arXiv research

Global implicit function theorem for Fréchet spaces, solving derivative loss problems.

Global inverse function theorem proved easily using Riemannian geometry.

Proves globally hyperbolic spacetimes via null distance completeness.

GONs improve predictions of maximizers from noisy black-box functions.

Global minima found for multidimensional scaling with penalties.

Global calculus for manifolds with boundary, solving evolution problems.

In this article, we introduce and study the notion of a complete special holonomy manifold $(X,ω)$ which is given by a global perturbation potential function, i.e., there is a function $f$ on $X$ such that $ω'=ω-\mathcal{L}_{\nabla f}ω$ is sufficiently small in $L^{\infty}$-norm. We establish some vanishing theorems on…

Bayesian optimization is a sample-efficient method for finding a global optimum of an expensive-to-evaluate black-box function. A global solution is found by accumulating a pair of query point and its function value, repeating these two procedures: (i) modeling a surrogate function; (ii) maximizing an acquisition funct…

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

Global optimization problems whose objective function is expensive to evaluate can be solved effectively by recursively fitting a surrogate function to function samples and minimizing an acquisition function to generate new samples. The acquisition step trades off between seeking for a new optimization vector where the…

The paper guarantees global stability for stochastic subgradient methods in nonsmooth nonconvex optimization.

Global optimization finds applications in a wide range of real world problems. The multi-start methods are a popular class of global optimization techniques, which are based on the ideas of conducting local searches at multiple starting points. In this work we propose a new multi-start algorithm where the starting poin…

SGD converges to global minimum for certain non-convex functions.

We give a topological condition for a generic sliced space to be globally hyperbolic, without any hypothesis on the lapse function, shift function and spatial metric.

Researchers examine global properties of a scalar curvature functional to solve the prescribed Ricci curvature problem.

ECP optimizes expensive functions without knowing Lipschitz constant.

This paper proposes a new global optimization algorithm using deep learning.

Gradients help find global optima in complex functions.

Deep neural networks perform well on local tasks but struggle with global tasks.

ProGO optimizes non-convex functions without gradients, outperforming existing methods.

The paper investigates how activation functions impact the training of Neural ODEs, leading to global convergence.

The folk questions in Lorentzian Geometry, which concerns the smoothness of time functions and slicings by Cauchy hypersurfaces, are solved by giving simple proofs of: (a) any globally hyperbolic spacetime $(M,g)$ admits a smooth time function $τ$ whose levels are spacelike Cauchy hyperfurfaces and, thus, also a smooth…

Machine learning algorithms typically perform optimization over a class of non-convex functions. In this work, we provide bounds on the fundamental hardness of identifying the global minimizer of a non convex function. Specifically, we design a family of parametrized non-convex functions and employ statistical lower bo…

A new scheme reduces global search cost by a square root factor.

Global EQG sums boundary states over manifold diffeomorphism classes.

Global existence and geometry of constant mass aspect function foliation in perturbed Schwarzschild spacetime studied.

The chapter explores globally hyperbolic spacetimes using topology and functional analysis.

Synthetic proof shows globally hyperbolic Lorentzian spaces with specific curvature are warped products.

For any prime power $q$ and any dimension $s$, a new construction of $(t,s)$-sequences in base $q$ using global function fields is presented. The construction yields an analog of Halton sequences for global function fields. It is the first general construction of $(t,s)$-sequences that is not based on the digital metho…

Single-timescale actor-critic finds globally optimal policy.

This paper studies GAIL's global convergence for general MDP and nonlinear rewards.

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…

We consider the problem of extending functions φ:\to S^n to functions u:B^{n+1}\to S^n for n=2,3. We assume φto belong to the critical space W^{1,n} and we construct a W^{1,(n+1,\infty)}-controlled extension u. The Lorentz-Sobolev space W^{1,(n+1,\infty)} is optimal for such controlled extension. Then we use such resul…

Efficient binary sampling method for global optimization of univariate functions with low regret.

Global harmonic maps into SU(1,1) constructed from Smyth potentials using DPW method.

Global approximation for piecewise linear paths via signatures.

The paper proves convergence to global optima for a class of distributed algorithms for nonconvex optimization in network-based multi-agent settings. Agents are permitted to communicate over a time-varying undirected graph. Each agent is assumed to possess a local objective function (assumed to be smooth, but possibly …

In this article, existence results concerning temporal functions with additional properties on a globally hyperbolic manifold are obtained. These properties are certain bounds on geometric quantities as lapse and shift. The results are linked to completeness properties and the existence of closed isometric embeddings i…

SGD converges globally to logistic loss minima for two-layer nets.

Efficient algorithm for global optimization of multivariate Lipschitz functions.

This paper gives a new proof that maximal, globally hyperbolic, flat spacetimes of dimension $n\geq 3$ with compact Cauchy hypersurfaces are globally foliated by Cauchy hypersurfaces of constant mean curvature, and that such spacetimes admit a globally defined constant mean curvature time function precisely when they a…

New method for global optimization of Gaussian processes reduces computational time.

HALO uses local Lipschitz constants to optimize functions efficiently.

Develops methods to calculate global index of real polynomials.

We introduce the gradient flow of the Seiberg-Witten functional on a compact, orientable Riemannian 4-manifold and show the global existence of a unique smooth solution to the flow. The flow converges uniquely in $C^\infty$ up to gauge to a critical point of the Seiberg-Witten functional.

RelatIF selects more intuitive training examples for explaining model predictions.

We study the error landscape of deep linear and nonlinear neural networks with the squared error loss. Minimizing the loss of a deep linear neural network is a nonconvex problem, and despite recent progress, our understanding of this loss surface is still incomplete. For deep linear networks, we present necessary and s…

The possibility of the global Lagrangian reduction of a mechanical system with symmetry is shown to be connected with the characteristic class of a principal fiber bundle of the configuration space over the factor manifold. It is proved that the reduced system is globally Lagrangian if and only if the product of the mo…