Paper proves existence of unique constant scalar curvature Kähler metric under certain conditions.
problem Existence of constant scalar curvature Kähler metrics on polarized manifolds.
method Direct proof using microscopic stability thresholds and conditions on the limit.
result Existence of a unique constant scalar curvature Kähler metric under specific conditions.
The notion of Berman-Gibbs stability was originally introduced by Robert Berman for Q-Fano varieties X. We show that the pair (X,−KX) is K-stable (resp. K-semistable) provided that X is Berman-Gibbs stable (resp. semistable).
GADD accelerates uniform-rate discrete diffusion models by 2 orders of magnitude.
problem Slow sampling in uniform-rate discrete diffusion models.
method Gibbs-based corrector (GADD) that constructs Gibbs posterior likelihoods directly from the concrete score function.
result Achieves an overall sampling complexity of O(polylog(ε−1)). New Gibbs sampling method improves MCMC efficiency.
problem Improving efficiency of Gibbs sampling.
method Non-uniform random scan with selection probability optimization.
result Non-uniform scan improves mixing time of Markov chain.
The pairwise influence matrix of Dobrushin has long been used as an analytical tool to bound the rate of convergence of Gibbs sampling. In this work, we use Dobrushin influence as the basis of a practical tool to certify and efficiently improve the quality of a discrete Gibbs sampler. Our Dobrushin-optimized Gibbs samp…
Extends probabilistic approach for Kahler-Einstein metrics on Fano manifolds.
problem Constructing Kahler-Einstein metrics on log Fano manifolds with non-discrete automorphism groups.
method Introduces Gibbs polystability and uses moment map constraint to break symmetry.
result Gibbs polystability conjectured to be equivalent to existence of Kahler-Einstein metric.
Uniform K-stability ensures existence of special metrics on toric manifolds.
problem Existence of conformally Kähler, Einstein-Maxwell metrics on toric manifolds.
method Introducing uniform K-stability and showing its equivalence to properness of relative K-energy.
result Uniform K-stability is necessary and sufficient for the existence of f-extremal metrics on toric manifolds. We prove some criteria for uniform K-stability of log Fano pairs. In particular, we show that uniform K-stability is equivalent to β-invariant having a positive lower bound. Then we study the relation between optimal destabilization conjecture and the conjectural equivalence between uniform K-stability and K-stabilit…
Proves uniform K-stability is open in Kähler cone.
problem Stability of Kähler metrics in complex geometry.
method Introduced new norm on test configurations and estimates for non-archimedean energy functionals.
result Uniform K-stability is an open condition in the Kähler cone.
We show relationships between uniform K-stability and plt blowups of log Fano pairs. We see that it is enough to evaluate certain invariants defined by volume functions for all plt blowups in order to test uniform K-stability of log Fano pairs. We also discuss the uniform K-stability of two log Fano pairs under crepant…
Holomorphic vector fields and anti-canonical divisors on complex manifolds are studied.
problem Existence of non-trivial holomorphic vector fields on compact complex manifolds.
method Vanishing result for measure preserving holomorphic vector fields, Gibbs stability, and log terminal singularities.
result No non-trivial holomorphic vector fields on compact complex manifolds with big anti-canonical line bundle.
Equivalence proven between divisorial stability and quotient log divisorial stability.
problem Equivalence of divisorial stability and log divisorial stability under finite group actions.
method Interpolation technique and equivariant divisorial stability construction.
result Equivariant divisorial stability of a polarized variety is equivalent to log divisorial stability of its quotient.
Uniform Ding stability implies existence of Kähler-Einstein metric on big anticanonical manifolds.
problem Existence of Kähler-Einstein metrics on manifolds with big anticanonical class.
method Developed a theory of Deligne functionals and slope formulas for singular metrics, proving a slope formula for the Ding functional in the big setting.
result Existence of a unique Kähler-Einstein metric implies uniform Ding stability.
We prove effective uniformization for nearly round 2-spheres and investigate their stability.
problem Proving effective uniformization for nearly round 2-spheres and their stability.
method Utilizing an identity related to the third-order differential of the conformal factor, and an isometric embedding of a round sphere into Euclidean space using an orthogonal basis of the first eigenspace of the Laplacian operator.
result We provide a simplified proof of effective uniformization and its stability.
The paper simplifies K-stability conditions for spherical varieties.
problem K-stability of polarized spherical varieties.
method Expressed K-stability in combinatorial terms, provided sufficient conditions.
result G-uniform K-stability provides a checkable condition for existence of constant scalar curvature metrics.
