical properties in the overparameterized setting.
We study the quantization of coupled Kähler-Einstein (CKE) metrics, namely we approximate CKE metrics by means of the canonical Bergman metrics, so called the ``balanced metrics''. We prove the existence and weak convergence of balanced metrics for the negative first Chern class, while for the positive first Chern clas…
GCNs converge and remain stable on large random graphs, revealing geometric insights.
problem Understanding the behavior of GCNs on large, sparse random graphs.
method Analysis of GCNs on random graph models with latent variables and geometric edge probabilities.
result GCNs converge to their continuous counterparts as graph size increases, and are stable to small graph deformations.
Schoen-Yau's zero mass theorem stability remains an open question.
problem Geometric stability of Schoen-Yau's zero mass theorem.
method Review of geometric stability, examples, and convergence notions.
result Open question on geometric stability of Schoen-Yau's zero mass theorem.