arXiv research

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…

New stability theorem for hyperbolic metrics without volume bounds.

Adapts a short argument to derive a stability theorem for smooth maps.

We discuss a Moser type argument to show when a deformation of a Lie group homomorphism and of a Lie subgroup is trivial. For compact groups we obtain stability results.

We introduce an inductive argument for proving birational superrigidity and K-stability of singular Fano complete intersections of index one, using the same types of information from lower dimensions. In particular, we prove that a hypersurface in $\mathbb{P}^{n+1}$ of degree $n+1$ with only ordinary singularities of m…

The purpose of this note is to point to a gap in an argument in our paper "Stabilization for the automorphisms of free groups with boundaries", and explain how to fill it.

This paper analyzes stability and generalization of Markov chain stochastic gradient methods.

The paper proves stability in compact finite dimensional Alexandrov spaces using equivariant Gromov--Hausdorff convergence.

In this paper, we shall show that a polarized algebraic manifold is K-stable if the polarization class admits a Kaehler metric of constant scalar curvature. This generalizes the results of Chen-Tian, Donaldson and Stoppa. (Parts of the arguments are based on a forthcoming paper "A stronger concept of K-stability." )

Stability of non-abelian X-ray transform proven in higher dimensions.

We formulate a notion of K-stability for Kähler manifolds, and prove one direction of the Yau-Tian-Donaldson conjecture in this setting. More precisely, we prove that the Mabuchi functional being bounded below (resp. coercive) implies K-semistability (resp. uniformly K-stable). In particular this shows that the existen…

The paper analyzes generalization bounds for NC-SC/NC-C stochastic minimax optimization.

We extend an argument of Stoppa to make some prgress towards a proof that Kähler-Einstein manifolds are "b-stable". We point out some algebro-geometric questions, involving finite generation, that arise.

New proof of homological stability for surface mapping classes.

We give a new proof of the fact that the condition of a Fano manifold admitting a Kähler-Einstein metric is Zariski-open (provided that the automorphism group is discrete). This proof does not use the characterisation involving stability. The arguments involve estimates of Futaki invariants obtained from a differential…

For each integer k > 1, Johnson gave a 3-manifold with Heegaard splittings of genera 2k and 2k-1 such that any common stabilization of these two surfaces has genus at least 3k-1. We modify his argument to produce a 3-manifold with two Heegaard splitings of genus 2k such that any common stabilization of them has genus a…

The purpose of this note is to show that classical cobordism arguments, which go back to the pioneering works of Mandelbaum and Moishezon, provide quick and unified proofs of any knot surgered compact simply-connected 4-manifold X_K becoming diffeomorphic to X after a single stabilization by connected summing with S^2 …

Study extends geodesic ray transform results to orientable surfaces.

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 …

Extends Palais' result on diffeomorphisms to homeomorphisms and bi-Lipschitz mappings.

The paper examines stability of Sobolev inequalities on manifolds with Ricci curvature bounds.

Study stabilizes translating solitons in hyperbolic space for MCF.

Sharp stability of Alexandrov's theorem for $C^1$ domains in the small-excess regime

Stability result for nearly isometric subspaces and Finsler surfaces.

We consider the heat flow of corotational harmonic maps from $\mathbb R^3$ to the three-sphere and prove the nonlinear asymptotic stability of a particular self-similar shrinker that is not known in closed form. Our method provides a novel, systematic, robust, and constructive approach to the stability analysis of self…

Study proves stability of big bang singularity in complex system.

We prove nonlinear stability for a large class of solutions to the Einstein equations with a positive cosmological constant and compact spatial topology in arbitrary dimensions, where the spatial metric is Einstein with either positive or negative Einstein constant. The proof uses the CMC Einstein flow and stability fo…

We study the spaces of polynomials stratified into the sets of polynomial with fixed number of roots inside certain semialgebraic region $Ω$, on its border, and at the complement to its closure. Presented approach is a generalisation, unification and development of several classical approaches to stability problems in …

We give an alternative proof of Madsen-Weiss' generalized Mumford conjecture. Our proof is based on ideas similar to Madsen-Weiss' original proof, but it is more geometrical and less homotopy theoretical in nature. At the heart of the argument is a geometric version of Harer stability, which we formulate as a theorem a…

Proves stability of Minkowski space for specific initial data.

Barrier methods classify minimal submanifolds in hyperkaehler spaces.

We consider the problem of learning stabilizable systems governed by nonlinear state equation $h_{t+1}=φ(h_t,u_t;θ)+w_t$. Here $θ$ is the unknown system dynamics, $h_t$ is the state, $u_t$ is the input and $w_t$ is the additive noise vector. We study gradient based algorithms to learn the system dynamics $θ$ from samp…

Homological stability for sequences of groups is often proved by studying the spectral sequence associated to the action of a typical group in the sequence on a highly-connected simplicial complex whose stabilizers are related to previous groups in the sequence. In the case of mapping class groups of manifolds, suitabl…

New analysis improves understanding of bilevel optimization stability and generalization.

Study shows formation of Kerr black holes with complete apparent horizons and proves Penrose inequalities.

We establish existence of the eta-invariant as well as of the Atiyah-Patodi-Singer and the Cheeger-Gromov rho-invariants for a class of Dirac operators on an incomplete edge space. Our analysis applies in particular to the signature, the Gauss-Bonnet and the spin Dirac operator. We derive an analogue of the Atiyah-Pato…

The paper derives uniform stability-based coverage bounds for conformal prediction methods.

Study on deformations of Lie groupoid morphisms and their properties.

Study on self-consuming generative models with diverse human curation, focusing on convergence and stability.

Proves Khovanov homology functoriality and positivity for gl2 webs.

Proves properness of K-moduli spaces for Fano varieties.

Given a holomorphic vector bundle $E$ on the twistor space $\mathrm{Tw}(M)$ of a simple hyperkähler manifold $M$, we view it as a family of bundles $\left\{E_I\right\}$ on the fibres $π^{-1}(I)$ of the twistor projection $π: \mathrm{Tw}(M) \to \mathbb{CP}^1$, and study the relationship between stability of $E$ and its …

New techniques prove bounded acyclicity results for semi-simplicial sets.

We prove that any finite energy geodesic ray with a finite Mabuchi slope is maximal in the sense of Berman-Boucksom-Jonsson, and reduce the proof of the uniform Yau-Tian-Donaldson conjecture for constant scalar curvature Kähler metrics to Boucksom-Jonsson's regularization conjecture about the convergence of non-Archime…

Improved Lasso estimator speeds up variable selection.

In this note, we observe that if $B$ is a ball in a Euclidean space with dimension $n$, $n\geq3$, then a stable CMC hypersurface $Σ$ with free boundary in $B$ satisfies \[ nA\leq L\leq nA\left( \frac{1+\sqrt{1+4(n+1)H^2}}{2} \right)\,, \] where $L$, $A$ and $H$ denote the length of $\partial Σ$, the area of $Σ$ and the…

Given a family of groups admitting a braided monoidal structure (satisfying mild assumptions) we construct a family of spaces on which the groups act and whose connectivity yields, via a classical argument of Quillen, homological stability for the family of groups. We show that stability also holds with both polynomial…

Sharp stability of Möbius group among sphere-valued maps proved in arbitrary dimensions.