arXiv research

In this note a proof is given for global existence and uniqueness of minimal surfaces of Lorentzian type from a cylinder into globally hyperbolic Lorentzian manifolds for given initial values up to the first derivatives.

We prove the existence of global minimizers of Allen-Cahn equation in dimensions $8$ and above. More precisely, given any strictly area-minimizing Lawson's cones, there are global minimizers whose nodal sets are asymptotic to the cones. As a consequence of Jerison-Monneau's program we establish the existence of many co…

The ultimate goal of optimization is to find the minimizer of a target function.However, typical criteria for active optimization often ignore the uncertainty about the minimizer. We propose a novel criterion for global optimization and an associated sequential active learning strategy using Gaussian processes.Our crit…

We give a locally minimal, but not globally minimal bridge position of a knot, that is, an unstabilized, nonminimal bridge position of a knot. It implies that a bridge position cannot always be simplified so that the bridge number monotonically decreases to the minimal.

Euler's elastica with monotone curvature is uniquely minimal.

The paper studies minimal surface flow and translating solitons, proving global solutions and convergence.

First order methods can take extremely long to find global minima of non-convex functions.

Extends global stability of Minkowski spacetime to minimal decay assumptions.

We study global Mumford-Shah minimizers in $\R^N$, introduced by Bonnet as blow-up limits of Mumford-Shah minimizers. We prove a new monotonicity formula for the energy of $u$ when the singular set $K$ is contained in a smooth enough cone. We then use this monotonicity to prove that for any reduced global minimizer $(u…

Study finds many nonplanar minimal spheres in elongated ellipsoids.

Study finds conditions for global minimizers on curved manifolds with fast diffusion and nonlocal interactions.

We derive a Bernstein type result for the special Lagrangian equation, namely, any global convex solution must be quadratic. In terms of minimal surfaces, the result says that any global minimal Lagrangian graph with convex potential must be a hyper-plane.

SGD methods fail to converge to global minimizers in deep neural networks with ReLU activation.

WSFN overcomes saddle points for non-convex functionals in Wasserstein space.

Since the pioneering work of Canham and Helfrich, variational formulations involving curvature-dependent functionals, like the classical Willmore functional, have proven useful for shape analysis of biomembranes. We address minimizers of the Canham-Helfrich functional defined over closed surfaces enclosing a fixed volu…

The paper finds curves minimizing elastic energy pinned at endpoints.

We prove a local splitting theorem for three-manifolds with mean convex boundary and scalar curvature bounded from below that contain certain locally area-minimizing free boundary surfaces. Our methods are based on those of Micallef and Moraru. We use this local result to establish a global rigidity theorem for area-mi…

Gradient descent with preconditioning finds global optima in overparameterized nonconvex factorization.

The past decade has witnessed a successful application of deep learning to solving many challenging problems in machine learning and artificial intelligence. However, the loss functions of deep neural networks (especially nonlinear networks) are still far from being well understood from a theoretical aspect. In this pa…

New algorithms minimize regret with global costs in online learning.

Extends Minkowski stability proof to minimal decay assumptions.

In this paper we consider the length minimizing properties of Hamiltonian paths generated by quasi-autonomous Hamiltonians on symplectically aspherical manifolds. Motivated by the work of L. Polterovich and M. Schwarz, we study the role of the fixed global extrema in the Floer complex of the generating Hamiltonian. Our…

New method creates vacuum data at minimal and borderline decay thresholds.

Minimal networks minimize length and mass in certain configurations.

Study on elastic curves pinned at the boundary, focusing on minimizers and their interaction with obstacles.

Learning new representations of input observations in machine learning is often tackled using a factorization of the data. For many such problems, including sparse coding and matrix completion, learning these factorizations can be difficult, in terms of efficiency and to guarantee that the solution is a global minimum.…

In this note, we study submanifold geometry of the Atiyah-Hitchin manifold, the double cover of the $2$-monopole moduli space. When the manifold is naturally identified as the total space of a line bundle over $S^2$, the zero section is a distinguished minimal $2$-sphere of considerable interest. In particular, there h…

Researchers found infinite links with specific bridge positions.

The paper creates symmetrical discrete minimal nets using Schwarz reflection.

We consider a Canham-Helfrich-type variational problem defined over closed surfaces enclosing a fixed volume and having fixed surface area. The problem models the shape of multiphase biomembranes. It consists of minimizing the sum of the Canham-Helfrich energy, in which the bending rigidities and spontaneous curvatures…

The paper proves the existence of area-minimizing hypersurfaces in AF manifolds of higher dimensions.

In a 2004 paper, Lindblad demonstrated that the minimal surface equation on $\mathbb{R}l^{1,1}$ describing graphical time-like minimal surfaces embedded in $\mathbb{R}^{1,2}$ enjoy small data global existence for compactly supported initial data, using Christodoulou's conformal method. Here we give a different, geometr…

Gradient descent finds global optima in ResNets with sufficient parameters.

Global minimizers exist for Tonelli Lagrangians on half-Lie groups.

Lightlike hypersurfaces in cone structures minimize time.

We describe local similarities and global differences between minimal surfaces in Euclidean 3-space and constant mean curvature 1 surfaces in hyperbolic 3-space. We also describe how to solve global period problems for constant mean curvature 1 surfaces in hyperbolic 3-space, and we give an overview of recent results o…

Geodesics in jet space are constructed from polynomials, with some yielding globally minimizing paths.

Proves a principle for one-phase Bernoulli problem minimizers.

This work addresses decentralized online optimization in non-stationary environments. A network of agents aim to track the minimizer of a global time-varying convex function. The minimizer evolves according to a known dynamics corrupted by an unknown, unstructured noise. At each time, the global function can be cast as…

Global minima found for multidimensional scaling with penalties.

Recurring international financial crises have adverse socioeconomic effects and demand novel regulatory instruments or strategies for risk management and market stabilization. However, the complex web of market interactions often impedes rational decisions that would absolutely minimize the risk. Here we show that, for…

Cone structures over minimal products can't be calibrated smoothly.

We consider the problem of minimizing $\int_{0}^L \sqrt{ξ^2 +K^2(s)}\, ds$ for a planar curve having fixed initial and final positions and directions. The total length $L$ is free. Here $s$ is the variable of arclength parametrization, $K(s)$ is the curvature of the curve and $ξ>0$ a parameter. This problem comes from…

In relativity, the energy of a moving particle depends on the observer, and the rest mass is the minimal energy seen among all observers. The Wang-Yau quasi-local mass for a surface in spacetime introduced in [7] and [8] is defined by minimizing quasi-local energy associated with admissible isometric embeddings of the …

In this note, we focus on smooth nonconvex optimization problems that obey: (1) all local minimizers are also global; and (2) around any saddle point or local maximizer, the objective has a negative directional curvature. Concrete applications such as dictionary learning, generalized phase retrieval, and orthogonal ten…

New method finds global minima using function evaluations and kernel approximations.

We give an a priori bound on the (n-7)-dimensional measure of the singular set for an area-minimizing n-dimensional hypersurface, in terms of the geometry of its boundary.

We construct a sequence of embedded minimal disks in a ball where the curvatures blow up only at the center. The sequence converges to a limit which is not smooth and not proper.