We show that uniform K-stability is a Zariski open condition in Q-Gorenstein families of Q-Fano varieties. To prove this result, we consider the behavior of the stability threshold in families. The stability threshold (also known as the delta-invariant) is a recently introduced invariant that is known to detect the K-s…
New stability criteria for Fano varieties using generalized b-divisors.
problem Characterizing uniform K-stability in Fano varieties. method Introducing a new function ildeδ and formalism for K-stability, proving stability conditions for Kähler-Einstein metrics. result Existence of a unique Kähler-Einstein metric implies uniform D-log K-stability when ildeδ(D)>1. New algorithms achieve uniform stability for empirical risk minimization.
problem Designing uniformly stable optimization algorithms for empirical risk minimization.
method Black-box conversion of smooth optimization algorithms and development of Mirror Descent for smooth optimization.
result Optimal algorithms with uniform stability and convergence rates for smooth optimization.
From the work of Dervan-Keller, there exists a quantization of the critical equation for the J-flow. This leads to the notion of J-balanced metrics. We prove that the existence of J-balanced metrics has a purely algebro-geometric characterization in terms of Chow stability, complementing the result of Dervan-Keller. We…
In this paper we prove that for toric varieties the uniform K-stability is the necessary condition for the existence of extremal metrics.
We apply a recent theorem of Li and the first author to give some criteria for the K-stability of Fano varieties in terms of anticanonical Q-divisors. First, we propose a condition in terms of certain anticanonical Q-divisors of given Fano variety, which we conjecture to be equivalent to the K-stability. We prove that …
We introduce a norm on the space of test configurations, which we call the minimum norm. We conjecture that uniform K-stability with respect to this norm is equivalent to the existence of a constant scalar curvature Kähler metric. This notion of uniform K-stability is analogous to coercivity of the Mabuchi functional. …
The paper proves conditions for existence of constant scalar curvature Kähler metrics with cone singularities.
problem Existence of constant scalar curvature Kähler metrics with cone singularities.
method Log K-polystability and G-uniform log K-stability are established. result Uniform log K-stability is achieved for normal varieties. Article proves effective conditions for existence of Kähler metrics.
problem Existence of extremal Kähler metrics on fibrations.
method Weighted uniform K-stability conditions derived from moment polytopes.
result Various effective conditions for K-stability verified.
The paper classifies test configurations and derives a criterion for uniform K-stability of certain algebraic varieties.
problem Uniform K-stability of G-varieties of complexity 1. method Classification of G-equivariant normal test configurations via combinatorial data and derivation of a criterion for uniform K-stability. result Derivation of a criterion for uniform K-stability in terms of combinatorial data.
We introduce uniform K-stability and its relationship with the coercivity property of the K-energy functional, for general polarized manifolds. Since the automorphism groups are not necessarily finite, size of the norm measuring uniformity should be reduced with respect to the group action. About this point we explain …
PAC-Bayes bounds for Gibbs posteriors derived via singular learning theory.
problem Generalization bounds for overparameterized models with data-dependent priors.
method Explicit non-asymptotic PAC-Bayes bounds using singular learning theory.
result Explicit posterior-averaged risk bounds for overparameterized models.
New method uses Coulomb gases for Monte Carlo integration with reduced errors.
problem Reducing integration errors in numerical algorithms.
method Using Gibbs measures with a large deviations approach.
result Preserves large deviation principle for improved integration.
Markov jump processes and continuous time Bayesian networks are important classes of continuous time dynamical systems. In this paper, we tackle the problem of inferring unobserved paths in these models by introducing a fast auxiliary variable Gibbs sampler. Our approach is based on the idea of uniformization, and sets…
The paper generalizes K-stability results to singular and weighted settings.
problem Generalizing K-stability to singular and weighted settings.
method Generalization of results in \cite{Li22a} to singular and weighted settings.
result The \(\mathbb{G}\)-uniform weighted K-stability for models implies \(\mathbb{G}\)-coercivity of the weighted Mabuchi functional.
Stability result for a popular algorithm in optimal transport.
problem Stability of the Iterative Proportional Fitting Procedure in time and metric.
method Uniform stability analysis in the 1-Wasserstein metric.
result Quantitative stability result for entropy-regularized Optimal Transport and Schrödinger bridges.
Proves constant scalar curvature Kähler metrics are very general.
problem Existence of constant scalar curvature Kähler metrics on smooth polarized varieties.
method Combining uniform arc K-stability and algebraic properties in families.
result The constant scalar curvature Kähler locus is very general.
