Local gaps in Ricci shrinkers depend only on dimension.
problem Understanding local properties of Ricci shrinkers.
method Proved local versions of Ricci curvature and entropy gap theorems.
result Local gaps depend only on dimension, not global entropy.
Local gap theorem for Ricci shrinkers ensures flatness if certain functionals are close to zero.
problem Understanding the global geometry of Ricci shrinkers from local information.
method Proving a local gap theorem using the local μ-functional. result Ricci shrinkers are flat if the local μ-functional is close to zero. Study gap phenomenon in flat manifolds with Ricci curvature.
problem Understanding curvature decay in flat manifolds.
method Construct solutions to Yamabe flow and analyze curvature decay.
result If curvature decays quickly, manifold must be flat.
In this paper we construct compact manifolds with fixed boundary geometry which admit Riemannian metrics of unit volume with arbitrarily large Steklov spectral gap. We also study the effect of localized conformal deformations that fix the boundary geometry. For instance, we prove that it is possible to make the spectra…
GNA optimally identifies the best arm with small gaps.
problem Best arm identification in fixed-budget settings.
method Generalized Neyman Allocation (GNA) for asymptotically locally minimax optimal BAI.
result GNA's worst-case bounds match the lower and upper bounds in the small-gap regime.
New method for spectral and Bergman kernels under local spectral gap condition.
problem Analyzing spectral and Bergman kernels for complex manifolds.
method Developed a new scaling method to study spectral and Bergman kernels.
result Established pointwise asymptotics of spectral and Bergman kernels.
In this paper, we prove a gap result for a locally conformally flat complete non-compact Riemannian manifold with bounded non-negative Ricci curvature and a scalar curvature average condition. We show that if it has positive Green function, then it is flat. This result is proved by setting up new global Yamabe flow. Ot…
Arithmetic spaces simplified to simplicial complexes.
problem Understanding the complexity of arithmetic locally symmetric spaces.
method Homotopy equivalence to a simplicial complex with linearly bounded simplices, using a strengthened Margulis collar lemma.
result Arithmetic locally symmetric spaces are homotopy equivalent to simplicial complexes with linearly bounded simplices.
Study calculates eigenvalues and eigenfunctions for spherical triangles and finds fundamental gap behavior.
problem Understanding eigenvalues and gaps in spherical triangles.
method Explicit computation of Dirichlet eigenvalues and eigenfunctions for spherical lunes and triangles.
result Fundamental gap of spherical triangles increases as the angle of the lune decreases.
A new framework reduces inconsistencies in chaotic surrogate modeling.
problem Consistency issues between probabilistic objectives and dynamical system dynamics.
method KAFFEE (Kalman-Aware Framework For Ergodic Emulation), a differentiable extended Kalman filter.
result KAFFEE mitigates the dynamic-probabilistic consistency gap, improving reconstruction and predictive scores.
In our previous work "Characterization of certain homorphic geodesic cycles on Hermitian locally symmetric manifolds of the noncompact type" in "Modern methods in Complex Analysis" Annals of Math. Studies 138 (1995) 85-118, we formulated a conjecture: the so called "gap phenomenon". The purpose of the article is two-fo…
Study potential computational gaps in symmetric binary perceptrons using fl-RDT.
problem Potential statistical-computational gaps in symmetric binary perceptrons.
method Parametric utilization of fully lifted random duality theory (fl-RDT).
result Observation of a computational gap SCG=αc−αa in SBP. Federated learning studies separate client data and distribution gaps.
problem Understanding performance differences in federated learning across different datasets.
method Proposed a framework to disentangle out-of-sample and participation gaps.
result Dataset synthesis strategy is crucial for realistic simulations of federated learning generalization.
New local-search methods close the gap in sparse tensor PCA.
problem Sparse tensor PCA underperforms compared to other methods.
method Proposes new local-search methods including greedy and random-threshold variants.
result Proves local-search methods close the gap to best known polynomial-time procedures.
Study shows how to count and equidistribute cusped Hitchin representations with entropy gaps.
problem Counting and equidistribution of cusped Hitchin representations.
method Renewal theorem of Kesseböhmer and Kombrink applied to count and equidistribute.
result Entropy gaps at infinity allow for counting and equidistribution results.
