arXiv research

In this paper we prove several results on the geometry of surfaces immersed in $\mathbf R^3$ with small or bounded $L^2$ norm of $|A|$. For instance, we prove that if the $L^2$ norm of $|A|$ and the $L^p$ norm of $H$, $p>2$, are sufficiently small, then such a surface is graphical away from its boundary. We also prove …

In four and higher dimensions, we show that any stationary admissible Yang-Mills field can be gauge transformed to a smooth field if the $L^2$ norm of the curvature is sufficiently small. There are three main ingredients. The first is Price's monotonicity formula, which allows us to assert that the curvature is small n…

Small Nijenhuis tensor found on compact manifolds.

We consider immersions admitting uniform graph representations over the affine tangent space over a ball of fixed radius r>0. We show that for sufficiently small C^0-norm of the graph functions, each graph function is smooth with small C^1-norm.

New algorithms reduce regret for convex bandits with small comparator norms.

We provide a necessary and sufficient condition that $L^p$-norms, $2<p<6$, of eigenfunctions of the square root of minus the Laplacian on 2-dimensional compact boundaryless Riemannian manifolds $M$ are small compared to a natural power of the eigenvalue $λ$. The condition that ensures this is that their $L^2$ norms ove…

New framework assesses regularization norms in ill-posed problems, revealing L2 instability and proposing adaptive fractional RKHS solutions.

Study on scalar curvature bounds and manifold topological complexity.

PSiLON Net uses $L_1$ weight normalization and 1-path-norm regularization for efficient learning and sparsity.

Unified algorithm for minimizing composite functions with flexible design.

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

Differential calculus on metric spaces is contained in the algebraic study of normed groupoids with $δ$-structures. Algebraic study of normed groups endowed with dilatation structures is contained in the differential calculus on metric spaces. Thus all algebraic properties of the small world of normed groups with dilat…

After appropriate normalizations an embedded disk whose second fundamental form has large norm contains a multi-valued graph, provided the L^P norm of the mean curvature is sufficiently small. This generalizes to non-minimal surfaces a well known result of Colding and Minicozzi.

An elementary proof found for the double bubble problem in a specific norm.

Noise injection before gradient steps helps in regularization for neural networks.

This paper presents a normalization mechanism called Instance-Level Meta Normalization (ILM~Norm) to address a learning-to-normalize problem. ILM~Norm learns to predict the normalization parameters via both the feature feed-forward and the gradient back-propagation paths. ILM~Norm provides a meta normalization mechanis…

We consider surfaces with boundary satisfying a sixth order nonlinear elliptic partial differential equation corresponding to extremising the $L^2$-norm of the gradient of the mean curvature. We show that such surfaces with small $L^2$-norm of the second fundamental form and satisfying so-called `flat boundary conditio…

Paper introduces a new $G^\star$ regret measure for online convex optimization with smooth losses.

Let $(M,g)$ be a noncompact complete $n$-manifold with harmonic curvature and positive Sobolev constant. Assume that $L_2$ norms of Weyl curvature and traceless Ricci curvature are finite. We prove that $(M,g)$ is Einstein if $n \ge 5$ and $L_{n/2}$ norms of Weyl curvature and traceless Ricci curvature are small enough…

We prove an inequality between the $L^{\infty}$-norm of the contact Hamiltonian of a positive loop of contactomorphims and the minimal Reeb period. This implies that there are no small positive loops on hypertight or Liouville fillable contact manifolds. Non-existence of small positive loops for overtwisted 3-manifolds…

Estimates for the norm of the second fundamental form, $|A|$, play a crucial role in studying the geometry of surfaces. In fact, when $|A|$ is bounded the surface cannot bend too sharply. In this paper we prove that for an embedded geodesic disk with bounded $L^2$ norm of $|A|$, $|A|$ is bounded at interior points, pro…

We propose a new point of view for regularizing deep neural networks by using the norm of a reproducing kernel Hilbert space (RKHS). Even though this norm cannot be computed, it admits upper and lower approximations leading to various practical strategies. Specifically, this perspective (i) provides a common umbrella f…

The Willmore conjecture states that any immersion F:T^2 -> R^n of a 2-torus into flat euclidean space satisfies $\int_{T^2} H^2\geq 2π^2$. We prove it under the condition that the L^p-norm of the Gaussian curvature is sufficiently small.

Hexagonal norm double bubble problem solved with minimal configurations.

CNN layers with large norms are still robust to adversarial attacks.

This work improves trace norm regularization for multi-task learning with limited data.

In this paper we present several curvature estimates and convergence results for solutions of the Ricci flow. The curvature estimates depend on smallness of certain local space-time integrals of the norm of the Riemann curvature tensor, while the convergence results require finiteness of space-time integrals of the nor…

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.

The study shows how to regularize weakly harmonic maps using Sobolev norms and Coulomb frames.

The study analyzes robustness of estimators in linear models with adversarial errors.

HBR improves normative modeling of neuroimaging data across multiple sites.

Batch normalization (batch norm) is often used in an attempt to stabilize and accelerate training in deep neural networks. In many cases it indeed decreases the number of parameter updates required to achieve low training error. However, it also reduces robustness to small adversarial input perturbations and noise by d…

Quantifies closeness of special Lagrangians under Floer conditions.

This note is devoted to Keller-Lieb-Thirring spectral estimates for Schrödinger operators on infinite cylinders: the absolute value of the ground state level is bounded by a function of a norm of the potential. Optimal potentials with small norms are shown to depend on a single variable. The proof is a perturbation arg…

We obtain a local volume growth for complete, noncompact Riemannian manifolds with small integral bounds and with Bach tensor having finite $L^2$ norm in dimension 4.

In this paper we present several curvature estimates for solutions of the Ricci flow which depend on smallness of certain local integrals of the norm of the Riemann curvature tensor.

Regularization can induce grokking in neural networks, improving generalization.

New insights into network generalization show learning rate affects both norm and sharpness.

Near-interpolating models grow norms quickly, affecting generalization.

Improves adversarial robustness by constraining logits with a bounded function.

On a closed balanced manifold, we show that if the Chern scalar curvature is small enough in a certain Sobolev norm then a slightly modified version of the Chern-Yamabe flow~\cite{Angella:2015aa} converges to a solution of the Chern-Yamabe problem. We also prove that if the Chern scalar curvature, on closed almost-Herm…

New algorithms improve approximation of matrix norms, with applications in statistics and machine learning.

We analyze the level sets of the norm of the Witten spinor in an asymptotically flat Riemannian spin manifold of positive scalar curvature. Level sets of small area are constructed. We prove curvature estimates which quantify that, if the total mass becomes small, the manifold becomes flat with the exception of a set o…

We develop a new approach to, and small extension of, results of Cheeger, Colding and Tian concerning the $L^{k/2}$ norm of the curvature of a Riemannian manifold Gromov-Hausdorff close to a codimension $k$ singularity.

We establish both local and global well-posedness for the heat flow of polyharmonic maps from $R^n$ to a compact Riemannian manifold without boundary for initial data with small BMO norms.

Small mass implies a bilipschitz diffeomorphism to flat space

Sparse methods for supervised learning aim at finding good linear predictors from as few variables as possible, i.e., with small cardinality of their supports. This combinatorial selection problem is often turned into a convex optimization problem by replacing the cardinality function by its convex envelope (tightest c…

We consider closed immersed hypersurfaces evolving by surface diffusion flow, and perform an analysis based on local and global integral estimates. First we show that a properly immersed stationary (ΔH \equiv 0) hypersurface in \R^3 or \R^4 with restricted growth of the curvature at infinity and small total tracefree c…