arXiv research

Global regularity proved for 4D Ricci flow with scalar curvature integral bound.

Study optimal transport on globally hyperbolic spacetimes, focusing on weak Kantorovich potentials' regularity.

Proves globally hyperbolic spacetimes via null distance completeness.

Global convergence of SGD proven for two-layer neural nets with regularization.

The paper studies global invertibility of maps on Finsler manifolds.

Recent works have shown that on sufficiently over-parametrized neural nets, gradient descent with relatively large initialization optimizes a prediction function in the RKHS of the Neural Tangent Kernel (NTK). This analysis leads to global convergence results but does not work when there is a standard $\ell_2$ regulari…

The study examines differential smoothness in specific Artin-Schelter regular algebras of dimension 5.

Establishes a Lorentzian Lasry-Lions regularization theorem for functions on globally hyperbolic spacetimes.

Non-convex regularizers usually improve the performance of sparse estimation in practice. To prove this fact, we study the conditions of sparse estimations for the sharp concave regularizers which are a general family of non-convex regularizers including many existing regularizers. For the global solutions of the regul…

On any complete Riemannian manifold $M$ and for all $p\in [2,\infty)$, we prove a family of second order $L^{p}$-interpolation inequalities that arise from the following simple $L^{p}$-estimate valid for every $u \in C^{\infty}(M)$: $\|\nabla u\|_{p}^p \leq \|u Δ_{p} u\|_1\in [0,\infty],$ where $Δ_p$ denotes the $p…

Under weak regularity assumptions, only, we develop a fully geometric theory of vacuum Einstein spacetimes with T2 symmetry, establish the global well-posedness of the initial value problem for Einstein's field equations, and investigate the global causal structure of the constructed spacetimes. Our weak regularity ass…

Low regularity spacetimes split into simpler structures.

The paper constructs Levi flat structures using structure sheaves and differential complexes.

We show that there are not pure $\mathcal{C}^5$ regular y-global Landsberg surfaced. The proof is based on the averaged connection associated with the linear Chern's connection and the classification of irreducibles holonomies of torsion-free affine connections. The structure consists on exausting all the possible case…

A new SSL method improves medical image classification using global latent mixing.

We are concerned with the global weak rigidity of the Gauss-Codazzi-Ricci (GCR) equations on Riemannian manifolds and the corresponding isometric immersions of Riemannian manifolds into the Euclidean spaces. We develop a unified intrinsic approach to establish the global weak rigidity of both the GCR equations and isom…

We are interested in global properties of systems of left-invariant differential operators on compact Lie groups: regularity properties, properties on the closedness of the range and finite dimensionality of their cohomology spaces, when acting on various function spaces e.g. smooth, analytic and Gevrey. Extending the …

Poisson learning doesn't solve graph semi-supervised learning issues.

Softmax policy gradient achieves global optimality in wide neural networks with entropy regularization.

Study of regularized least squares in RKKS with indefinite kernels.

Generalizes global hyperbolicity to higher signatures and proves compactness.

Concave regularization methods provide natural procedures for sparse recovery. However, they are difficult to analyze in the high dimensional setting. Only recently a few sparse recovery results have been established for some specific local solutions obtained via specialized numerical procedures. Still, the fundamental…

Proposes new attribution methods for trees with regularization.

Let $Σ$ be a connected, oriented surface with punctures and negative Euler characteristic. We introduce regular globally hyperbolic anti-de Sitter structures on $Σ\times \mathbb{R}$ and provide two parameterisations of their deformation space: as an enhanced product of two copies of the Fricke space of $Σ$ and as the b…

Unified framework reveals regularization mechanism in deep ReLU networks via convex optimization.

Paper studies smoothness of bi-conformal heat flow on 4-manifolds.

Several data analysis techniques employ similarity relationships between data points to uncover the intrinsic dimension and geometric structure of the underlying data-generating mechanism. In this paper we work under the model assumption that the data is made of random perturbations of feature vectors lying on a low-di…

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

Solves geodesic equations on special Kähler manifolds, proving global regularity.

We provide adaptive inference methods, based on $\ell_1$ regularization, for regular (semi-parametric) and non-regular (nonparametric) linear functionals of the conditional expectation function. Examples of regular functionals include average treatment effects, policy effects, and derivatives. Examples of non-regular f…

A new L-BFGS method tackles large-scale optimization with fewer evaluations.

New flow method solves Christoffel-Minkowski problem.

Global existence and convergence of heat flow for p-harmonic maps.

We study the evolution equations for a regularized version of Dirac-harmonic maps from closed Riemannian surfaces. We establish the existence of a global weak solution for the regularized problem, which is smooth away from finitely many singularities. Moreover, we discuss the convergence of the evolution equations and …

Regularization plays a crucial role in supervised learning. Most existing methods enforce a global regularization in a structure agnostic manner. In this paper, we initiate a new direction and propose to enforce the structural simplicity of the classification boundary by regularizing over its topological complexity. In…

Spectral methods are popular in detecting global structures in the given data that can be represented as a matrix. However when the data matrix is sparse or noisy, classic spectral methods usually fail to work, due to localization of eigenvectors (or singular vectors) induced by the sparsity or noise. In this work, we …

Proves flows of two-convex Lagrangians are regular, global, and converge.

Develops a novel global pooling framework using optimal transport.

Path regularization reveals convex optimization in deep ReLU networks.

Two important goals of high-dimensional modeling are prediction and variable selection. In this article, we consider regularization with combined $L_1$ and concave penalties, and study the sampling properties of the global optimum of the suggested method in ultra-high dimensional settings. The $L_1$-penalty provides th…

New algorithm improves convergence of gradient boosting trees.

The paper extends completeness notions to low-regularity spacetimes.

We first obtain the interior $C^{1,1}$-regularity and solvability for the degenerate real Monge-Ampère equation in a bounded, $C^3$-smooth and strictly convex domain in $\mathbb R^d$ ($d\ge2$), assuming that the boundary data is only globally $C^{1,1}$, and the $d$-th root of the nonnegative right-hand side is globally…

Theoretical comparison of three invariance approaches in deep linear networks.

This research smooths out fluid equations to avoid sudden shocks.

Study shows policy gradient convergence for entropy-regularized MDPs with neural nets in mean-field regime.

We give a formal and complete characterization of the explicit regularizer induced by dropout in deep linear networks with squared loss. We show that (a) the explicit regularizer is composed of an $\ell_2$-path regularizer and other terms that are also re-scaling invariant, (b) the convex envelope of the induced regula…

We investigate the initial value problem for the Einstein-Euler equations of general relativity under the assumption of Gowdy symmetry on T3, and we construct matter spacetimes with low regularity. These spacetimes admit, both, impulsive gravitational waves in the metric (for instance, Dirac mass curvature singularitie…