Global regularity proved for 4D Ricci flow with scalar curvature integral bound.
problem Global regularity of 4D Ricci flow with integral scalar curvature bound.
method Extended Ge-Jiang's result to include integral bound on scalar curvature.
result Global ε \varepsilon ε -regularity for 4D Ricci flow with integral scalar curvature bound. Global bounds for mean curvature flow gradients are derived.
problem Global regularity of mean curvature flows in higher dimensions.
method Derives global bounds for Hölder norms of gradients of solutions.
result Global bounds for Hölder norms of gradients are derived.
Global optimization for hybrid system identification problems.
problem Switching linear regression and bounded-error estimation in hybrid systems.
method Branch-and-bound strategy with efficient lower bounds for continuous optimization.
result Global optimality is always guaranteed with scalable algorithms.
New method certifies global robustness of neural networks efficiently.
problem Adversarial examples threaten certifiably robust neural networks.
method Formalized global robustness, adapted widely-used architectures with efficient global Lipschitz bounds.
result Certifiable robust models achieve state-of-the-art verifiable accuracy with negligible costs.
Lorentzian Ptolemy inequality linked to curvature bounds.
problem Global timelike sectional curvature bounds in Lorentzian geometry.
method Investigation of Ptolemy inequality in globally hyperbolic spacetimes.
result Equivalence of Lorentzian Ptolemy inequality to curvature bound.
Develops methods to calculate global index of real polynomials.
problem Calculating the global index of real polynomials.
method Two methods: via atypical fibres and Milnor arcs clusters.
result Derives upper bounds for the global index, refining Durfee's degree-based bound.
We prove global and local upper bounds for the Hessian of log positive solutions of the heat equation on a Riemannian manifold. The metric is either fixed or evolves under the Ricci flow. These upper bounds supplement the well-known global lower bound.
A new algorithm optimizes Gaussian process posterior mean functions efficiently.
problem Optimizing Gaussian process posterior mean functions over hyperrectangles is challenging due to nonlinearity and nonconvexity.
method PALM-Mean, a piecewise-analytic lower-bounding framework embedded in reduced-space spatial branch-and-bound.
result PALM-Mean improves scalability for large datasets compared to general-purpose solvers.
Synthetic proof shows globally hyperbolic Lorentzian spaces with specific curvature are warped products.
problem Synthetic proof of rigidity for globally hyperbolic Lorentzian spaces.
method Synthetic geometry and warped product analysis.
result Spaces with specific curvature and distance realizer are warped products.
I show that if a geodesic space has curvature bounded below locally in the sense of Alexandrov then its completion has the same lower curvature bound globally.
Global optimization for low-rank matrix recovery from noisy measurements.
problem Low-rank matrix recovery from noisy measurements.
method Factorized parametrization, curvature bound, stochastic gradient descent.
result Global convergence guarantee for stochastic gradient descent from random initialization.
Global stability bounds for matrix frames in phase retrieval problems.
problem Phase retrieval for matrix frames in various applications.
method Computable global stability bounds for the quasi-linear analysis map β, using Whitney stratification of positive semidefinite matrices of low rank.
result Novel conditions for a frame to be generalized phase retrievable.
New Alexandrov-Patchwork construction for Lorentzian spaces with curvature bounds.
problem Understanding finite diameter constraints in Lorentzian geometry.
method Constructing Alexandrov-Patchwork and proving Bonnet-Myers theorem for Lorentzian spaces.
result Lorentzian spaces with curvature bounds have finite diameter.
New framework for DNN training guarantees convergence to global minimum.
problem Training deep neural networks to converge to global minimum.
method Reformulated minimization problem with recursive algorithmic framework, using bounded style assumptions.
result Convergence to an ε-(global) minimum with O(1/ε^3) gradient computations.
First order methods can take extremely long to find global minima of non-convex functions.
problem Finding global minimizers of non-convex functions.
method Designing a family of non-convex functions and using statistical lower bounds for parameter estimation.
result First order methods can take exponential time to converge to a global minimizer.
Global existence of Yamabe flows on hyperbolic space proved without curvature bounds.
problem Global existence of Yamabe flows on hyperbolic space without completeness or curvature bounds.
method Instantaneously complete initial metrics, no curvature bounds required.
result Global existence of Yamabe flows on hyperbolic space of arbitrary dimension m ≥ 3 m\geq3 m ≥ 3 . New algorithm optimizes Hölder continuous functions efficiently.
problem Optimizing Hölder continuous multivariate functions.
method Uses a query creation rule for global optimization, avoiding proxy functions.
result Achieves an average regret bound of $O(T^{- racα{n}})$ for Hölder exponent α α α . Global methods outperform local in forecasting groups of time series, even in heterogeneous datasets.
problem Forecasting groups of time series, especially in heterogeneous datasets.
method Local methods consider each series separately, global methods fit a single model to all series.
result Global methods can outperform local methods in forecasting groups of time series, even in heterogeneous datasets.