The paper derives uniform stability-based coverage bounds for conformal prediction methods.
problem Establishing theoretical guarantees for conformal prediction methods.
method Uniform stability perspective applied to full-conformal, jackknife+, and CV+ prediction regions.
result Coverage bounds for finite-dimensional models derived using a concentration argument.
We give a complete criterion for the existence of generalized Kähler Einstein metrics on toric Fano manifolds from view points of a uniform stability in a sense of GIT and the properness of a functional on the space of Kähler metrics.
Decomposes J-energy into simpler intersection numbers for stability analysis.
problem Analyzing J-stability in algebraic geometry.
method Proves a decomposition formula for J-energy and shows equivalence of stability conditions.
result Equivalence of J-stability and K-stability for surfaces under pseudoeffective conditions.
New algorithms sample convex bodies using Markov chains and restricted Gaussian oracles.
problem Sampling uniformly from convex bodies efficiently.
method Markov chain Monte Carlo with proximal sampler and restricted Gaussian oracle.
result Efficient implementation of RGO for uniform sampling on convex bodies.
Extended Q-learning stability and convergence analysis.
problem Stability and convergence of Q-learning with linear function approximation.
method Analysis of (ε,κ)-tamed Gibbs policy, and other Q-learning variants. result Far more structure is required for convergence of Q-learning.
We establish the essentially optimal form of Donaldson's geodesic stability conjecture regarding existence of constant scalar curvature Kähler metrics. We carry this out by exploring in detail the metric geometry of Mabuchi geodesic rays, and the uniform convexity properties of the space of Kähler metrics.
New stability bounds for SGD on nonsmooth convex losses.
problem Understanding stability of SGD on nonsmooth convex losses.
method Sharp upper and lower bounds for SGD and full-batch GD on nonsmooth convex losses.
result SGD can be less stable but still useful for generalization bounds.
Accelerated gradient method's stability deteriorates exponentially with steps.
problem Algorithmic stability of Nesterov's accelerated gradient method.
method Analysis of two notions of algorithmic stability for Nesterov's accelerated gradient method.
result Stability of Nesterov's accelerated method deteriorates exponentially with the number of gradient steps.
In the present paper and the companion paper [8] a probabilistic (statistical mechanical) approach to the study of canonical metrics and measures on a complex algebraic variety X is introduced. On any such variety with positive Kodaira dimension a canonical (birationally invariant) random point processes is defined and…
Geometric interpretation of Fock-Goncharov positivity and disk stabilization in symmetric space.
problem Understanding Fock-Goncharov positivity and its geometric implications.
method Geometric interpretation and bending deformations of Fuchsian representations.
result Stabilization of a uniform Finsler quasi-convex disk in the symmetric space.
The present paper provides a new generic strategy leading to non-asymptotic theoretical guarantees on the Leave-one-Out procedure applied to a broad class of learning algorithms. This strategy relies on two main ingredients: the new notion of Lq stability, and the strong use of moment inequalities. Lq stability e…
Geodesic rays prove key aspects of cscK metrics existence and stability.
problem Existence and stability of constant scalar curvature Kähler metrics.
method Reduction to regularization conjecture and analysis of geodesic rays.
result Uniform K-stability and JKX-stability are sufficient for cscK metrics existence. Paper develops a new geometric framework for Kerr stability.
problem Uniform decay properties of Kerr solutions.
method Geometric framework for Teukolsky equation in nonlinear Kerr spacetime.
result First nonlinear version of Chandrasekhar transformation.
Proves finitely generated associated graded rings for valuations on log Fano pairs.
problem Stability thresholds of log Fano pairs.
method Proves finite generation of associated graded rings for valuations.
result Log Fano pairs are uniformly K-stable if their stability threshold is less than a certain value.
The paper studies dynamical properties in semigroups modulo ideals.
problem Analyzing shadowing, expansivity, and stability in semigroups with ideals.
method Investigates shadowing, expansivity, and stability properties in uniform transformation semigroups modulo an ideal.
result Establishes that if a semigroup exhibits shadowing and expansivity modulo an ideal, it is also topologically stable modulo that ideal.
Study shows how non-uniform scaling affects persistence diagrams.
problem Stability of persistence diagrams under non-uniform scaling.
method Explicit bounds on bottleneck distance derived for Euclidean scaling.
result Explicit bounds on the stability of persistence diagrams under non-uniform scaling.