arXiv research

Uniform convexity in divisible domains leads to hyperbolic geometry.

Uniform diffusion approximation for SGD in non-convex settings.

New algorithms sample convex bodies using Markov chains and restricted Gaussian oracles.

We characterize convex cocompact subgroups of the mapping class group of a surface in terms of uniform convergence actions on the zero locus of the limit set. We also construct subgroups that act as uniform convergence groups on their limit sets, but are not convex cocompact.

Uniform spectral gap for convex cocompact hyperbolic surfaces and expanders.

This work establishes uniform convergence of subdifferentials in stochastic optimization.

We establish the essentially optimal form of Donaldson's geodesic stability conjecture regarding existence of constant scalar curvature Kähler metrics. We carry this out by exploring in detail the metric geometry of Mabuchi geodesic rays, and the uniform convexity properties of the space of Kähler metrics.

The paper provides uniform length estimates for trajectories on flat cone surfaces.

New sampler improves uniform sampling over convex bodies with fewer queries.

Counterexamples show failure of uniform laws of large numbers for subdifferentials.

New stability bounds for SGD on nonsmooth convex losses.

Study shows uniform-time chaos propagation in mean field Langevin dynamics.

Geometric interpretation of Fock-Goncharov positivity and disk stabilization in symmetric space.

New method turns optimization algorithms into uniformly stable learning algorithms for non-Euclidean norms.

The paper proves rigidity and uniformization theorems for infinite circle patterns and convex polyhedra in hyperbolic 3-space.

Uniform counting formulas for orthogeodesics in Kleinian groups converge.

Unified stability bounds for noisy SGD across convex and non-convex losses.

Study eigenvalues for special curvature equations on star-shaped surfaces.

Let $U\subseteq\mathbb{R}^{n}$ be open and convex. We show that every (not necessarily Lipschitz or strongly) convex function $f:U\to\mathbb{R}$ can be approximated by real analytic convex functions, uniformly on all of $U$. In doing so we provide a technique which transfers results on uniform approximation on bounded …

Generalizes smoothness conditions for optimization methods.

Characterizes Anosov representations and strongly convex cocompact groups with eigenvalue gaps.

With the help of hyper-ideal circle pattern theory, we have developed a discrete version of the classical uniformization theorems for surfaces represented as finite branched covers over the Riemann sphere as well as compact polyhedral surfaces with non-positive curvature. We show that in the case of such surfaces discr…

Given a monotone convex function on the space of essentially bounded random variables with the Lebesgue property (order continuity), we consider its extension preserving the Lebesgue property to as big solid vector space of random variables as possible. We show that there exists a maximum such extension, with explicit …

Let $Γ$ be a Zariski dense convex cocompact subgroup contained in an arithmetic lattice of $\operatorname{SO}(n, 1)^{\circ}$. We prove uniform exponential mixing of the geodesic flow for congruence covers of the hyperbolic manifold $Γ\backslash \mathbb{H}^n$ avoiding finitely many prime ideals. This extends the work of…

We consider the problem of minimizing the relative perimeter under a volume constraint in an unbounded convex body $C\subset \mathbb{R}^{n+1}$, without assuming any further regularity on the boundary of $C$. Motivated by an example of an unbounded convex body with null isoperimetric profile, we introduce the concept of…

We obtain the classical Hanner inequalities by the Bellman function method. These inequalities give sharp estimates for the moduli of convexity of Lebesgue spaces. Easy ideas from differential geometry help us to find the Bellman function using neither "magic guesses" nor calculations.

We consider stochastic gradient descent algorithms for minimizing a non-smooth, strongly-convex function. Several forms of this algorithm, including suffix averaging, are known to achieve the optimal $O(1/T)$ convergence rate in expectation. We consider a simple, non-uniform averaging strategy of Lacoste-Julien et al. …

Unified high-probability regret bounds for online convex optimisation with randomised gradient estimators.

Accelerated gradient method's stability deteriorates exponentially with steps.

New proof shows symmetry for certain curved surfaces in higher dimensions.

Discrete conformal maps on surfaces with vertex decorations are studied.

Entropy rigidity proven for 3D and higher convex projective manifolds.

New bounds for SGLD show error decreases with more data.

For a sequence of nonnegative random variables, we provide simple necessary and sufficient conditions to ensure that each sequence of its forward convex combinations converges in probability to the same limit. These conditions correspond to an essentially measure-free version of the notion of uniform integrability.

Unique floating and buoyancy surfaces identify convex polytopes.

Compactness proven for manifolds with nonnegative Ricci curvature and uniformly convex boundary.

We consider strictly convex hypersurfaces which are evolving by the non-parametric logarithmic Gauss curvature flow subject to a Neumann boundary condition. Solutions are shown to converge smoothly to hypersurfaces moving by translation. In particular, for bounded domains we prove that convex functions with prescribed …

New methods improve convergence in non-convex non-smooth learning problems.

Most high-dimensional estimation and prediction methods propose to minimize a cost function (empirical risk) that is written as a sum of losses associated to each data point. In this paper we focus on the case of non-convex losses, which is practically important but still poorly understood. Classical empirical process …

We investigate 1) the rate at which refined properties of the empirical risk---in particular, gradients---converge to their population counterparts in standard non-convex learning tasks, and 2) the consequences of this convergence for optimization. Our analysis follows the tradition of norm-based capacity control. We p…

We provide a constructive, variational proof of Rivin's realization theorem for ideal hyperbolic polyhedra with prescribed intrinsic metric, which is equivalent to a discrete uniformization theorem for spheres. The same variational method is also used to prove a discrete uniformization theorem of Gu et al. and a corres…

We study the evolution of convex complete non-compact graphs by positive powers of Gauss curvature. We show that if the initial complete graph has a local uniform convexity, then the graph evolves by any positive power of Gauss curvature for all time. In particular, the initial graph is not necessarily differentiable.

Extends Langevin dynamics for constrained domains.

The paper studies complex Finsler metrics and their equivalence to the Kobayashi metric.

In this paper, we propose a novel uniform generalization bound on the time and inverse temperature for stochastic gradient Langevin dynamics (SGLD) in a non-convex setting. While previous works derive their generalization bounds by uniform stability, we use Rademacher complexity to make our generalization bound indepen…

The paper simplifies K-stability conditions for spherical varieties.

We study solutions of the mean curvature flow which are defined for all negative curvature times, usually called ancient solutions. We give various conditions ensuring that a closed convex ancient solution is a shrinking sphere. Examples of such conditions are: a uniform pinching condition on the curvatures, a suitable…

The study extends curvature bounds to non-smooth spaces and proves stability of mean curvature.