Develop intrinsic consensus-based optimization framework on Riemannian manifolds with bounded curvature.
problem Nonconvex optimization on manifolds
method Intrinsic consensus-based optimization on Riemannian manifolds with bounded curvature
result Global convergence of the mean-field equation toward a global minimizer of the objective function.
New algorithms improve privacy in bandit problems with partial information.
problem Privacy constraints in multi-armed bandit problems with partial reward information.
method Proposed a generic framework for designing ε ε ε -global DP extensions of UCB and KL-UCB algorithms. result AdaP-KLUCB algorithm achieves optimal regret bound under ε ε ε -global DP constraints. New method for global optimization of Gaussian processes reduces computational time.
problem Nonconvex optimization problems with Gaussian processes trained on few data points.
method Reduced-space formulation with branch-and-bound solver and McCormick relaxations.
result Significantly reduced computational time compared to state-of-the-art methods.
Improved regret bounds for DP-KLUCB and DP-IMED in Bernoulli bandits.
problem Minimizing regret in stochastic bandits under ε-global Differential Privacy.
method Developed DP versions of KLUCB and IMED, proving tighter lower bounds and matching upper bounds.
result DP-KLUCB and DP-IMED achieve asymptotically optimal regret under ε-global DP.
Global existence of Yamabe flow on non-compact manifolds with unbounded initial curvature.
problem Global existence of Yamabe flow on non-compact manifolds with unbounded initial curvature.
method Assumption of conformally equivalent initial metric to a complete background metric with bounded scalar curvature and positive Yamabe invariant.
result Global existence of Yamabe flow without requiring initial curvature bounds.
The study sets limits on heat equation solutions' Hessians on curved spaces.
problem Bounding Hessians of positive solutions to heat equations on Kähler manifolds.
method Global and local upper bounds for Hessian matrices under curvature constraints.
result Improved bounds on Hessians for Riemannian manifolds with lower sectional curvature.
A new UCB algorithm tackles global optimization in stochastic process bandits.
problem Global optimization in stochastic process bandits.
method UCB algorithm based on generic chaining for continuous domains.
result Theoretical bounds and optimality of the algorithm for Gaussian processes.
Meta algorithm solves multivariate optimization using univariate optimizers.
problem Multivariate global optimization problems.
method Meta algorithm combining univariate global optimizers.
result Meta algorithm provides robust regret guarantees.
Let G be a group acting on the plane by orientation-preserving homeomorphisms. We show that if for some k>0 there is a ball of radius r > k/\sqrt{3} such that each point x in the ball satisfies |gx -hx| < k for all g, h in G, and the action of G satisfies a nonwandering hypothesis, then the action has a global fixed po…
Study shows how discrete graph curvature relates to manifold curvature.
problem Relating discrete graph curvature to intrinsic manifold curvature.
method Continuum limits of Ollivier's Ricci curvature on data clouds.
result Random geometric graphs inherit global curvature properties of manifolds.
Lie group integrators improve global error estimates.
problem Global error estimates for Lie group integrators.
method Relate local error to global error, derive from bounds.
result Lie-Butcher theory proves global error estimates for Lie group integrators.
GBML with deep nets converges globally and generalizes well.
problem Theoretical guarantees for few-shot learning with deep nets.
method Proving global convergence and generalization bounds for GBML with over-parameterized DNNs.
result GBML with over-parameterized DNNs converges globally to the optimum at a linear rate and achieves good generalization.
In the first part, we derive a sharp gradient estimate for the log of Dirichlet heat kernel and Poisson heat kernel on domains, and a sharpened local Li-Yau gradient estimate that matches the global one. In the second part, without explicit curvature assumptions, we prove a global upper bound for the fundamental soluti…
Optimized AIS scheme reduces bias and MSE for general proposals.
problem Performing Monte Carlo integration with general proposals.
method Global optimization of χ²-divergence using stochastic gradient Langevin dynamics.
result Explicit theoretical guarantees for uniform-in-time MSE reduction.
Global calculus for manifolds with boundary, solving evolution problems.
problem Global solvability of evolution problems on manifolds with boundary.
method Established global functional calculus and Gårding inequality for pseudo-differential operators without local coordinates.
result Global solvability for a class of evolution problems.
