arXiv research

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

We consider a wide range of regularized stochastic minimization problems with two regularization terms, one of which is composed with a linear function. This optimization model abstracts a number of important applications in artificial intelligence and machine learning, such as fused Lasso, fused logistic regression, a…

New algorithms optimize convex functions with high-order derivatives.

We show that for acylindrically hyperbolic groups $Γ$ (with no nontrivial finite normal subgroups) and arbitrary unitary representation $ρ$ of $Γ$ in a (nonzero) uniformly convex Banach space the vector space $H^2_b(Γ;ρ)$ is infinite dimensional. The result was known for the regular representations on $\ell^p(Γ)$ with …

We prove that for the mean curvature flow of two-convex hypersurfaces the intrinsic diameter stays uniformly controlled as one approaches the first singular time. We also derive sharp $L^{n-1}$-estimates for the regularity scale of the level set flow with two-convex initial data. Our proof relies on a detailed analysis…

Uniform convexity in divisible domains leads to hyperbolic geometry.

We consider compact convex hypersurfaces contracting by functions of their curvature. Under the mean curvature flow, uniformly convex smooth initial hypersurfaces evolve to remain smooth and uniformly convex, and contract to points after finite time. The same holds if the initial data is only weakly convex or non-smoot…

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

We prove that any complete immersed globally orientable uniformly 2-convex translating soliton $Σ\subset \mathbb{R}^{n+1}$ for the mean curvature flow is locally strictly convex. It follows that a uniformly 2-convex entire graphical translating soliton in $\mathbb{R}^{n+1},\, n\geq 3$ is the axisymmetric "bowl soliton…

We establish the equivalence between the family of closed uniformly regular Riemannian manifolds and the class of complete manifolds with bounded geometry.

We show that for any group $G$ that is hyperbolic relative to subgroups that admit a proper affine isometric action on a uniformly convex Banach space, then $G$ acts properly on a uniformly convex Banach space as well.

This article generalizes the work of Ballmann and Światkowski to the case of Reflexive Banach spaces and uniformly convex Busemann spaces, thus giving a new fixed point criterion for groups acting on simplicial complexes.

Solves a specific Dirichlet problem for Lagrangian mean curvature equations.

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

In this paper, we present a generic framework to extend existing uniformly optimal convex programming algorithms to solve more general nonlinear, possibly nonconvex, optimization problems. The basic idea is to incorporate a local search step (gradient descent or Quasi-Newton iteration) into these uniformly optimal conv…

Study on anisotropic curvature flow for noncompact convex hypersurfaces.

Sharp generalization of boundary regularity for area minimizing currents with arbitrary multiplicity.

Study shows convergence speed for Fekete points on specific sets.

Classifies 3-manifolds with uniformly positive scalar curvature.

Study shows Stochastic Mirror Descent optimizes convex problems with infinite noise variance.

Constructs uniformly positive scalar curvature metrics on open manifolds

In this paper we investigate the strict convexity and the differentiability properties of the stable norm, which corresponds to the homogenized surface tension for a periodic perimeter homogenization problem (in a regular and uniformly elliptic case). We prove that it is always differentiable in totally irrational dire…

Formula connects length and correlation functions via ghost polygons and Poisson bracket.

New algorithm uniformly samples high-dimensional convex bodies efficiently.

Study uniformly differentiable graphs in Carnot groups, proving area formulas.

The paper studies curvature flows in hyperbolic space and proves convergence to spheres under certain conditions.

Given a hyperbolic domain, the nearest point retraction is a conformally natural homotopy equivalence from the domain to the boundary of the convex core of its complement. Marden and Markovic showed that if the domain is uniformly perfect, then there exists a conformally natural quasiconformal map which admits a bounde…

Flow of convex hypersurfaces in hyperbolic space converges to geodesic spheres.

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 …

The paper studies constraint maps with singularities and free boundaries, proving continuity near singularities and optimality.

Study asymptotic behavior of Weingarten surfaces at infinity.

In this paper we study the covering numbers of the space of convex and uniformly bounded functions in multi-dimension. We find optimal upper and lower bounds for the $ε$-covering number of $\C([a, b]^d, B)$, in the $L_p$-metric, $1 \le p < \infty$, in terms of the relevant constants, where $d \geq 1$, $a < b \in \mathb…

The paper studies a flow of convex hypersurfaces expanding by their support and curvature functions.

In this paper, we consider noncompact ancient solutions to the mean curvature flow in $\mathbb{R}^{n+1}$ ($n \geq 3$) which are strictly convex, uniformly two-convex, and noncollapsed. We prove that such an ancient solution is a rotationally symmetric translating soliton.

The paper analyzes the convergence rates of Q-learning with entropy regularization and linear function approximation.

Existence of harmonic maps near projections in hyperbolic spaces for large convex sets.

Paper establishes tight lower bounds for minimizing certain smooth and convex functions.

We show how Lasry-Lions's result on regularization of functions defined on $\mathbb{R}^n$ or on Hilbert spaces by sup-inf convolutions with squares of distances can be extended to (finite or infinite dimensional) Riemannian manifolds $M$ of bounded sectional curvature. More specifically, among other things we show that…

New adapted renormalized volume for hyperbolic 3-manifolds with compressible boundary.

We prove that any translating soliton for the mean curvature flow which is noncollapsed and uniformly 2-convex must be the rotationally symmetric bowl soliton. In particular, this proves a conjecture of White and Wang, in the 2-convex case in arbitrary dimension.

In this paper we prove the probabilistic continuous complexity conjecture. In continuous complexity theory, this states that the complexity of solving a continuous problem with probability approaching 1 converges (in this limit) to the complexity of solving the same problem in its worst case. We prove the conjecture ho…

We show that the nearest point retraction is a uniform quasi-isometry from the Thurston metric on a hyperbolic domain in the Riemann sphere to the boundary of the convex hull of its complement. As a corollary, one obtains explicit bounds on the quasi-isometry constant of the nearest point retraction with respect to the…

Study shows unique tangent cones for area-minimizing currents at boundary points.

We study Hessian fully nonlinear uniformly elliptic equations and show that the second derivatives of viscosity solutions of those equations (in 12 or more dimensions) can blow up in an interior point of the domain. We prove that the optimal interior regularity of such solutions is no more than C^{1+ε}, showing the opt…

The notion of nonpositive curvature in Alexandrov's sense is extended to include p-uniformly convex Banach spaces. Infinite dimensional manifolds of semi-negative curvature with a p-uniformly convex tangent norm fall in this class on nonpositively curved spaces, and several well-known results, such as existence and uni…

Algorithmic stability is a classical approach to understanding and analysis of the generalization error of learning algorithms. A notable weakness of most stability-based generalization bounds is that they hold only in expectation. Generalization with high probability has been established in a landmark paper of Bousque…

Establishes smooth Ricci flows from convex surfaces in 3D space.

We establish continuous maximal regularity results for parabolic differential operators acting on sections of tensor bundles on Riemannian manifolds. As an application, we show that solutions to the Yamabe flow instantaneously regularize and become real analytic in space and time. The regularity result is obtained by i…