The paper studies eigenvalues in gaps of the essential spectrum of a Bochner-Schrödinger operator.
problem Eigenvalue distribution in gaps of the essential spectrum of the Bochner-Schrödinger operator.
method Trace asymptotics formula and Weyl type asymptotic formula for eigenvalue counting function.
result The spectrum of Hp in the gap is discrete. New rigidity theorem for sharp spectral gap in nonnegatively curved spaces.
problem Rigidity of sharp spectral gap in nonnegatively curved spaces.
method Mixing Sobolev theory and singular 1D-localization.
result Rigidity of λ=diam2π2 in compact RCD(0,N) spaces. Graph Neural Networks and Guided Local Search improve TSP solutions.
problem Finding optimal solutions to the Traveling Salesperson Problem quickly.
method Hybrid approach combining Graph Neural Networks and Guided Local Search.
result Significant reduction in optimality gap for TSP solutions.
Improves time series classification with forest proximities.
problem Time series classification accuracy and efficiency.
method PF-GAP, an extension of RF-GAP proximities to proximity forests, combined with Multi-Dimensional Scaling and Local Outlier Factors.
result Forest proximities show stronger connection between misclassified points and outliers.
New Dynkin condition for manifolds with boundary yields bi-Lipschitz equivalence and spectral properties.
problem Characterizing smooth Riemannian manifolds with boundary.
method Introducing a Dynkin-type condition and proving its equivalence to a weighted manifold.
result Bi-Lipschitz equivalence and various spectral properties of manifolds with boundary.
Develops correlation number for specific potentials and Hitchin representations.
problem Analyzing correlation numbers for potentials with entropy gaps and Hitchin representations.
method Defines a correlation number for pairs of cusped Hitchin representations and explores its connection to the Manhattan curve.
result Establishes a connection between the correlation number and the Manhattan curve, revealing rigidity properties.
Study gaps and clusters in eigenvalues of magnetic Laplacian on manifolds.
problem Understanding gaps and clusters in eigenvalues of magnetic Laplacian on manifolds.
method Analyzes high tensor powers of Hermitian line bundles with non degenerate curvature, proving Riemann-Roch numbers for eigenvalue clusters and describing spectral projectors.
result Clusters and gaps in eigenvalues are described by Riemann-Roch numbers and have pointwise kernel descriptions.
This paper improves neural tangent kernels for better generalization and local elasticity.
problem Performance gap between neural tangent kernels and real-world neural networks.
method Introduces label-aware kernels using Hoeffding decomposition.
result Models trained with proposed kernels simulate NNs better in terms of generalization and local elasticity.
New theorem improves spectral gap for sampling from mixture distributions.
problem Sampling from multimodal distributions with simulated tempering.
method Introduced a decomposition theorem for the restricted spectral gap of simulated tempering.
result Lower bound on the restricted spectral gap for mixture distributions.
We introduce a novel class of localized atomic environment representations, based upon the Coulomb matrix. By combining these functions with the Gaussian approximation potential approach, we present LC-GAP, a new system for generating atomic potentials through machine learning (ML). Tests on the QM7, QM7b and GDB9 biom…
Sharp convergence theorem for Yang-Mills flow on ALE manifolds proved.
problem Proving convergence of Yang-Mills flow on ALE gravitational instantons.
method Noncompact version of the 'parabolic gap theorem'.
result Sharp convergence theorem for Yang-Mills flow on ALE 4-manifolds.
Establishes refined singularity estimate for nonnegative n-superharmonic functions in locally conformally flat manifolds.
problem Analyzing volume growth and verifying Cohn-Vossen inequality in locally conformally flat manifolds.
method Refined singularity estimate and characterization of volume growth.
result Analytically characterizes volume growth and verifies Cohn-Vossen inequality.
New framework tightens certified robustness gaps in machine learning models.
problem Persistent gap between theoretical certified robustness and empirical accuracy.
method Leverages Lipschitz continuity and novel confidence intervals.
result Improves robust accuracy, compressing the gap between theory and practice.
Differential privacy(DP) has now become a standard in case of sensitive statistical data analysis. The two main approaches in DP is local and central. Both the approaches have a clear gap in terms of data storing,amount of data to be analyzed, analysis, speed etc. Local wins on the speed. We have tested the state of th…
This article announces the completion of the classification of rank 4 locally projective polytopes and their quotients. There are seventeen universal locally projective polytopes (nine nondegenerate). Amongst their 441 quotients are a further four (nonuniversal) regular polytopes, and 152 nonregular but section regular…
Rewiring GNNs to optimize community and feature alignment improves their performance.
problem Improving GNNs' performance by addressing over-squashing and generalization issues.
method Three rewiring strategies: ComMa, FeaSt, and ComFy, targeting community structure, node labels, and their alignment.
result Rewiring strategies enhance GNNs' performance by optimizing label-community alignment.
