arXiv research

Gradient descent with logistic loss can interpolate deep networks with smoothed ReLU activations under certain conditions.

Proves rigidity for specific initial data sets under the dominant energy condition.

New method improves optimization algorithms without Lipschitz smoothness.

Smooth maps bound Betti numbers of zero sets.

Generalizes smoothness conditions for optimization methods.

Smooth dec initial data sets may not extend to smooth spacetimes.

The (global) Lipschitz smoothness condition is crucial in establishing the convergence theory for most optimization methods. Unfortunately, most machine learning and signal processing problems are not Lipschitz smooth. This motivates us to generalize the concept of Lipschitz smoothness condition to the relative smoothn…

The paper proves gradient estimates for nonlinear parabolic equations on smooth metric measure spaces.

Smoothed analysis of complexity bounds and condition numbers has been done, so far, on a case by case basis. In this paper we consider a reasonably large class of condition numbers for problems over the complex numbers and we obtain smoothed analysis estimates for elements in this class depending only on geometric inva…

Generative models use kernel smoothing for conditioning on small example sets.

Study abelianization of Lie algebroids and groupoids, providing conditions for existence.

The study explores conditions for $h$-cobordisms between smooth 4-manifolds.

Establishes necessary and sufficient conditions for smooth triviality of Lie subalgebras and Lie ideals, and proves Moser's trick for foliations.

New method improves simulation efficiency in high dimensions.

Quasispheres can be approximated by smooth spheres.

Eigenvalue estimate for the Dirac-Witten operator is given on bounded domains (with smooth boundary) of spacelike hypersurfaces satisfying the dominant energy condition, under four natural boundary conditions (MIT, APS, modified APS, and chiral conditions). This result is a generalisation of Friedrich's inequality for …

We use gauge theoretic and algebraic methods to examine sufficient conditions for smooth points on the moduli space of flat connections on a compact manifold and on the character variety of a finitely generated and presented group. We give a complete proof of the slice theorem for the action of the group of gauge trans…

We prove that the essential smoothness of the gravitational metric at shock waves in GR, a PDE regularity issue for weak solutions of the Einstein equations, is determined by a geometrical condition which we introduce and name the {\it Riemann-flat condition}. The Riemann-flat condition determines whether or not the es…

In this paper we prove that there exists a smooth classical solution to the HJB equation for a large class of constrained problems with utility functions that are not necessarily differentiable or strictly concave. The value function is smooth if admissible controls satisfy an integrability condition or if it is contin…

Study on elasticity with mixed boundary conditions, proving spectral asymptotics.

This paper shows that the time $t$ map of the averaged Euler equations, with Dirichlet, Neumann, and mixed boundary conditions is canonical relative to a Lie-Poisson bracket constructed via a non-smooth reduction for the corresponding diffeomorphism groups. It is also shown that the geodesic spray for Neumann and mixed…

Paper finds lower bounds for eigenvalues of Bi-drifted Laplacian on smooth metric measure spaces.

We study a hybrid conditional gradient - smoothing algorithm (HCGS) for solving composite convex optimization problems which contain several terms over a bounded set. Examples of these include regularization problems with several norms as penalties and a norm constraint. HCGS extends conditional gradient methods to cas…

Derives smooth homogeneous structures for low-rank tensors.

New method combines CATE and CQTE to estimate treatment effects across different quantiles.

For a smooth strictly pseudoconvex hypersurface in a complex manifold, we give a necessary and sufficient condition for being CR-diffeomorphic to a real-analytic CR manifold. Our condition amounts to a holomorphic extension property for the canonically associated function expressing $2$-jets of the formal Segre varieti…

Smooth actions on manifolds can be globally defined under certain conditions.

We prove that compact Cauchy horizons in a smooth spacetime satisfying the null energy condition are smooth. As an application, we consider the problem of determining when a cobordism admits Lorentzian metrics with certain properties. In particular, we prove a result originally due to Tipler without the smoothness hypo…

This paper studies continuum-armed bandits under Besov smoothness conditions and derives minimax rates.

In this paper we discuss the smoothness conditions for metrics on a cohomogeneity one manifold, i.e. metrics invariant under a Lie group whose generic orbits are hypersurfaces. Along these hypersurfaces one describes the metrics in terms of a collection of functions defined along a geodesic normal to the hypersurfaces.…

New method for adaptive estimation and inference in econometric models without knowing smoothness.

Proves compact Cauchy horizons have constant surface gravity under null energy condition.

We show that, under a certain condition, contact 5-manifolds can `coarsely' distinguish smooth structures on compact Stein 4-manifolds via contact open books. We also give a simple sufficient condition for an infinite family of Stein 4-manifolds to have an infinite subfamily of pairwise non-diffeomorphic Stein 4-manifo…

On a smooth complete Riemannian spin manifold with smooth compact boundary, we demonstrate that the Atiyah-Singer Dirac operator $\mathrm{D}_{\mathcal B}$ in $\mathrm{L}^{2}$ depends Riesz continuously on $\mathrm{L}^{\infty}$ perturbations of local boundary conditions ${\mathcal B}$. The Lipschitz bound for the map ${…

In this paper we extend Y.Eliashberg's $h$-principle to arbitrary generic smooth maps of smooth manifolds. Namely, we prove a necessary and sufficient condition for a continuous map of smooth manifolds of the same dimension to be homotopic to a generic map with a prescribed Thom-Boardman singularity $Σ^I$ at each point…

Timelike curvature and Brunn-Minkowski inequality linked in non-smooth spacetimes.

We study the relationship between many natural conditions that one can put on a diffeological vector space: being fine or projective, having enough smooth (or smooth linear) functionals to separate points, having a diffeology determined by the smooth linear functionals, having fine finite-dimensional subspaces, and hav…

Researchers develop weighted GJMS operators for smooth metric measure spaces.

We consider differentiable maps in the setting of Abstract Differential Geometry and we study the conditions that ensure the uniqueness of differentials in this setting. In particular, we prove that smooth maps between smooth manifolds admit a unique differential, coinciding with the usual one. Thus smooth manifolds fo…

Conditional COT-GAN predicts sequences using past data and kernel smoothing.

We establish conditions for a continuous map of nonzero degree between a smooth closed manifold and a negatively curved manifold of dimension greater than four to be homotopic to a smooth cover, and in particular a diffeomorphism when the degree is one. The conditions hold when the volumes or entropy-volumes of the two…

Local smoothing of metrics with small curvature, removing Ricci curvature condition.

New algorithm tackles nonconvex machine learning problems with adaptive normalization and independent sampling.

Investigates Darboux rectifying curves on smooth surfaces.

Novel approach for estimating conditional expectations using Bayesian quadrature.

Adapts a short argument to derive a stability theorem for smooth maps.

Unified proof of smooth fibration theorems for collapsed manifolds.

In this paper we investigate H-minimal graphs of lower regularity. We show that noncharactersitic C^1 H-minimal graphs whose components of the unit horizontal Gauss map are in W^{1,1} are ruled surfaces with C^2 seed curves. In a different direction, we investigate ways in which patches of C^1 H-minimal graphs can be g…