New stability notion ensures generalization in adaptive settings for linear queries.
problem Ensuring generalization in adaptive settings for linear queries.
method Introducing a new stability notion based on observing outputs and quantifying their effect on posterior data sample elements.
result The new stability notion is both necessary and sufficient for generalization in adaptive settings.
Study compares different stability notions in Kähler geometry.
problem Comparing various stability notions in Kähler geometry.
method Introduce and study geodesic stability on rays with specific singularity types.
result Equivalence of some stability notions under certain conditions.
Introduces stability conditions for polarized varieties, linking to K-stability.
problem Stability conditions for polarized varieties.
method Analogue of Bridgeland's stability for polarized varieties, Z-stability, Z-critical Kähler metrics.
result Polarized varieties with certain stability conditions admit Z-critical Kähler metrics.
Geometric invariant theory introduces stability conditions mirroring abelian category theory.
problem Stability conditions in geometric invariant theory.
method Axiomatic notion of central charge and stability condition on schemes and stacks.
result Introduction of stability conditions for polarized schemes and smooth projective varieties.
We explore in some detail the notion of algorithmic stability as a viable framework for analyzing the generalization error of learning algorithms. We introduce the new notion of training stability of a learning algorithm and show that, in a general setting, it is sufficient for good bounds on generalization error. In t…
Study on generalization for data-dependent hypothesis sets.
problem Understanding generalization in hypothesis sets dependent on data.
method Learning guarantee based on transductive Rademacher complexity and hypothesis set stability.
result Generalization bound for data-dependent hypothesis sets.
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. We introduce the notion of T-stability for torsion-free Higgs sheaves as a natural generalization of the notion of T-stability for torsion-free coherent sheaves over compact complex manifolds. We prove similar properties to the classical ones for Higgs sheaves. In particular, we show that only saturated flags of to…
We investigate a semi-continuity property for stability conditions for sheaves that is important for the problem of variation of the moduli spaces as the stability condition changes. We place this in the context of a notion of stability previously considered by the authors, called multi-Gieseker-stability, that general…
We review the notion of Gieseker stability for torsion-free Higgs sheaves. This notion is a natural generalization of the classical notion of Gieseker stability for torsion-free coherent sheaves. We prove some basic properties that are similar to the classical ones for torsion-free coherent sheaves over projective alge…
Paper derives exponential bounds for learning risk using stable hypothesis.
problem Tackles the gap between optimal and suboptimal generalization bounds.
method Uses recent advances in concentration inequalities and a weaker stability notion.
result Derives an exponential tail bound for the concentration of the estimated risk.
Study on stability of constant mean curvature hypersurfaces in Riemannian manifolds.
problem Stability of constant higher mean curvature hypersurfaces in Riemannian manifolds.
method Introduced a new notion of stability and used two stability operators to relate it to the first eigenvalues. Applied to Space Forms and proved non-stability for certain hypersurfaces.
result Embedded rotational spheres with constant k-mean curvature in HnxR or SnxR are not stable.
New stability criteria for vector bundles linked to Hermite-Einstein geometry.
problem Stability of higher-rank vector bundles and their moduli spaces.
method Introducing m-positivity and a smooth function for coherent subbundles, linking to Hermite-Einstein geometry. result Hermite-Einstein bundles are uniformly semi-stable, and new stability conditions are established.
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…
Study on disk configurations in strips shows stability patterns.
problem Understanding stability patterns in disk configurations in strips.
method Finite presentation of rational homology groups, representation stability.
result Disk configuration space exhibits first-order representation stability.
Stability is a key aspect of data analysis. In many applications, the natural notion of stability is geometric, as illustrated for example in computer vision. Scattering transforms construct deep convolutional representations which are certified stable to input deformations. This stability to deformations can be interp…
Stability of capillary hypersurfaces with higher order mean curvature.
problem Stability of capillary hypersurfaces with constant higher order mean curvature.
method Generalization of classical stability theory for capillary hypersurfaces.
result Results on stability for capillary hypersurfaces with higher order mean curvature.
