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.
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.
Paper introduces a differentiable regularizer for condition number to improve neural network stability.
problem Maintaining numerical stability in neural networks to ensure reliable and performant models.
method Introduces a novel differentiable regularizer for the condition number of weight matrices.
result Derives a differentiable formula for the gradient of the regularizer, promoting matrices with low condition numbers.
New bounds and examples for sphere unknotting numbers.
problem Comparing unknotting numbers for 2-spheres in 4-space.
method Algebraic and geometric techniques.
result Stabilization number is bounded above by one more than Casson-Whitney number.
New findings show that some knotted surfaces remain distinct even after many stabilizations.
problem Understanding the behavior of knotted surfaces after internal stabilizations.
method Analyzing the properties of smoothly knotted surfaces and their behavior under internal stabilizations.
result There is no upper bound on the number of internal stabilizations required for some knotted surfaces to remain distinct.
New criterion selects optimal number of clusters based on stability.
problem Challenges in selecting optimal number of clusters in non-parametric clustering.
method Proposes a stability-based validation criterion combining between-cluster and within-cluster stability.
result Empirically demonstrates effectiveness in selecting optimal number of clusters.
Paper analyzes stability of discrete-time hypercomplex-valued Hopfield neural networks.
problem Stability of discrete-time hypercomplex-valued Hopfield neural networks.
method Introduces real-part associative hypercomplex number systems and B-projection functions to ensure stability. result Stability analysis of several discrete-time hypercomplex-valued Hopfield-type neural networks confirmed.
This work analyzes the stability of graph filters under large perturbations.
problem Stability of graph filters under large edge rewires.
method Proves a bound on stability using frequency response and community structure.
result Graph filter stability depends on perturbation to community structure.
In this paper, we study the Lagrangian F-stability of closed Lagrangian self-shrinkers immersed in complex Euclidean space. We show that any closed Lagrangian self-shrinker with first Betti number greater than one is Lagrangian F-unstable. In particular, any two-dimensional embedded closed Lagrangian self-shrinker is L…
This paper enhances stability selection by evaluating overall results robustness and identifying optimal regularization values.
problem Improving the robustness and reliability of high-dimensional variable selection.
method Developed a stability estimator to evaluate stability of stability selection results, calibrating key parameters.
result Identified optimal regularization value and improved stability of variable selection.
Homological stability proved for handlebody mapping class groups.
problem Homological stability for handlebody mapping class groups.
method Categorical framework developed by Randal-Williams and Wahl, allowing for any number of marked discs and boundary points.
result Homology of handlebody groups stabilizes with respect to genus and number of marked discs for all finite degree coefficient systems.
Analyzes how many people can receive stable income in a pooled annuity fund.
problem Quantifying the trade-off between income stability and the number of members in a pooled annuity fund.
method Investment returns held constant, systematic longevity risk omitted. Derived an analytical expression for income stability.
result The number of fund members who receive stable income is independent of the mortality model.
Upper bounds on revised first Betti number and torus stability for RCD spaces.
problem Bounding the revised first Betti number and stability of RCD spaces.
method Proving an upper bound on the rank of the abelianised revised fundamental group and establishing torus stability.
result Spaces with saturated upper bound on revised first Betti number are mGH-close to flat tori.
The paper analyzes the stability of parameter recovery in Linear Structural Equation Models.
problem Understanding the stability of parameter recovery in Linear Structural Equation Models (LSEM).
method Condition number analysis for bow-free LSEM and random input parameter analysis.
result For bow-free LSEM, parameter recovery is stable under certain conditions and with high probability.
Study examines how cluster number affects short-text clustering, introducing a stability metric.
problem Challenges in finding meaningful clusters in short-text data.
method Introduces a stability metric to determine cluster robustness and visualizes cluster subdivisions.
result Choosing a cluster number involves balancing informativeness and complexity, not seeking a single 'optimal' solution.
