arXiv research

Upper bounds for volume spectrum depend on volume, dimension, and a conformal invariant.

Improved bounds on combining hypothesis classes for binary functions.

Upper bound on singular set dimension for area-minimizing currents.

Study finds conditions for operator fields to be in strictly upper triangular form in small dimensions.

We prove an asymptotic analog of the classical Hurewicz theorem on mappings which lower dimension. This theorem allows us to find sharp upper bound estimates for the asymptotic dimension of groups acting on finite dimensional metric spaces and allows us to prove a useful extension theorem for asymptotic dimension. As a…

The paper proves curvature-related dimension bounds for manifolds.

Generic homeos on complex manifolds have full metric mean dimension.

We establish in this paper an upper bound on the second eigenvalue of n-dimensional spheres in the conformal class of the round sphere. This upper bound holds in all dimensions and is asymptotically sharp as the dimension increases.

In this note we show that in metric measure spaces satisfying the reduced curvature-dimension condition CD*(K,N) we always have geodesics in the Wasserstein space of probability measures that satisfy the critical convexity inequality of CD*(K,N) also for intermediate times and in addition the measures along these geode…

Quadratic bounds found for graph dimensions.

This research sets limits on how complex multi-class learning problems can be.

Given a closed manifold M, we prove the upper bound of (n+d)/2 for the length of a product of systoles that can form a curvature-free lower bound for the total volume of M, in the spirit of M. Gromov's systolic inequalities. Here n is the dimension of M, while d is the is the cohomological dimension of its fundamental …

This paper considers affine analogues of the isoperimetric inequality in the sense of piecewise linear topology. Given a closed polygon P embedded in R^d having n edges, we give upper and lower bounds for the minimal number of triangles needed to forma triangulated embedded orientable surface in R^d having P as its geo…

On a closed weighted Riemannian manifold with nonnegative Bakry-Émery Ricci curvature, it is shown that the ratio of the $k$-th to first eigenvalues of the weighted Laplacian is dominated by $641k^2$, using an argument via the Cheeger constant. While improving the previous exponential upper bound, the order of $k$ here…

Bray's football theorem (\cite{bray2009penrose}) is a weakening of Bishop theorem in dimension 3. It gives a sharp volume upper bound for a three dimensional manifold with scalar curvature larger than $n(n-1)$ and Ricci curvature larger than $\varepsilon$. This paper extends Bray's football theorem in high dimensions, …

Upper bounds found for Seiberg-Witten moduli spaces under specific conditions.

In this work we study the quantitative relation between the recursive teaching dimension (RTD) and the VC dimension (VCD) of concept classes of finite sizes. The RTD of a concept class $\mathcal C \subseteq \{0, 1\}^n$, introduced by Zilles et al. (2011), is a combinatorial complexity measure characterized by the worst…

The study bounds dimensions and proves existence of holomorphic sections on Kähler Ricci shrinkers.

A fundamental theorem of Wolfe isometrically identifies the space of flat differential forms of dimension $m$ in $\mathbb{R}^n$ with the space of flat $m$-cochains, that is, the dual space of flat chains of dimension $m$ in $\mathbb{R}^n$. The main purpose of the present paper is to generalize Wolfe's theorem to the se…

The study finds limits on dimensions of certain scales and fields for conformal manifolds.

The paper connects GNNs to VC dimension theory to study their generalization performance.

Improved upper bound for equivariant Yamabe invariant in 3D.

Study robust learning without knowing perturbation sets, using interactions with attackers.

Computes upper bounds for instanton knot homology.

Theorems prove upper bounds for foliations on closed Alexandrov spaces.

Sharp upper bounds found for Steklov eigenvalues of a specific hypersurface.

Extends Tanaka theory to supergeometry for upper bounds on supersymmetry.

New research determines the optimal sample complexity for multiclass and list learning.

Upper bounds for Steklov eigenvalues of warped products are derived.

The paper strengthens a theorem on crossings under linear perturbations with Hausdorff measure estimates.

Proposes CLUB for reliable MI minimization in high dimensions.

This is a survey on upper and lower bounds for finite group actions on bounded surfaces, 3-dimensional handlebodies and closed handles, handlebodies in arbitrary dimensions and finite graphs (the common feature of these objects is that all have free fundamental group).

We determine all Chern numbers of smooth complex projective varieties of dimension at least four which are determined up to finite ambiguity by the underlying smooth manifold. We also give an upper bound on the dimension of the space of linear combinations of Chern numbers with that property and prove its optimality in…

New algorithm for learning functions with bounds on error and sample complexity.

New functionals defined for free boundary minimal submanifolds in higher dimensions.

Bounds on conformal dimension for certain Coxeter group boundaries.

We give upper bounds, linear in rank, to the topological dimensions of the Gromov boundaries of the intersection graph, the free factor graph and the cyclic splitting graph of a finitely generated free group.

uHMC achieves fast mixing in high dimensions with gradient evaluations.

Lower bounds for geodesic ball volume in 3D with Ricci curvature constraints.

We study isometries in the contact sub-pseudo-Riemannian geometry. In particular we give an upper bound on the dimension of the isometry group of a general sub-pseudo-Riemannian manifold and prove that the maximal dimension is attained for the left invariant structures on the Heisenberg group.

This work proves lower bounds on a greedy teaching set construction algorithm.

We introduce higher-order Poincar'e constants for compact weighted manifolds and estimate them from above in terms of subsets. These estimates imply upper bounds for eigenvalues of the weighted Laplacian and the first nontrivial eigenvalue of the $p$-Laplacian. In the case of the closed eigenvalue problem and the Neuma…

Improved uniform convergence bound with fat-shattering dimension reduces sample complexity gap.

Relates geodesic integrals to Killing tensors, exploring their dimensions.

We prove that the group of isometries of a metric measure space that satisfies the Riemannian curvature condition, $RCD^*(K,N),$ is in fact a Lie group. We obtain an optimal upper bound on its dimension and classify the spaces where this maximal dimension is achieved.

Upper bound found for divergence-free Killing 2-tensors on manifolds.

Optimal sampling bounds for various classification losses under different regularization terms.

New bounds on SGD's final iterate convergence rate in constant dimension.