We define a new notion of "b-stability" for a polarised algebraic variety, adapted to the existence problem for Kahler-Einstein metrics on Fano manifolds.
This paper analyzes stability of decision trees and logistic regression.
problem Stability of decision trees and logistic regression is analyzed to understand their performance and sensitivity.
method Two stability notions (hypothesis and pointwise hypothesis stability) are derived for decision trees and logistic regression. The stability of decision trees depends on the number of leaves, while for logistic regression, it depends on the smallest eigenvalue of the Hessian matrix. Upper bounds on generalization error are constructed.
result Logistic regression is not a stable learning algorithm.
In this paper, we introduce notions of nonlinear stabilities for a relative ample line bundle over a holomorphic fibration and define the notion of a geodesic-Einstein metric on this line bundle, which generalize the classical stabilities and Hermitian-Einstein metrics of holomorphic vector bundles. We introduce a Dona…
Solves modified conjecture for Fano manifolds using Ding stability.
problem Finding Kähler-Einstein metrics on Fano manifolds.
method Interprets Ding semistability and solves modified conjecture.
result Solves modified conjecture for coupled Kähler-Einstein metrics on Fano manifolds.
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).
New stability theorem for non-hyperbolic group actions.
problem Structural stability of non-hyperbolic group actions.
method Introducing 'meandering hyperbolicity' for group actions on geodesic metric spaces.
result Meandering-hyperbolic actions are structurally stable.
Introduces relative stability conditions on triangulated categories.
problem Stability conditions in triangulated categories.
method Definition and deformation of relative stability conditions.
result Deformation of relative stability conditions via gluing stability conditions.
This work explores the trade-offs between stability and accuracy in statistical estimation.
problem Understanding the statistical cost of algorithmic stability.
method Statistical decision-theoretic perspective, focusing on worst-case and average-case stability.
result Optimal stable estimators for mean estimation and regression settings are developed, revealing trade-offs between stability and accuracy.
Study stability of surfaces in spacetimes, proving new estimates and theorems.
problem Stability of surfaces in spacetime and their applications.
method Variational techniques, Christodoulou-Yau estimate, Cohn-Vossen inequality, global theorem, capillary stability, area inequality, diameter estimate.
result Established new estimates and theorems for stable surfaces in spacetime.
Introduces stability for families of K-polystable varieties and connects it to optimal symplectic connections.
problem Forming correct moduli for fibrations and understanding their geometry.
method Introduces a new stability condition for fibrations and relates it to the existence of optimal symplectic connections.
result Proves that the existence of an optimal symplectic connection implies semistability of the fibration.
We characterize language generation with stability and breadth, proving impossibility results.
problem Characterizing and proving impossibility results for language generation with stability and breadth.
method Analysis of existing notions of breadth and stability, proving lower bounds.
result Proven impossibility of generating with higher perplexity or lower hallucination rate for stable generators.
Extends classical stability results to new geometric settings.
problem Stability of holomorphic vector bundles on complex manifolds.
method Introduces (ω,Ω)-Hermite-Einstein and (ω,Ω)-stable conditions. result Generalised Hermite-Einstein condition implies (ω,Ω)-semi-stability. The paper proves stability in compact finite dimensional Alexandrov spaces using equivariant Gromov--Hausdorff convergence.
problem Stability in compact finite dimensional Alexandrov spaces.
method Equivariant Gromov--Hausdorff convergence and almost commutative diagrams.
result Stability result in compact finite dimensional Alexandrov spaces.
The paper relaxes the stability condition to boost confidence in generalization for randomized learning algorithms.
problem The tension between uniform stability and L2-stability in generalization bounds. method Establishes in-expectation first moment generalization error bounds for L2-stable randomized learning algorithms and uses subbagging to achieve near-tight exponential bounds. result Improves generalization bounds for convex and non-convex optimization problems with SGD.
