arXiv research

Define the 1-handle stabilization distance between two surfaces properly embedded in a fixed 4-dimensional manifold to be the minimal number of 1-handle stabilizations necessary for the surfaces to become ambiently isotopic. For every nonnegative integer $m$ we find a pair of 2-knots in the 4-sphere whose stabilization…

Single stabilization is not always enough to make exotic surfaces isotopic.

The paper defines new knot genera and finds bounds for stabilization distances.

Kim-Milman flow map stable under regular target measures

Proves stability of cone-volume measure with nearly constant density.

Suppose $K$ is a knot in $S^3$ with bridge number $n$ and bridge distance greater than $2n$. We show that there are at most ${2n\choose n}$ distinct minimal genus Heegaard splittings of $S^3\setminusη(K)$. These splittings can be divided into two families. Two splittings from the same family become equivalent after at …

Adapts Stein's method for geometric inequalities, addressing boundary terms.

The study bounds distances in simplicial complexes and defines new invariants for 3-manifolds and handlebody-knots.

Paper proves stability of positive mass theorem for specific types of manifolds.

We consider the set of connected surfaces in the 4-ball with boundary a fixed knot in the 3-sphere. We define the stabilization distance between two surfaces as the minimal $g$ such that we can get from one to the other using stabilizations and destabilizations through surfaces of genus at most $g$. Similarly, we consi…

In this note, we show that the solution to the Dirichlet problem for the minimal surface system in any codimension is unique in the space of distance-decreasing maps. This follows as a corollary of the following stability theorem: if a minimal submanifold $Σ$ is the graph of a (strictly) distance-decreasing map, then $…

Two algorithms estimate Wasserstein distance matrices from few entries for manifold learning.

The study bounds the stability of Gaussian mixtures under small perturbations.

Compactness theorem for timed-metric spaces established.

We show that the number of stabilizations needed to interchange the handlebodies of a Heegaard splitting of a closed 3-manifold by an isotopy is bounded below by the smaller of twice its genus or half its Hempel distance. This is a combinatorial version of a proof by Hass, Thompson and Thurston of a similar theorem, bu…

A distance-squared function is one of the most significant functions in the application of singularity theory to differential geometry. Moreover, distance-squared mappings are naturally extended mappings of distance-squared functions, wherein each component is a distance-squared function. In this paper, compositions of…

We show that if a Heegaard splitting is the result of stabilizing a high distance Heegaard splitting exactly once then its mapping class group is finitely generated.

Study on surfaces of genus g≥1 in 3D contact sub-Riemannian manifolds, proving finiteness or infiniteness of induced distance.

Researchers modify $d_p$ distance to handle long, thin splines.

We analyze critical points of the Sliced Wasserstein Distance for optimization stability.

Stability of positive mass theorem for hyperbolic manifolds studied.

Study shows how non-uniform scaling affects persistence diagrams.

The paper examines stability of the Sobolev inequality in metric spaces with curvature dimension conditions.

We study the stability of the Positive Mass Theorem using the Intrinsic Flat Distance. In particular we consider the class of complete asymptotically flat rotationally symmetric Riemannian manifolds with nonnegative scalar curvature and no interior closed minimal surfaces whose boundaries are either outermost minimal h…

The paper improves decision tree stability for health care applications.

The paper converts metric bounds to distance function Hölder bounds and proves compactness theorems.

Sharp stability result for maps near infinitely concentrated minimisers.

New stability theory for Sinkhorn semigroups with explicit decay rates.

Research shows surfaces close to planes in Hausdorff distance.

The paper proves stability of manifolds with boundary under volume and distance constraints.

Persistence diagrams (PDs) play a key role in topological data analysis (TDA), in which they are routinely used to describe topological properties of complicated shapes. PDs enjoy strong stability properties and have proven their utility in various learning contexts. They do not, however, live in a space naturally endo…

This paper shows how to estimate distances in latent space of random graphs using entropic OT.

Study on reducing dimensionality in high-dimensional regression with kernel methods and stability analysis.

Kevin Hartshorn showed that if a three-dimensional manifold $M$ admits a Heegaard surface $Σ$ with Hempel distance $d$ then every incompressible surface in $M$ has genus at least $\frac{d}{2}$. Scharlemann-Tomova generalized this, proving that in such a manifold, every other Heegaard surface for $M$ of genus $g' < \fra…

We study the boundary rigidity problem for compact Riemannian manifolds with boundary $(M,g)$: is the Riemannian metric $g$ uniquely determined, up to an action of diffeomorphism fixing the boundary, by the distance function $ρ_g(x,y)$ known for all boundary points $x$ and $y$? We prove in this paper global uniqueness …

Wasserstein distance plays increasingly important roles in machine learning, stochastic programming and image processing. Major efforts have been under way to address its high computational complexity, some leading to approximate or regularized variations such as Sinkhorn distance. However, as we will demonstrate, regu…

Stable density-based clustering via multiparameter persistence.

We consider rotationally symmetric spaces with low regularity, which we regard as integral currents spaces or manifolds with Sobolev regularity and are assumed to have nonnegative scalar curvature. Relying on the flat distance and on Sobolev norms, we establish several nonlinear stability estimates about the ``distance…

The paper proves stability for scalar-flat metrics on manifolds with boundary.

The paper proves stability of eigenvalue inequalities on surfaces.

Backpropagation-free RL method trains layers using local signals.

Quantitative stability for nearly minimizing Yamabe metrics.

Away from the central axis, we prove the stability of the Positive Mass Theorem in the $W^{1,p}$ sense for asymptotically flat axisymmetric manifolds with nonnegative scalar curvature satisfying some additional technical assumptions. We also derive estimates for the volumes of regions, the areas of axisymmetric surface…

Paper tackles GAN instability in audio and speech signals using a new similarity metric.

This paper relates parameter distance to gradient breakdown for a broad class of nonlinear compositional functions. The analysis leads to a new distance function called deep relative trust and a descent lemma for neural networks. Since the resulting learning rule seems to require little to no learning rate tuning, it m…

The paper examines how to test if two learning algorithms produce similar outcomes.

Unified treatment of stability problems in geometry and analysis.

We modify an approach of Johnson to define the distance of a bridge splitting of a knot in a 3-manifold using the dual curve complex and pants complex of the bridge surface. This distance can be used to determine a complexity, which becomes constant after a sufficient number of stabilizations and perturbations, yieldin…