arXiv research

Infinite width ReLU networks can approximate functions with bounded Euclidean norm.

Study improves bounds on p-covectors and proves stable systolic inequalities.

The paper examines slopes and their norms in exceptional Dehn fillings.

The study provides bounds for geodesic diameter in Euclidean space.

New algorithms optimize convex functions with high-order derivatives.

The paper examines torsions in Minkowskian product of Finsler metrics.

Uniform convergence of interpolators proven for Gaussian data.

New method turns optimization algorithms into uniformly stable learning algorithms for non-Euclidean norms.

A compact Riemannian manifold may be immersed into Euclidean space by using high frequency Laplace eigenfunctions. We study the geometry of the manifold viewed as a metric space endowed with the distance function from the ambient Euclidean space. As an application we give a new proof of a result of Burq-Lebeau and othe…

The higher order singular value decomposition (HOSVD) of tensors is a generalization of matrix SVD. The perturbation analysis of HOSVD under random noise is more delicate than its matrix counterpart. Recently, polynomial time algorithms have been proposed where statistically optimal estimates of the singular subspaces …

The paper proves the stability of a 3-ball under curvature constraints.

The paper introduces a new method for tail bounds of random vectors and matrices.

Characterizes functions representable by infinite-width ReLU networks with bounded weights.

The paper studies scalar curvature of self-shrinkers and proves curvature bounds.

Muon optimizes Transformer training with heavy-tailed data, achieving optimal sample complexity.

Study on extremizers for Sobolev inequality on curved manifolds.

Study classifies 3D self-shrinkers with constant second form norm.

Study rigidity of minimal Legendrian submanifolds in spheres via eigenvalues.

RATQ is a new quantizer for optimizing noisy gradients in machine learning.

New algorithm achieves optimal privacy and efficiency in non-Euclidean convex optimization.

Non-Euclidean BPM extends optimization theory to non-Euclidean norms.

Sharp bounds for max-sliced Wasserstein distances derived for empirical distributions.

Study on neural networks' sample complexity with one hidden layer.

We study sequences of conformal deformations of a smooth closed Riemannian manifold of dimension $n$, assuming uniform volume bounds and $L^{n/2}$ bounds on their scalar curvatures. Singularities may appear in the limit. Nevertheless, we show that under such bounds the underlying metric spaces are pre-compact in the Gr…

New bounds for SA with arbitrary norm contractions and Markovian noise.

The paper extends manifold learning to arbitrary norms, improving molecular motion mapping.

We show the existence of a global unique and analytic solution for the mean curvature flow, the surface diffusion flow and the Willmore flow of entire graphs for Lipschitz initial data with small Lipschitz norm. We also show the existence of a global unique and analytic solution to the Ricci-DeTurck flow on euclidean s…

We determine the Hausdorff limit-set of the Euclidean hypersurfaces with large $λ_1$ or small extrinsic radius. The result depends on the $L^p$ norm of the curvature that is assumed to be bounded a priori, with a critical behaviour for $p$ equal to the dimension minus 1.

It is well known that the description of topological and geometric properties of bisectors in normed spaces is a non-trivial subject. In this paper we introduce the concept of bounded representation of bisectors in finite dimensional real Banach spaces. This useful notion combines the concepts of bisector and shadow bo…

The goal of this article is to establish estimates involving the Yamabe minimal volume, mixed minimal volume and some topological invariants on compact 4-manifolds. In addition, we provide topological sphere theorems for compact submanifolds of spheres and Euclidean spaces, provided that the full norm of the second fun…

Paper proves inequalities for differential forms in Euclidean and Heisenberg groups.

In this work we prove convergence results of sequences of Riemannian $4$-manifolds with almost vanishing $L^2$-norm of a curvature tensor and a non-collapsing bound on the volume of small balls. In Theorem 1.1, we consider a sequence of closed Riemannian $4$-manifolds, whose $L^2$-norm of the Riemannian curvature tenso…

Generalizes momentum methods using Hamiltonian dynamics.

Study shows horofunction compactification's topology matches dual norm's unit ball.

Researchers classify 3D self-shrinkers in 4D space.

The study classifies complete self-shrinkers in Euclidean space.

Develops a new asymptotic efficiency theory for non-Euclidean parameter spaces.

This work addresses the convergence of SGD's final iterate without restrictive assumptions.

Classical integral geometry takes place in Euclidean space, but one can attempt to imitate it in any other metric space. In particular, one can attempt this in R^n equipped with the metric derived from the p-norm. This has, in effect, been investigated intensively for 1<p<\infty, but not for p=1. We show that integral …

The study classifies harmonic cubic polynomials in up to 4 dimensions.

The paper connects polygon spaces with quotient spaces using spin actions and normed division algebras.

In this paper, we study the problem of approximately computing the product of two real matrices. In particular, we analyze a dimensionality-reduction-based approximation algorithm due to Sarlos [1], introducing the notion of nuclear rank as the ratio of the nuclear norm over the spectral norm. The presented bound has i…

Formula proves monotonicity for anisotropic minimal hypersurfaces.

New framework improves robustness of implicit neural networks.

We investigate isometric immersions of disks with constant negative curvature into $\mathbb{R}^3$, and the minimizers for the bending energy, i.e. the $L^2$ norm of the principal curvatures over the class of $W^{2,2}$ isometric immersions. We show the existence of smooth immersions of arbitrarily large geodesic balls i…

EF21-Muon optimizes deep learning with error feedback, improving efficiency and accuracy.

Gradient descent near stability threshold shows sharpness oscillations.

Gradient descent near stability threshold exhibits sharpness oscillations.