arXiv research

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

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

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

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

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

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

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

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

Generalizes momentum methods using Hamiltonian dynamics.

The study provides bounds for geodesic diameter in Euclidean space.

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

The study classifies complete self-shrinkers in Euclidean space.

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.

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…

Formula proves monotonicity for anisotropic minimal hypersurfaces.

New algorithms optimize convex functions with high-order derivatives.

The paper examines torsions in Minkowskian product of Finsler metrics.

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.

Estimates latent norms and Gram matrices for graphs on Euclidean balls.

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

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

The problem of joint feature selection across a group of related tasks has applications in many areas including biomedical informatics and computer vision. We consider the l2,1-norm regularized regression model for joint feature selection from multiple tasks, which can be derived in the probabilistic framework by assum…

New optimization method combines gradient clipping and non-Euclidean smoothness.

The paper classifies 2D complete $λ$-surfaces in 3D space.

The paper explores centroids and static equilibrium points in non-Euclidean geometries.

New algorithms for efficient matrix profile computation using various Euclidean distances.

We show that the surface area preserving mean curvature flow in Euclidean space exists for all time and converges exponentially to a round sphere, if initially the L^2-norm of the traceless second fundamental form is small (but the initial hypersurface is not necessarily convex).

The L 1-Sobolev inequality states that the L n/(n--1)-norm of a compactly supported function on Euclidean n-space is controlled by the L 1-norm of its gradient. The generalization to differential forms (due to Lanzani & Stein and Bourgain & Brezis) is recent, and states that a the L n/(n--1)-norm of a compactly support…

The study classifies complete Lagrangian self-shrinkers in 4D space.

The spectral $k$-support norm enjoys good estimation properties in low rank matrix learning problems, empirically outperforming the trace norm. Its unit ball is the convex hull of rank $k$ matrices with unit Frobenius norm. In this paper we generalize the norm to the spectral $(k,p)$-support norm, whose additional para…

New Muon and Momo variants improve neural network optimization robustness.

The paper analyzes and improves a deep learning optimization technique using matrix gradient orthogonality.

Study rigidity of minimal Legendrian submanifolds in spheres via eigenvalues.

Uniform convergence of interpolators proven for Gaussian data.

Using sparse-inducing norms to learn robust models has received increasing attention from many fields for its attractive properties. Projection-based methods have been widely applied to learning tasks constrained by such norms. As a key building block of these methods, an efficient operator for Euclidean projection ont…

Paper proves uniqueness of minimal hypersurfaces in specific domains.

We study singular monopoles on open subsets in the $3$-dimensional Euclidean space. We give two characterizations of Dirac type singularities. One is given in terms of the growth order of the norms of sections which are invariant by the scattering map. The other is given in terms of the growth order of the norms of the…

We consider the empirical risk minimization problem for linear supervised learning, with regularization by structured sparsity-inducing norms. These are defined as sums of Euclidean norms on certain subsets of variables, extending the usual $\ell_1$-norm and the group $\ell_1$-norm by allowing the subsets to overlap. T…

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

We obtain a Chern-Osserman type equality of a complete properly immersed surface in Euclidean space, provided the L^2-norm of the second fundamental form is finite. Also, by using a monotonicity formula, we prove that if the L^2-norm of mean curvature of a noncompact surface is finite, then it has at least quadratic ar…

We show that the problem of tiling the Euclidean plane with a finite set of polygons (up to translation) boils down to prove the existence of zeros of a non-negative convex function defined on a finite-dimensional simplex. This function is a generalisation, in the framework of branched surfaces, of the Thurston semi-no…

In this paper, we give pinching Theorems for the first nonzero eigenvalue $λ$ of the Laplacian on the compact hypersurfaces of the Euclidean space. Indeed, we prove that if the volume of $M$ is 1 then, for any $ε>0$, there exists a constant $C\_ε$ depending on the dimension $n$ of $M$ and the $L\_{\infty}$-norm of the …

New method accelerates steepest descent for convex optimization.

The study improves norms of spectral projectors on specific surfaces.