Compactifies stability space for A2 category, introducing q-deformed rational numbers.
problem Stability conditions in triangulated categories and their compactifications.
method Embedding into an infinite-dimensional projective space, using B3 braid group action. result Two orbits in the boundary correspond to q-deformed rational numbers. Let T be a separating incompressible torus in a 3-manifold M. Assuming that a genus g Heegaard splitting V∪SW can be positioned nicely with respect to T (e.g. V∪SW is strongly irreducible), we obtain an upper bound on the number of stabilizations required for V∪SW to become isotopic to a…
In this paper we present a new proof of the homological stability of the moduli space of closed surfaces in a simply connected background space K, which we denote by Sg(K). The homology stability of surfaces in K with an arbitrary number of boundary components, Sg,n(K) was studied by the authors in \cite{…
Proposes a method to predict cluster number and cluster representatives using cluster stability analysis.
problem Determining the number of clusters in a dataset.
method Analyzes cluster stability using Monte-Carlo simulation to predict cluster number and find cluster representatives.
result Significant improvement in predicting cluster numbers and cluster composition in large datasets.
We develop efficient algorithms to estimate the stability of Ordinary Least Squares regression results.
problem Measuring the stability of regression conclusions in low dimensions.
method Efficient algorithms for estimating the minimum number of samples that need to be removed to change a regression conclusion.
result We can estimate stability up to a factor of 3 better than the greedy heuristic and certify stability even for dropping a majority of samples.
New stabilizing number defined for knots, linking bounds in 4D.
problem Defining a new measure for knot boundaries in 4D.
method Defining stabilizing number sn(K), bounding it by signatures, Casson-Gordon invariants, and 4-genus. result Found examples where stabilizing number is less than 4-genus.
In variable or graph selection problems, finding a right-sized model or controlling the number of false positives is notoriously difficult. Recently, a meta-algorithm called Stability Selection was proposed that can provide reliable finite-sample control of the number of false positives. Its benefits were demonstrated …
Polynomial invariant of quandles counts random link colorings.
problem Counting Q-colorings of random braids in quandles. method Average number of Q-colorings for large n. result The average number of Q-colorings coincides with a polynomial PQ. Nearly Kähler 6-manifolds are unstable under Ricci flow.
problem Stability of nearly Kähler 6-manifolds under Ricci flow.
method Analysis of Betti numbers and entropy of Perelman.
result Strict nearly Kähler 6-manifolds are dynamically unstable.
Study on cohomology of level 4 braid group, proving representation stability.
problem Investigating cohomology of level 4 braid group.
method Exact formula for first Betti number, representation stability, representation theory development.
result Representation stability in level 4 braid group.
The paper calculates asymptotic Betti numbers and homology multiplicities for graph configuration spaces.
problem Understanding the homology of ordered configuration spaces of graphs.
method Explicit formulas for asymptotic Betti numbers and homology multiplicities in characteristic zero.
result Explicit formulas for asymptotic multiplicities in homology of irreducible representations of the symmetric group.
UCB algorithm provides stable sample means for sequential data.
problem Challenges in inferential tasks with sequential data.
method Stability property of UCB algorithm for multiarmed bandit problems.
result UCB algorithm ensures asymptotically normal sample means.
New invariant for classifying 4-manifolds up to cobordism.
problem Classifying closed, oriented topological 4-manifolds up to s-cobordism. method Introducing a stable range invariant after stabilization by a fixed number of S2imesS2. result A new invariant for classifying 4-manifolds up to s-cobordism. Exotic submanifolds in 4-manifolds remain exotic after stabilizations.
problem Constructing exotic codimension-1 submanifolds in 4-manifolds.
method Constructing pairs of exotic codimension-1 submanifolds with diffeomorphic complements and showing they remain exotic after stabilizations.
result Exotic submanifolds remain exotic after any number of stabilizations.
This is full length article (draft version) where problem number of topics in Topic Modeling is discussed. We proposed idea that Renyi and Tsallis entropy can be used for identification of optimal number in large textual collections. We also report results of numerical experiments of Semantic stability for 4 topic mode…
Study on stability of Sasaki structures under deformations.
problem Stability of Sasaki structures under transverse holomorphic deformations.
method Analysis of transverse Kähler holonomy groups and stability properties.
result Stability of ${\oldmathcal S}$ under certain conditions on Sasaki manifolds.