Effective Yau-Tian-Donaldson conjecture for spherical varieties.
problem Finding effective K-stability criteria for spherical varieties.
method Formulated an effective variant of the Yau-Tian-Donaldson conjecture and reviewed effective K-stability criteria for spherical varieties.
result Effective K-stability criteria can be computed given combinatorial data.
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.
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…
We introduce a strengthening of K-stability, based on filtrations of the homogeneous coordinate ring. This allows for considering certain limits of families of test-configurations, which arise naturally in several settings. We prove that if a manifold with no automorphisms admits a cscK metric, then it satisfies this s…
We show that for any two Heegaard splittings of genus p and q for the same closed 3-manifold, there is a common stabilization of genus at most 3/2 p + 2q - 1. One may compare this to recent examples of Heegaard splittings whose smallest common stabilizations have genus at least p+q or p+1/2q depending on the…
New rigidity results for scalar curvature with stabilized conditions.
problem Establishing rigidity for scalar curvature with stabilized conditions.
method Construction of foliations and development of a monotone quantity using Ricci flow and heat equation.
result Generalized classical scalar curvature rigidity results to the \(T^{
times}\)-stabilized setting.
We define a notion of stability for chiral ring of four dimensional N=1 theory by introducing test chiral rings and generalized a maximization. We conjecture that a chiral ring is the chiral ring of a superconformal field theory if and only if it is stable. We then study N=1 field theory derived from D3 branes probing …
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.
The paper studies stability of discrete planar curves using variational methods.
problem Stability of discrete planar curves under area constraints.
method Unified interpretation of discrete curvatures, determination of equilibrium curves, stability analysis.
result Equilibrium curves for the length functional under area-constraint conditions are determined and their stability is studied.
We introduce a notion of algorithmic stability of learning algorithms---that we term \emph{argument stability}---that captures stability of the hypothesis output by the learning algorithm in the normed space of functions from which hypotheses are selected. The main result of the paper bounds the generalization error of…
We introduce a notion of stability for sheaves with respect to several polarisations that generalises the usual notion of Gieseker-stability. We prove, under a boundedness assumption, which we show to hold on threefolds or for rank two sheaves on base manifolds of arbitrary dimension, that semistable sheaves have a pro…
We introduce a strong notion of quasiconvexity in finitely generated groups, which we call stability. Stability agrees with quasiconvexity in hyperbolic groups and is preserved under quasi-isometry for finitely generated groups. We show that the stable subgroups of mapping class groups are precisely the convex cocompac…
An introduction is provided to some current research trends in stability in geometric invariant theory and the problem of Kaehler metrics of constant scalar curvature. Besides classical notions such as Chow-Mumford stability, the emphasis is on several new stability conditions, such as K-stability, Donaldson's infinite…
Flexible approach for normal approximations in geometric and topological statistics.
problem Normal approximation for complex statistics not expressible as sums of score functions.
method Flexible add-one cost operator combined with strong stabilization theory.
result Established normal approximation results for geometric and topological statistics.
Parabolic structures with rational weights encode certain iterated blowups of geometrically ruled surfaces. In this paper, we show that the three notions of parabolic polystability, K-polystability and existence of constant scalar curvature Kähler metrics on the iterated blowup are equivalent, for certain polarizations…
Survey on GIT for vector bundles and Harder-Narasimhan filtration.
problem Constructing moduli spaces of holomorphic vector bundles.
method Geometric Invariant Theory (GIT) and Harder-Narasimhan filtration.
result Correspondences between GIT and vector bundle stability.
Survey various symmetry notions for toric varieties.
problem Understanding different types of symmetries in toric varieties.
method Exploring algebraic, complex, representation, combinatorial, convex, and geometric stability perspectives.
result Establishes relationships between different symmetry notions.