Derives a new global gradient estimate for graph functions.
problem Estimating the gradient of functions on graphs.
method Derives a new global gradient estimate for positive functions on graphs.
result The new gradient estimate is independent of previous estimates and can be applied to heat kernel bounds.
Global weak solutions found for Landau-Lifshitz equations into compact Lie algebras.
problem Existence of global weak solutions to Landau-Lifshitz equations into compact Lie algebras.
method Followed Arnold's ideas and used test functions and approximate equations.
result Existence of global weak solutions for the Cauchy problems of Landau-Lifshitz equations.
Global optimization algorithm finds sparse mixed membership matrix factorization's global optimum.
problem Sparse mixed membership matrix factorization problems with local optima.
method Derives a global optimization algorithm for sparse mixed membership matrix factorization.
result Guaranteed ε ε ε -global optimum across random initializations and multiple modes. The paper provides gradient estimates for solutions on manifolds with integral Ricci bounds.
problem Global regularity estimates for solutions of Δ u = f Δu = f Δ u = f on Riemannian manifolds. method Proves L p L^p L p -gradient estimates under integral Ricci bounds and constructs a counterexample. result Optimal constant lower bounds on Ricci curvature are shown in the pointwise sense.
Efficient algorithm for global optimization of multivariate Lipschitz functions.
problem Global optimization of multivariate Lipschitz continuous functions.
method Proposes an efficient minimax optimal algorithm using a predetermined query creation rule.
result Achieves an average regret bound of O ( L n T − 1 n ) O(L\sqrt{n}T^{-\frac{1}{n}}) O ( L n T − n 1 ) , minimax optimal. New bounds for SMC show its advantage over MCMC in multimodal distributions.
problem Estimating expectations under multimodal distributions with slow global mixing.
method Proves finite sample complexities for SMC with local mixing times, addressing bias through sequential resampling.
result SMC provides fully polynomial time approximation for multimodal problems.
Study curve flows with a global forcing term, proving distance comparison and convexity.
problem Analyzing the behavior of curves under curve shortening flow with a global forcing term.
method Distance comparison principle, finite time exclusion of singularities, convexity and convergence analysis.
result Convexity and smooth exponential convergence to a circle for closed curves.
New conditions found for essential tori in 3-manifolds to intersect Heegaard splittings.
problem Understanding intersections of essential tori with Heegaard splittings in 3-manifolds.
method Analyzing essential tori in 3-manifolds and their intersections with strongly irreducible Heegaard splittings.
result Conditions for a global bound on the number of curves in the intersection of essential tori and Heegaard splittings.
Study finds conditions for global minimizers on curved manifolds with fast diffusion and nonlocal interactions.
problem Existence of global minimizers for a free energy functional on negatively curved manifolds.
method Investigation of Carlson-Levin type inequalities for Cartan-Hadamard manifolds.
result Establishes necessary and sufficient conditions for the existence of global energy minimizers.
New method finds global optima in variational inference.
problem Uncertainty in finding global optima in variational inference.
method Deterministic optimization algorithm for variational inference.
result Always converges to globally optimal variational lower bound.
Global existence and convergence of pluriclosed flow on Oeljeklaus-Toma manifolds.
problem Global existence and convergence of pluriclosed flow on specific complex manifolds.
method Established global existence with arbitrary initial data and Gromov-Hausdorff convergence of blowdown limits.
result Gromov-Hausdorff convergence of blowdown limits to a torus under conjectural bounds.
New diffusions help globally optimize non-convex functions.
problem Optimizing non-convex functions globally.
method Euler discretization of Langevin diffusion.
result Different diffusions optimize different convex and non-convex functions.
We characterize those spacetimes which admit a isometric (or conformal) embedding in some Lorentz-Minkowski space L^N. In particular, any globally hyperbolic spacetime can be isometrically embedded in L^N. This is proven by a result of its own interest: the construction of a smooth time function whose gradient is bound…
We introduce a holomorphic sheaf E on a Sasaki manifold and study two new notions of stability for E along the Sasaki-Ricci flow related to the `jumping up' of the number of global holomorphic sections of E at infinity. First, we show that if the Mabuchi K-energy is bounded below, the transverse Riemann tensor is bound…
Study well-posedness of SPDE on Riemannian manifolds with rough initial conditions.
problem Well-posedness of parabolic Anderson model on Riemannian manifolds with rough initial conditions.
method Construct intrinsic Gaussian noises, explore global geometry, use Feynman-Kac formula.
result Show well-posedness with non-positive curvature and conditions on α α α .