Graph braid groups' complexity stabilizes for most graphs.
problem Stabilization of topological complexity in graph braid groups.
method Geometric lower bounds on configuration spaces.
result Topological complexity stabilizes for most graphs.
The paper examines curvature and stability in quasi-geostrophic motions using spherical harmonics.
problem Analyzing the curvature and stability of quasi-geostrophic motions.
method Utilizing spherical harmonics and structure constants, the curvature of the L2 metric on the central extension is computed. result A lower bound for weather prediction error in a simplified model is suggested.
A popular method for selecting the number of clusters is based on stability arguments: one chooses the number of clusters such that the corresponding clustering results are "most stable". In recent years, a series of papers has analyzed the behavior of this method from a theoretical point of view. However, the results …
In this paper, we introduce a new concept of stability for cross-validation, called the (β,ϖ)-stability, and use it as a new perspective to build the general theory for cross-validation. The (β,ϖ)-stability mathematically connects the generalization ability and the stability of…
Homological stability for unordered configuration spaces of connected manifolds was discovered by Th. Church and extended by O. Randal-Williams and B. Knudsen: Hi(Ck(M);Q) is constant for k≥f(i). We characterize the manifolds satisfying strong stability: H∗(Ck(M);Q) is constant for $k\gg…
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…
This paper extends homological stability results for configuration spaces of manifolds.
problem Homological stability of configuration spaces of manifolds.
method Analyzing the cohomology of configuration spaces of manifolds, focusing on stability in odd and even degrees.
result The stable range for homology groups of configuration spaces depends on the dimension of the manifold and the number of configuration points.
Novel approach ensures stability of compact schemes for variable PDEs.
problem Ensuring stability of compact schemes for variable coefficient PDEs.
method Difference equation approach to derive stability conditions.
result Derives sufficient condition for unconditional stability.
We present a new proof of Reidemeister and Singer's Theorem that any two Heegaard splittings of the same 3-manifold have a common stabilization. The proof leads to an upper bound on the minimal genus of a common stabilization in terms of the number of negative slope inflection points and type-two cusps in a Rubinstein-…
The virtual Betti number conjecture states that any hyperbolic three-manifold has a finite cover with positive first Betti number. We show that this would follow if it were known that the derived series of the fundamental group G of a hyperbolic three-manifold satisfies a certain stability property. The stability pro…
New criterion for cylinder stability in curved spaces.
problem Stability of cylinders in curved spaces.
method Extending Plateau-Rayleigh criterion to curved spaces and proving existence of instability threshold.
result Existence of a positive number L0 for cylinder instability in E(κ,τ) spaces. We address a special case of the Stabilization Problem for Heegaard splittings, establishing an upper bound on the number of stabilizations required to make a Heegaard splitting of a Haken 3-manifold isotopic to an amalgamation along an essential surface. As a consequence we show that for any positive integer n there…
New method improves feature selection by integrating stability paths.
problem Improving feature selection with tighter false positive control.
method Integrating stability paths to strengthen theoretical bounds on E(FP).
result Significantly more true positives with same E(FP) control.
Finite p-group actions on manifolds have limited stabilizer subgroups.
problem Understanding the structure of stabilizer subgroups in group actions on manifolds.
method Bounding the index of a subgroup H in a finite p-group G acting on a compact manifold M, ensuring a controlled number of stabilizers.
result The existence of a subgroup H with a controlled index and limited stabilizers.
This paper analyzes the stability and generalization of triplet learning algorithms.
problem Lack of theoretical understanding of triplet learning's generalization performance.
method Stability analysis and high-probability generalization bounds for triplet learning algorithms.
result Established general high-probability generalization bound for triplet learning algorithms.
Single stabilization is not always enough to make exotic surfaces isotopic.
problem Determining if a single stabilization is sufficient to make exotic surfaces isotopic.
method Study of stabilization distance with satellite operations using Floer theoretic techniques.
result Found examples of exotic disks in the four-ball with arbitrarily large stabilization distance.
Study shows minimizing the norm of the ERM solution stabilizes kernel ridge-less regression.
problem Stability of kernel ridge-less regression.
method Minimizing the norm of the ERM solution to minimize CV stability.
result Interpolating solution with minimum norm minimizes CV stability.