arXiv research

Global Morse index theorem applied to Jacobi fields on CMC surfaces.

We show that the definition of global hyperbolicity in terms of the compactness of the causal diamonds and non-total imprisonment can be extended to spacetimes with continuous metrics, while retaining all of the equivalences to other notions of global hyperbolicity. In fact, global hyperbolicity is equivalent to the co…

New algorithm optimizes Hölder continuous functions efficiently.

Study on stability and continuity of solutions to complex Monge-Ampère equations on compact Hermitian manifolds.

Continuous-time distributed mirror descent with integral feedback converges to global optimum.

The paper proves Lipschitz continuity of cut times in spacetimes.

Efficient algorithm for global optimization of multivariate Lipschitz functions.

SBS uses SVGD to optimize continuous functions globally.

We prove local Holder continuity of quasi-n-harmonic mappings from Euclidean domains into metric spaces with non-positive curvature in the sense of Alexandrov. We also obtain global Holder continuity of such mappings from bounded Lipschitz domains.

Proves Hölder continuity of complex Monge-Ampère solutions.

Paper solves DRO for continuous distributions with iterative algorithms.

Foundation models fail to preserve continuous geometry, identified as the Geometric Alignment Tax.

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

Sparse-mode DMD disambiguates local and global modes in spatiotemporal data.

Continuous-time model shows insider trading constraints impact market dynamics.

ECP optimizes expensive functions without knowing Lipschitz constant.

Globally normalized neural sequence models are considered superior to their locally normalized equivalents because they may ameliorate the effects of label bias. However, when considering high-capacity neural parametrizations that condition on the whole input sequence, both model classes are theoretically equivalent in…

Let $(M,g)$ be a globally symmetric space of noncompact type, of arbitrary rank, and $Δ$ its Laplacian. We prove the existence of a meromorphic continuation of the resolvent $(Δ-\ev)^{-1}$ across the continuous spectrum to a Riemann surface multiply covering the plane. The methods are purely analytic and are adapted fr…

CAM-GAN improves GANs for continual learning with efficient feature map transformations.

Continuity of Kähler-Einstein potentials at singularities proven.

We consider Aubry-Mather theory for a subclass of class A spacetimes, i.e. compact vicious spacetimes with globally hyperbolic Abelian cover. In this subclass, called class A_1, we obtain improved results on timelike maximizers and Lipschitz continuity of the time separation of the Abelian cover on the i.g. optimal sub…

We consider rough metrics on smooth manifolds and corresponding Laplacians induced by such metrics. We demonstrate that globally continuous heat kernels exist and are Hölder continuous locally in space and time. This is done via local parabolic Harnack estimates for weak solutions of operators in divergence form with b…

A new method for hierarchical clustering using continuous embeddings and optimization.

Example of spacetime with causal bubbling, splitting into timelike and spacelike parts.

Hybrid systems are characterized by having an interaction between continuous dynamics and discrete events. The contribution of this paper is to provide hybrid systems with a novel geometric formulation so that controls can be added. Using this framework we describe some new global controllability tests for hybrid contr…

New insights into continual learning for deep models, showing convergence issues but local linear solutions.

FS&P uses birth-death process to ensure global convergence of stochastic conic particle gradient descent.

Study continuity of phi-invariant for degenerating graphs.

In this paper, we prove global second derivative estimates for solutions of the Dirichlet problem for the Monge-Ampere equation when the inhomogeneous term is only assumed to be Holder continuous. As a consequence of our approach, we also establish the existence and uniqueness of globally smooth solutions to the second…

Paper proposes a new strategy to improve initial performance of federated models.

Proposes DMOC for more nuanced neural network robustness.

Understanding the global optimality in deep learning (DL) has been attracting more and more attention recently. Conventional DL solvers, however, have not been developed intentionally to seek for such global optimality. In this paper we propose a novel approximation algorithm, BPGrad, towards optimizing deep models glo…

The maximal analytic Schwarzschild spacetime is manifestly inextendible as a Lorentzian manifold with a twice continuously differentiable metric. In this paper, we prove the stronger statement that it is even inextendible as a Lorentzian manifold with a continuous metric. To capture the obstruction to continuous extens…

Proves existence of optimal shallow neural networks with ReLU activation.

We prove that the global minimum of the backpropagation (BP) training problem of neural networks with an arbitrary nonlinear activation is given by the ridgelet transform. A series of computational experiments show that there exists an interesting similarity between the scatter plot of hidden parameters in a shallow ne…

Proves convergence of mean curvature flow on cylinders with unique continuation.

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…

For a bounded domain equipped with a piecewise Lipschitz continuous Riemannian metric g, we consider harmonic map from $(Ω, g)$ to a compact Riemannian manifold $(N,h)\subset\mathbb R^k$ without boundary. We generalize the notion of stationary harmonic map and prove the partial regularity. We also discuss the global Li…

Lie groupoid equivariant neural networks are a new type of neural network.

Sufficient condition for log-continuity of complex Monge-Ampère solutions.

A federated model learns shared archetypes from heterogeneous clients in continual learning.

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

Continuing our previous work (arXiv:1509.07981v1), we derive another global gradient estimate for positive functions, particularly for positive solutions to the heat equation on finite or locally finite graphs. In general, the gradient estimate in the present paper is independent of our previous one. As applications, i…

The paper analyzes the current state of the world economy and offers a short-term forecast of its development. Our analysis of log-periodic oscillations in the DJIA dynamics suggests that in the second half of 2017 the United States and other more developed countries could experience a new recession, due to the third p…

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

Paper introduces a continuous convexity measure for compact sets.

We are analysing the convexity and continuity properties of the Mabuchi functional along weak geodesics. The key technical point in our paper is the global approximation of weak geodesics obtained via a well-chosen family of Monge-Ampère equations.

The problem of pricing Bermudan options using Monte Carlo and a nonparametric regression is considered. We derive optimal non-asymptotic bounds for a lower biased estimate based on the suboptimal stopping rule constructed using some estimates of continuation values. These estimates may be of different nature, they may …