In this paper we study the Riesz transform on complete and connected Riemannian manifolds M with a certain spectral gap in the L2 spectrum of the Laplacian. We show that on such manifolds the Riesz transform is Lp bounded for all p∈(1,∞). This generalizes a result by Mandouvalos and Marias and extend…
KZImputer improves time series data quality with adaptive imputation for short to medium-sized gaps.
problem Missing data in time series analysis.
method Adaptive imputation method for univariate time series with tailored strategies for different gap positions.
result KZImputer achieves strong performance, especially for high missingness rates and high-sparsity regimes.
Detecting and recovering labels in binomial logistic mixtures is challenging due to an information gap.
problem Detecting and recovering labels in binomial logistic mixtures
method Propose two feasibility-aware inference procedures
result Avoid misleading component selections and improve label probability calibration
We study a class of fourth-order geometric problems modelling Willmore surfaces, conformally constrained Willmore surfaces, isoperimetrically constrained Willmore surfaces, bi-harmonic surfaces in the sense of Chen, among others. We prove several local energy estimates and derive a global gap lemma.
To integrate strategic, tactical and operational decisions, the two-stage optimization has been widely used to guide dynamic decision making. In this paper, we study the two-stage stochastic programming for complex systems with unknown response estimated by simulation. We introduce the global-local metamodel assisted t…
New classifier combines locally linear kernels for fast and accurate non-linear classification.
problem Developing a fast and accurate non-linear classifier.
method Combines locally linear classifiers using a ℓ1 Multiple Kernel Learning (MKL) problem with scalable MKL training for streaming kernels. result The resulting classifier achieves high accuracy with fast inference time.
The study uses Random Matrix Theory to identify structural changes in stock markets during shocks.
problem Understanding structural changes in stock markets during exogenous shocks.
method Random Matrix Theory and complexity gap analysis.
result The complexity gap collapses during shocks, indicating strong synchronization, and widens before shocks, signaling a rich structure.
Ricci flow controls curvature on manifolds with bounds.
problem Controlling curvature on manifolds with given bounds.
method Ricci flow with curvature bounds and entropy controls.
result Global curvature control at positive times for manifolds.
New accelerators for EM improve convergence speed in complex mixture models.
problem Improving the convergence speed of the EM algorithm for complex mixture models.
method Derive a new operator connecting global descent and local convergence, and use it to develop two acceleration strategies.
result Two new acceleration strategies (G-Accelerator and Geo-Adaptive) significantly improve EM algorithm performance.
Paper analyzes convergence of GDA for nonconvex-nonconcave minimax problems.
problem Understanding convergence of GDA for nonconvex-nonconcave minimax problems.
method Local convergence analysis of GDA with stepsize ratio Θ(κ).
result Stepsize ratio of Θ(κ) is necessary and sufficient for local convergence of GDA to a Stackelberg Equilibrium.
Study shows polynomial-width neural networks can closely approximate infinite-width networks in polynomial time.
problem Approximating dynamics of polynomial-width neural networks with infinite-width networks.
method Bounding approximation gap through a differential equation governed by mean-field dynamics, considering local Hessian.
result Polynomially many neurons are sufficient to closely approximate mean-field dynamics.
LSCI provides locally adaptive prediction sets for operator models with tighter coverage.
problem Generating robust, calibrated uncertainty quantification for operator models.
method Local Sliced Conformal Inference (LSCI) for operator models.
result LSCI yields tighter prediction sets with stronger adaptivity compared to conformal baselines.
Paper develops a new local convexity condition for non-isolated minima in non-convex optimization.
problem Lack of theory for non-isolated minima in non-convex optimization.
method Formulates a new local convexity condition and studies SGD convergence under this condition.
result Shows SGD can converge locally under the new condition.
We obtain option pricing formulas for stock price models in which the drift and volatility terms are functionals of a continuous history of the stock prices. That is, the stock dynamics follows a nonlinear stochastic functional differential equation. A model with full memory is obtained via approximation through a stoc…
Paper tackles non-stationary bandits with various examples.
problem Non-stationary stochastic bandit problem with specific cases.
method Proposes a single algorithm for multiple non-stationary bandit problems.
result Unified solution for four different bandit problems.
LASE improves local network structure visualization by targeting locally low-dimensional regions.
problem Global spectral embedding fails to capture local geometric features in sparse, transitive networks.
method Local Adjacency Spectral Embedding (LASE) using weighted spectral decomposition.
result LASE reveals locally low-dimensional structure, improving local reconstruction and visualization.
Unique submaximal symmetry found for certain parabolic geometries.
problem Determining the next realizable symmetry dimension in parabolic geometries.
method Analyzing submaximally symmetric structures of type (G,P) for specific Lie groups. result Local uniqueness of submaximally symmetric structures established.